HYBRID METHODS FOR RADIATION TRANSPORT USING INTEGRAL DEFERRED CORRECTION

By

Michael M. Crockatt

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Computational Mathematics, Science and Engineering – Doctor of Philosophy

2018

HYBRID METHODS FOR RADIATION TRANSPORT USING INTEGRAL DEFERRED CORRECTION

ABSTRACT

By

Michael M. Crockatt

This thesis provides roughly three contributions to the study of defect correction methods and hy-

brid methods for radiation transport. First, a modification of traditional integral deferred correction

(IDC) time-integration schemes is proposed that significantly reduces the storage requirements of the

methods. These methods, which we call low-storage IDC or LS-IDC methods, require storing only one

copy of each stage vector throughout the iteration process, whereas traditional IDC methods require

two copies of each vector. The convergence and stability properties of the methods are examined in a

variety of settings, and both analytical and experimental results are provided. A nonlinear ODE and a

linear transport equation are used to compare the accuracy and storage requirements of LS-IDC integra-

tors with other fully implicit schemes including diagonally implicit Runge-Kutta (DIRK) and space-time

discontinuous Galerkin (STDG) methods. The results demonstrate that LS-IDC methods have similar

accuracy but significantly reduced memory requirements compared to other fully implicit methods.

Second, extensions of a collision-based hybrid method for time-dependent radiation transport sim-

ulations are discussed. The hybrid methods are constructed by splitting the radiation flux into collided

and uncollided components to which low- and high-resolution discrete ordinates approximations are

applied, respectively. The use of arbitrarily high-order numerical approximations is emphasized, with

particular attention paid to high-order time-integration schemes. A range of time-integrators are con-

sidered including DIRK, STDG, IDC, and LS-IDC methods of up to fifth-order accuracy. Convergence

results in one spatial dimension are provided, and it is found that while the hybrid methods exhibit con-

vergence stagnation and order reduction in certain scenarios, the overall accuracy of the hybrid approx-

imations is comparable to standard discrete ordinates approximations in many cases. A test problem

in two-dimensional x y-geometry consisting of a mock-up of a standard hohlraum configuration is used

to compare the computational efficiency of the hybrid methods with standard discrete ordinates meth-

ods. It is observed that replacing a standard discrete ordinates method using an angular quadrature of

order N with a hybrid discrete ordinates method using angular quadratures of order 2N and N /2 for the

uncollided and collided fluxes, respectively, usually reduces solve time by a factor of 2 or more and in

many cases also yields a reduction in solution error by a factor of up to 2. However, it is noted that the

specified hybrid methods require approximately 4.25 times the storage of the corresponding standard

discrete ordinates methods.

Finally, two mechanisms for increasing the effectiveness of the hybrid methods are presented. The

first is a reconstruction procedure for mapping between arbitrary discrete ordinates quadratures within

the context of these hybrid methods. This approach, called Nyström reconstruction, is shown to be sig-

nificantly more accurate than previous reconstruction methods. When applied to the two-dimensional

hohlraum problem, it is observed that replacing a standard discrete ordinates method using an angular

quadrature of order N with a hybrid discrete ordinates method using a Nyström reconstruction proce-

dure and angular quadratures of order N and N /4 for the uncollided and collided fluxes, respectively,

consistently reduces solve time by a factor of between 4 and 8 while increasing memory requirements

by only 6% and producing little to no increase in solution error. Lastly, variations of hybrid methods us-

ing IDC integrators are described in which the hybrid approach is written as a two-grid iterative method

in angle that is combined with an IDC time-integration scheme. It is demonstrated that the resulting

methods are able to iteratively reduce the error due to the application of discrete ordinates quadratures

of different resolutions to the collided and uncollided components.

Copyright by
MICHAEL M. CROCKATT
2018

ACKNOWLEDGEMENTS

I would like to express my appreciation for the many individuals I have had the pleasure of interacting

with throughout my graduate studies. In particular, I would like to thank my advisor Andrew Christlieb

for his patience and guidance: I would never have been able to complete this thesis without his consis-

tent support. I would like to express my gratitude to Cory Hauck for hosting me at Oak Ridge National

Laboratory during the past five summers: I have greatly valued his mentorship and I have learned a great

deal during my summers in Oak Ridge. I would also like to thank Kris Garrett, from whom I have learned

a great deal about radiation transport, numerical methods, and scientific computing over the last several

years. Additionally, I would like to thank all of the members of my thesis committee for lending their

time, their ears, and their expertise to improving my thesis and increasing my understanding of how ra-

diation transport models are used in practical settings. Finally, I would like to thank all of the students,

postdocs, researchers, and support staff that I have interacted with at Michigan State University and Oak

Ridge National Laboratory for fostering a welcoming research environment that I am grateful to have

been a part of.

This work was supported in part by Michigan State University through computational resources

provided by the Institute for Cyber-Enabled Research, and by Oak Ridge National Laboratory (ORNL)

and Oak Ridge Associated Universities (ORAU) through the ORAU/ORNL High Performance Computing

(HPC) Grant Program. This thesis and the numerical software developed during the course of this work

were written using free and open-source software on a GNU/Linux system. Software development for

this work extensively leveraged the GNU toolchain distributed by the Free Software Foundation, and

made use of a number of numerical libraries including the Portable, Extensible Toolkit for Scientific

Computation (PETSc), the Armadillo C++ library for linear algebra & scientific computing, and the Boost

C++ libraries. This thesis was typeset using the TeX Live distribution of the LATEX typesetting system, with

figures generated using the gnuplot graphing utility.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Radiation transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1
Structure of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2
1.2.3
Standard methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Hybrid methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Defect correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integral deferred correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.1
1.3.2

1
1
4
4
6
10
12
14
14
17

CHAPTER 2

LOW-STORAGE INTEGRAL DEFERRED CORRECTION METHODS FOR SCIEN-
TIFIC COMPUTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
18
19
19
20
22
24
24
27
30
32
36
37

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
2.2 Method descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integral deferred correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2
2.2.3
Low-storage integral deferred correction . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Convergence and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Convergence in the limit of small timestep size . . . . . . . . . . . . . . . . . . . . .
2.3.2 Convergence for arbitrary timestep sizes . . . . . . . . . . . . . . . . . . . . . . . . .
Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A non-stiff, non-linear ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 Numerical tests

2.4.1
2.4.2

CHAPTER 3

3.1.1
3.1.2
3.1.3

3.1 The transport equation and discrete ordinates methods

. . . . . . . . . . . . . . . . . . . .
The transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A first-order discrete ordinates method . . . . . . . . . . . . . . . . . . . . . . . . . .
First-order hybrid discrete ordinates methods . . . . . . . . . . . . . . . . . . . . . .
3.2 High-order time discretizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Diagonally implicit Runge-Kutta methods . . . . . . . . . . . . . . . . . . . . . . . .
Space-time discontinuous Galerkin methods . . . . . . . . . . . . . . . . . . . . . .
3.2.2
Integral deferred correction methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3
3.2.4
Low-storage integral deferred correction methods . . . . . . . . . . . . . . . . . . .
Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specifics of high-order time discretizations . . . . . . . . . . . . . . . . . . . . . . .
3.3.1
3.3.2
Angular quadratures and relabeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HIGH-ORDER HYBRID METHODS FOR RADIATION TRANSPORT . . . . . . . . . . 42
42
42
43
45
48
50
54
59
64
66
67
73

3.3

vi

3.4 Numerical tests

3.3.3

Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 One spatial dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2
Two spatial dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79
82
82
88

CHAPTER 4

4.1

NYSTRÖM RECONSTRUCTION AND TWO-GRID DEFECT CORRECTION FOR
HYBRID METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.1.1 Nyström interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Two-grid iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.1.2
4.2 Method descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.2.1 Nyström reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.2.2
4.2.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.3 Numerical tests for Nyström methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.3.1 One spatial dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Two spatial dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.3.2
4.4 Numerical tests for two-grid methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.4.1 One spatial dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.4.2
Two spatial dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Two-grid IDC methods
Two-grid LS-IDC methods

CHAPTER 5

CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Low-storage integral deferred correction methods for scientific computing . . . . . . . . . 230
. . . . . . . . . . . . . . . . . . . . . . . 231

5.1
5.2 High-order hybrid methods for radiation transport

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

vii

LIST OF TABLES

Table 2.1: Contraction factors for IDC and LS-IDC methods in the stiff limit using different num-

bers and types of quadrature nodes with implicit Euler substeps.

. . . . . . . . . . . . .

29

Table 2.2: Number of angular flux vectors required by each class of time integration scheme
for solving the transport equation (1.2) using the discrete ordinates method. Here N
represents the order of accuracy of the method, which may differ from the number of
quadrature nodes or stages used by the method.

. . . . . . . . . . . . . . . . . . . . . . .

39

viii

LIST OF FIGURES

Figure 2.1: Contraction factors for IDC and LS-IDC methods along the negative real axis using

different numbers and types of quadrature nodes with implicit Euler substeps.

. . . .

31

Figure 2.2: Boundaries of the regions of absolute stability for IDC and LS-IDC methods using
implicit Euler substeps on (right biased) Gauss-Radau quadrature nodes with vari-
ous numbers of correction iterations. Regions of absolute stability lie to the left of
the origin.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Figure 2.3: Boundaries of the regions of absolute stability for IDC and LS-IDC methods using
implicit Euler substeps on four Gauss-Lobatto quadrature nodes with various num-
bers of correction iterations. Regions of absolute stability lie to the left of the origin.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Figure 2.4: Boundaries of the regions of absolute stability for IDC and LS-IDC methods using
implicit Euler substeps on five Gauss-Lobatto quadrature nodes with various num-
bers of correction iterations. Regions of absolute stability lie to the left of the origin.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Figure 2.5: Convergence of IDC, LS-IDC, and collocation integrators using three Gauss-Radau
nodes, four Gauss-Lobatto nodes, and five right-biased equispaced nodes for the IVP
(2.50). Errors are computed with respect to the ∞-norm at time t = 10. The number
of correction iterations used with each IDC and LS-IDC method is indicated in the
key of each plot.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Figure 2.6: Layout and sample solution for the lattice test problem for the transport equation

(1.2). Color scale of sample solution is log10.

. . . . . . . . . . . . . . . . . . . . . . . . .

38

Figure 2.7: Memory usage vs. L2(d(cid:126)x) error in the scalar flux distribution for the lattice test problem. 41

Figure 3.1: Butcher tableaus of SDIRK2 and SDIRK3 schemes. . . . . . . . . . . . . . . . . . . . . . .

70

Figure 3.2: Butcher tableaus of Kværnø7 and ARK8 schemes.

. . . . . . . . . . . . . . . . . . . . . .

71

Figure 3.3: L2 convergence studies of hybrid-Ia and hybrid-IIa methods based on SDIRK2 and
SDIRK3 integrators for a sequence of smooth 1D problems under a diffusive scaling
with σt = σs = 1, spatial domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for vari-
ous values of ε approaching 0. The error in the scalar flux of each solution is ap-
proximated by comparing to a reference solution using 218 spatial cells and an S32
discrete ordinates approximation in angle. Point marks for hybrid solutions whose
convergence is unhindered lie within the black boxes.

. . . . . . . . . . . . . . . . . . .

84

ix

Figure 3.4: L2 convergence studies of hybrid-Ia and hybrid-IIa methods based on Kværnø7 and
ARK8 integrators for a sequence of smooth 1D problems under a diffusive scaling
with σt = σs = 1, spatial domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for vari-
ous values of ε approaching 0. The error in the scalar flux of each solution is ap-
proximated by comparing to a reference solution using 218 spatial cells and an S32
discrete ordinates approximation in angle. Point marks for hybrid solutions whose
convergence is unhindered lie within the black boxes.

. . . . . . . . . . . . . . . . . . .

Figure 3.5: L2 convergence studies of hybrid-a methods based on STDG integrators for a se-
quence of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial
domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various values of ε approaching
0. The error in the scalar flux of each solution is approximated by comparing to a
reference solution using 218 spatial cells and an S32 discrete ordinates approxima-
tion in angle. Point marks for hybrid solutions whose convergence is unhindered lie
within the black boxes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.6: L2 convergence studies of hybrid-I methods based on IDC5 and LS-IDC5 integrators
for a sequence of smooth 1D problems under a diffusive scaling with σt = σs = 1,
spatial domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various values of ε ap-
proaching 0. The error in the scalar flux of each solution is approximated by com-
paring to a reference solution using 218 spatial cells and an S32 discrete ordinates
approximation in angle. Point marks for hybrid solutions whose convergence is un-
hindered lie within the black boxes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

86

87

Figure 3.7: Layout and sample solution for the hohlraum test problem. Color scale of solution

is log10.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Figure 3.8: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid SDIRK2 methods using
CL quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . .

∞

Figure 3.9: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid SDIRK2 methods using
TN quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . .

∞

Figure 3.10: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid SDIRK3 methods using
CL quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . .

∞

Figure 3.11: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid SDIRK3 methods using
TN quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . .

∞

x

92

93

94

95

Figure 3.12: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid Kværnø7 methods using
CL quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . .

∞

Figure 3.13: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid Kværnø7 methods using
TN quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . .

∞

Figure 3.14: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid ARK8 methods using CL
quadratures applied to the hohlraum test problem. The solve time (in seconds) and
the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . .

∞

Figure 3.15: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid ARK8 methods using TN
quadratures applied to the hohlraum test problem. The solve time (in seconds) and
the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . .

∞

96

97

98

99

Figure 3.16: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid STDG3 methods using
CL and TN quadratures applied to the hohlraum test problem. The solve time (in
seconds) and the L

error are included for each solution.

∞

. . . . . . . . . . . . . . . . . 100

Figure 3.17: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid STDG5 methods using
CL and TN quadratures applied to the hohlraum test problem. The solve time (in
seconds) and the L

error are included for each solution.

∞

. . . . . . . . . . . . . . . . . 101

Figure 3.18: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid IDC3 methods using CL
quadratures applied to the hohlraum test problem. The solve time (in seconds) and
the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . 102

∞

Figure 3.19: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid IDC3 methods using TN
quadratures applied to the hohlraum test problem. The solve time (in seconds) and
the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . 103

∞

Figure 3.20: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid IDC5 methods using CL
quadratures applied to the hohlraum test problem. The solve time (in seconds) and
the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . 104

∞

xi

Figure 3.21: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid IDC5 methods using TN
quadratures applied to the hohlraum test problem. The solve time (in seconds) and
the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . 105

∞

Figure 3.22: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid LS-IDC3 methods using
CL quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 106

∞

Figure 3.23: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid LS-IDC3 methods using
TN quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 107

∞

Figure 3.24: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid LS-IDC5 methods using
CL quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 108

∞

Figure 3.25: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid LS-IDC5 methods using
TN quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 109

∞

Figure 3.26: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the SDIRK2 integrator
applied to the holraum test problem. Hybrid methods using angular quadratures
that differ in order by a factor of 4 are shown in red and other hybrid methods are
shown in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . 110

Figure 3.27: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the SDIRK3 integrator
applied to the holraum test problem. Hybrid methods using angular quadratures
that differ in order by a factor of 4 are shown in red and other hybrid methods are
shown in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . 111

Figure 3.28: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the Kværnø7 integrator
applied to the holraum test problem. Hybrid methods using angular quadratures
that differ in order by a factor of 4 are shown in red and other hybrid methods are
shown in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 3.29: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the ARK8 integrator ap-
plied to the holraum test problem. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

xii

Figure 3.30: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the STDG3 and STDG5
integrators applied to the holraum test problem. Hybrid methods using angular
quadratures that differ in order by a factor of 4 are shown in red and other hybrid
methods are shown in blue. Note the differences in scales. . . . . . . . . . . . . . . . . . 114

Figure 3.31: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the IDC3 integrator ap-
plied to the holraum test problem. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 3.32: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the IDC5 integrator ap-
plied to the holraum test problem. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 3.33: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the LS-IDC3 integrator
applied to the holraum test problem. Hybrid methods using angular quadratures
that differ in order by a factor of 4 are shown in red and other hybrid methods are
shown in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 3.34: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the LS-IDC5 integrator
applied to the holraum test problem. Hybrid methods using angular quadratures
that differ in order by a factor of 4 are shown in red and other hybrid methods are
shown in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 3.35: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black and gray) and hybrid (red and blue) discrete ordinates methods based on
STDG integrators and various angular resolutions applied to the hohlraum test prob-
lem. CL (gray and blue) and TN (black and red) angular quadratures are considered.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figure 3.36: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black and gray) and hybrid (red and blue) discrete ordinates methods based on
DIRK integrators and various angular resolutions applied to the hohlraum test prob-
lem. CL (gray and blue) and TN (black and red) angular quadratures are considered.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Figure 3.37: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black and gray) and hybrid (red and blue) discrete ordinates methods based on
DIRK integrators and various angular resolutions applied to the hohlraum test prob-
lem. CL (gray and blue) and TN (black and red) angular quadratures are considered.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

xiii

Figure 3.38: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black and gray) and hybrid (red and blue) discrete ordinates methods based on IDC
integrators and various angular resolutions applied to the hohlraum test problem.
CL (gray and blue) and TN (black and red) angular quadratures are considered.

. . . . 124

Figure 3.39: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black and gray) and hybrid (red and blue) discrete ordinates methods based on LS-
IDC integrators and various angular resolutions applied to the hohlraum test prob-
lem. CL (gray and blue) and TN (black and red) angular quadratures are considered.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Figure 4.1: L2 convergence studies of hybrid STDG methods for a sequence of smooth 1D prob-
lems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time
0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar flux of
each solution is approximated by comparing to a reference solution using 218 spatial
cells and an S32 discrete ordinates approximation in angle. Point marks for hybrid
solutions whose convergence is unhindered lie within the black boxes.

. . . . . . . . . 151

Figure 4.2: L2 convergence studies of SDIRK2 hybrid methods for a sequence of smooth 1D
problems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final
time 0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar
flux of each solution is approximated by comparing to a reference solution using 218
spatial cells and an S32 discrete ordinates approximation in angle. Point marks for
hybrid solutions whose convergence is unhindered lie within the black boxes.

. . . . . 152

Figure 4.3: L2 convergence studies of SDIRK3 hybrid methods for a sequence of smooth 1D
problems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final
time 0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar
flux of each solution is approximated by comparing to a reference solution using 218
spatial cells and an S32 discrete ordinates approximation in angle. Point marks for
hybrid solutions whose convergence is unhindered lie within the black boxes.

. . . . . 153

Figure 4.4: L2 convergence studies of Kværnø7 hybrid methods for a sequence of smooth 1D
problems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final
time 0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar
flux of each solution is approximated by comparing to a reference solution using 218
spatial cells and an S32 discrete ordinates approximation in angle. Point marks for
hybrid solutions whose convergence is unhindered lie within the black boxes.

. . . . . 154

Figure 4.5: L2 convergence studies of ARK8 hybrid methods for a sequence of smooth 1D prob-
lems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time
0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar flux of
each solution is approximated by comparing to a reference solution using 218 spatial
cells and an S32 discrete ordinates approximation in angle. Point marks for hybrid
solutions whose convergence is unhindered lie within the black boxes.

. . . . . . . . . 155

xiv

Figure 4.6: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid SDIRK2 methods using
CL quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 158

∞

Figure 4.7: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid SDIRK2 methods using
TN quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 159

∞

Figure 4.8: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid SDIRK3 methods using
CL quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 160

∞

Figure 4.9: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid SDIRK3 methods using
TN quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 161

∞

Figure 4.10: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid Kværnø7 methods using
CL quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 162

∞

Figure 4.11: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid Kværnø7 methods using
TN quadratures applied to the hohlraum test problem. The solve time (in seconds)
and the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . 163

∞

Figure 4.12: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid ARK8 methods using CL
quadratures applied to the hohlraum test problem. The solve time (in seconds) and
the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . 164

∞

Figure 4.13: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid ARK8 methods using TN
quadratures applied to the hohlraum test problem. The solve time (in seconds) and
the L

error are included for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . 165

∞

Figure 4.14: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid STDG3 methods using
CL and TN quadratures applied to the hohlraum test problem. The solve time (in
seconds) and the L

error are included for each solution.

∞

. . . . . . . . . . . . . . . . . 166

xv

Figure 4.15: Comparison of scalar flux density (top rows) and deviation of scalar flux density from
reference solution (bottom rows) of hybrid and non-hybrid STDG5 methods using
CL and TN quadratures applied to the hohlraum test problem. The solve time (in
seconds) and the L

error are included for each solution.

∞

. . . . . . . . . . . . . . . . . 167

Figure 4.16: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the SDIRK2 integrator
applied to the holraum test problem. Hybrid methods using angular quadratures
that differ in order by a factor of 4 are shown in red and other hybrid methods are
shown in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . 168

Figure 4.17: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the SDIRK3 integrator
applied to the holraum test problem. Hybrid methods using angular quadratures
that differ in order by a factor of 4 are shown in red and other hybrid methods are
shown in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . 169

Figure 4.18: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the Kværnø7 integrator
applied to the holraum test problem. Hybrid methods using angular quadratures
that differ in order by a factor of 4 are shown in red and other hybrid methods are
shown in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . 170

Figure 4.19: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on the ARK8 integrator ap-
plied to the holraum test problem. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Figure 4.20: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and
hybrid (red and blue) discrete ordinates methods based on STDG3 and STDG5 inte-
grators applied to the holraum test problem. Hybrid methods using angular quadra-
tures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue. Note the differences in scales.

. . . . . . . . . . . . . . . . . . . . . . 172

Figure 4.21: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black and gray), hybrid-a (red and blue), and hybrid-c (pink and teal) discrete or-
dinates methods based on STDG integrators and using various angular quadrature
resolutions applied to the hohlraum test problem. CL (gray, blue, and teal) and TN
(black, red, and pink) angular quadratures are considered.

. . . . . . . . . . . . . . . . 175

Figure 4.22: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black and gray), hybrid-Ia (red and blue), and hybrid-Ic (pink and teal) discrete or-
dinates methods based on DIRK integrators and using various angular quadrature
resolutions applied to the hohlraum test problem. CL (gray, blue, and teal) and TN
(black, red, and pink) angular quadratures are considered.

. . . . . . . . . . . . . . . . 176

xvi

Figure 4.23: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black and gray), hybrid-IIa (red and blue), and hybrid-IIc (pink and teal) discrete
ordinates methods based on DIRK integrators and using various angular quadrature
resolutions applied to the hohlraum test problem. CL (gray, blue, and teal) and TN
(black, red, and pink) angular quadratures are considered.

. . . . . . . . . . . . . . . . 177

Figure 4.24: L2 convergence studies of IDC5 hybrid-IIb and hybrid-IIc S32,2 methods for a se-
quence of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial
domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various values of ε approaching
0. The error in the scalar flux of each solution is approximated by comparing to a
reference solution using 218 spatial cells and an S32 discrete ordinates approxima-
tion in angle. Point marks for hybrid solutions whose convergence is unhindered lie
within the black boxes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 4.25: L2 convergence studies of LS-IDC5 hybrid-IIb and hybrid-IIc S32,2 methods for a
sequence of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial
domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various values of ε approaching
0. The error in the scalar flux of each solution is approximated by comparing to a
reference solution using 218 spatial cells and an S32 discrete ordinates approxima-
tion in angle. Point marks for hybrid solutions whose convergence is unhindered lie
within the black boxes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Figure 4.26: L2 convergence studies of IDC5 hybrid-IIb and hybrid-IIc S32,4 methods for a se-
quence of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial
domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various values of ε approaching
0. The error in the scalar flux of each solution is approximated by comparing to a
reference solution using 218 spatial cells and an S32 discrete ordinates approxima-
tion in angle. Point marks for hybrid solutions whose convergence is unhindered lie
within the black boxes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Figure 4.27: L2 convergence studies of LS-IDC5 hybrid-IIb and hybrid-IIc S32,4 methods for a
sequence of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial
domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various values of ε approaching
0. The error in the scalar flux of each solution is approximated by comparing to a
reference solution using 218 spatial cells and an S32 discrete ordinates approxima-
tion in angle. Point marks for hybrid solutions whose convergence is unhindered lie
within the black boxes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Figure 4.28: Deviation of scalar flux density from reference solution for hybrid-IIb IDC3 methods
using CL quadratures and different numbers of correction iterations applied to the
hohlraum test problem. The solve time (in seconds) and the L
error are included
for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

∞

Figure 4.29: Deviation of scalar flux density from reference solution for hybrid-IIc IDC3 methods
using CL quadratures and different numbers of correction iterations applied to the
hohlraum test problem. The solve time (in seconds) and the L
error are included
for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

∞

xvii

Figure 4.30: Deviation of scalar flux density from reference solution for hybrid-IIb IDC3 methods
using TN quadratures and different numbers of correction iterations applied to the
hohlraum test problem. The solve time (in seconds) and the L
error are included
for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

∞

Figure 4.31: Deviation of scalar flux density from reference solution for hybrid-IIc IDC3 methods
using TN quadratures and different numbers of correction iterations applied to the
hohlraum test problem. The solve time (in seconds) and the L
error are included
for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

∞

Figure 4.32: Deviation of scalar flux density from reference solution for hybrid-IIb LS-IDC3 meth-
ods using CL quadratures and different numbers of correction iterations applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are in-
cluded for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

∞

Figure 4.33: Deviation of scalar flux density from reference solution for hybrid-IIc LS-IDC3 meth-
ods using CL quadratures and different numbers of correction iterations applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are in-
cluded for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

∞

Figure 4.34: Deviation of scalar flux density from reference solution for hybrid-IIb LS-IDC3 meth-
ods using TN quadratures and different numbers of correction iterations applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are in-
cluded for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

∞

Figure 4.35: Deviation of scalar flux density from reference solution for hybrid-IIc LS-IDC3 meth-
ods using TN quadratures and different numbers of correction iterations applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are in-
cluded for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

∞

Figure 4.36: Deviation of scalar flux density from reference solution for hybrid-IIb IDC5 methods
using CL quadratures and different numbers of correction iterations applied to the
hohlraum test problem. The solve time (in seconds) and the L
error are included
for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

∞

Figure 4.37: Deviation of scalar flux density from reference solution for hybrid-IIc IDC5 methods
using CL quadratures and different numbers of correction iterations applied to the
hohlraum test problem. The solve time (in seconds) and the L
error are included
for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

∞

Figure 4.38: Deviation of scalar flux density from reference solution for hybrid-IIb IDC5 methods
using TN quadratures and different numbers of correction iterations applied to the
hohlraum test problem. The solve time (in seconds) and the L
error are included
for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

∞

xviii

Figure 4.39: Deviation of scalar flux density from reference solution for hybrid-IIc IDC5 methods
using TN quadratures and different numbers of correction iterations applied to the
hohlraum test problem. The solve time (in seconds) and the L
error are included
for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

∞

Figure 4.40: Deviation of scalar flux density from reference solution for hybrid-IIb LS-IDC5 meth-
ods using CL quadratures and different numbers of correction iterations applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are in-
cluded for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

∞

Figure 4.41: Deviation of scalar flux density from reference solution for hybrid-IIc LS-IDC5 meth-
ods using CL quadratures and different numbers of correction iterations applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are in-
cluded for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

∞

Figure 4.42: Deviation of scalar flux density from reference solution for hybrid-IIb LS-IDC5 meth-
ods using TN quadratures and different numbers of correction iterations applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are in-
cluded for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

∞

Figure 4.43: Deviation of scalar flux density from reference solution for hybrid-IIc LS-IDC5 meth-
ods using TN quadratures and different numbers of correction iterations applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are in-
cluded for each solution.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

∞

Figure 4.44: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIb (red and blue) discrete ordinates methods based on the IDC3
integrator using CL quadratures applied to the hohlraum test problem with various
numbers of correction iterations. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Figure 4.45: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIc (red and blue) discrete ordinates methods based on the IDC3
integrator using CL quadratures applied to the hohlraum test problem with various
numbers of correction iterations. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Figure 4.46: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIb (red and blue) discrete ordinates methods based on the IDC3
integrator using TN quadratures applied to the hohlraum test problem with various
numbers of correction iterations. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

xix

Figure 4.47: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIc (red and blue) discrete ordinates methods based on the IDC3
integrator using TN quadratures applied to the hohlraum test problem with various
numbers of correction iterations. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Figure 4.48: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIb (red and blue) discrete ordinates methods based on the IDC5
integrator using CL quadratures applied to the hohlraum test problem with various
numbers of correction iterations. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Figure 4.49: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIc (red and blue) discrete ordinates methods based on the IDC5
integrator using CL quadratures applied to the hohlraum test problem with various
numbers of correction iterations. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Figure 4.50: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIb (red and blue) discrete ordinates methods based on the IDC5
integrator using TN quadratures applied to the hohlraum test problem with various
numbers of correction iterations. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Figure 4.51: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIc (red and blue) discrete ordinates methods based on the IDC5
integrator using TN quadratures applied to the hohlraum test problem with various
numbers of correction iterations. Hybrid methods using angular quadratures that
differ in order by a factor of 4 are shown in red and other hybrid methods are shown
in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Figure 4.52: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIb (red and blue) discrete ordinates methods based on the LS-
IDC3 integrator using CL quadratures applied to the hohlraum test problem with
various numbers of correction iterations. Hybrid methods using angular quadra-
tures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Figure 4.53: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIc (red and blue) discrete ordinates methods based on the LS-
IDC3 integrator using CL quadratures applied to the hohlraum test problem with
various numbers of correction iterations. Hybrid methods using angular quadra-
tures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

xx

Figure 4.54: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIb (red and blue) discrete ordinates methods based on the LS-
IDC3 integrator using TN quadratures applied to the hohlraum test problem with
various numbers of correction iterations. Hybrid methods using angular quadra-
tures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Figure 4.55: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIc (red and blue) discrete ordinates methods based on the LS-
IDC3 integrator using TN quadratures applied to the hohlraum test problem with
various numbers of correction iterations. Hybrid methods using angular quadra-
tures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Figure 4.56: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIb (red and blue) discrete ordinates methods based on the LS-
IDC5 integrator using CL quadratures applied to the hohlraum test problem with
various numbers of correction iterations. Hybrid methods using angular quadra-
tures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Figure 4.57: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIc (red and blue) discrete ordinates methods based on the LS-
IDC5 integrator using CL quadratures applied to the hohlraum test problem with
various numbers of correction iterations. Hybrid methods using angular quadra-
tures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Figure 4.58: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIb (red and blue) discrete ordinates methods based on the LS-
IDC5 integrator using TN quadratures applied to the hohlraum test problem with
various numbers of correction iterations. Hybrid methods using angular quadra-
tures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

Figure 4.59: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid-IIc (red and blue) discrete ordinates methods based on the LS-
IDC5 integrator using TN quadratures applied to the hohlraum test problem with
various numbers of correction iterations. Hybrid methods using angular quadra-
tures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Figure 4.60: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid (blue, teal, and green) discrete ordinates methods based on the
IDC3 integrator using CL quadratures applied to the hohlraum test problem with
various numbers of correction iterations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 222

xxi

Figure 4.61: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid (red, pink, and orange) discrete ordinates methods based on the
IDC3 integrator using TN quadratures applied to the hohlraum test problem with
various numbers of correction iterations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Figure 4.62: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid (blue, teal, and green) discrete ordinates methods based on the
IDC5 integrator using CL quadratures applied to the hohlraum test problem with
various numbers of correction iterations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Figure 4.63: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid (red, pink, and orange) discrete ordinates methods based on the
IDC5 integrator using TN quadratures applied to the hohlraum test problem with
various numbers of correction iterations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Figure 4.64: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid (blue, teal, and green) discrete ordinates methods based on the
LS-IDC3 integrator using CL quadratures applied to the hohlraum test problem with
various numbers of correction iterations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Figure 4.65: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid (red, pink, and orange) discrete ordinates methods based on the
LS-IDC3 integrator using TN quadratures applied to the hohlraum test problem with
various numbers of correction iterations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Figure 4.66: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid (blue, teal, and green) discrete ordinates methods based on the
LS-IDC5 integrator using CL quadratures applied to the hohlraum test problem with
various numbers of correction iterations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Figure 4.67: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard
(black) and hybrid (red, pink, and orange) discrete ordinates methods based on the
LS-IDC5 integrator using TN quadratures applied to the hohlraum test problem with
various numbers of correction iterations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Figure 5.1: Layout and sample solution for the inhomogeneous sphere problem. Color scale of

solution is log10.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Figure 5.2: L2-convergence studies of hybrid-Ic SDIRK3 methods and hybrid-IIc LS-IDC3 meth-
ods applied to the inhomogeneous sphere problem with a final time of 8 and a CFL
of 128. All solutions are computed using TN quadratures.

. . . . . . . . . . . . . . . . . 238

xxii

LIST OF ALGORITHMS

Algorithm 3.1: Hybrid Euler timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Algorithm 3.2: Nonhybrid DIRK timestep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Algorithm 3.3: Hybrid-Ia DIRK timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Algorithm 3.4: Hybrid-IIa DIRK timestep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Algorithm 3.5: Hybrid-a STDG timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Algorithm 3.6: Nonhybrid IDC timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Algorithm 3.7: Hybrid-Ia IDC timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Algorithm 3.8: Hybrid-Ib IDC timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Algorithm 3.9: Nonhybrid LS-IDC timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Algorithm 3.10: Hybrid-Ia LS-IDC timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Algorithm 3.11: Hybrid-Ib LS-IDC timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Algorithm 4.1: Hybrid Euler timestep with Nyström reconstruction.

. . . . . . . . . . . . . . . . . . 134

Algorithm 4.2: Hybrid-I DIRK timestep with Nyström reconstruction.

. . . . . . . . . . . . . . . . . 136

Algorithm 4.3: Hybrid-II DIRK timestep with Nyström reconstruction. . . . . . . . . . . . . . . . . . 137

Algorithm 4.4: Hybrid STDG timestep with Nyström reconstruction. . . . . . . . . . . . . . . . . . . 139

Algorithm 4.5: Hybrid-Ia and hybrid-Ic IDC timestep.

. . . . . . . . . . . . . . . . . . . . . . . . . . 141

Algorithm 4.6: Hybrid-II IDC and LS-IDC prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Algorithm 4.7: Hybrid-II IDC correction.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Algorithm 4.8: Hybrid-II LS-IDC correction.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xxiii

CHAPTER 1

INTRODUCTION

1.1 Motivation

The behavior of many systems in science and engineering can be described by a diverse array of

models of various levels of complexity. Many models rely on certain assumptions about the state of a

system in order to reduce the complexity of the model. When these assumptions are satisfied, or at the

very least approximately satisfied, it is possible to describe the dynamics of a system that are relevant to

a particular situaion using a less complex model. However, applying a model to a system that does not

satisfy the assumptions required by that model often results in behavior that deviates significantly from

that of the system under consideration.

The assumptions that are used to construct various models are often related to certain scales within

the system. More specifically, these assumptions often describe relative differences between the mag-

nitudes of certain physical features or the lengths of time over which certain processes occur. In many

cases, different models for a given system can be arranged in sequence based on the characteristic scales

for which assumptions are made, with “microscale” models for small characteristic scales on one end

and “macroscale” models for large characteristic scales on the other [89]. The determination of which

phenomena are accurately captured by a given model is usually governed by the relative differences be-

tween the assumed scales of the model and the characteristic scales on which each phenomenon occurs.

Using the most detailed model available might be desirable from an idealistic perspective, but this is

generally not realistically feasible. While one may be able to write out a set of mathematical expressions

to describe the behavior of a system using a given model, solving the resulting system of equations can be

exceptionally difficult. Exact, analytic solutions are known only for relatively simple models or reduced

scenarios that are significantly less complex than those found in most practical applications. Solving

more complex models under realistic conditions requires using alternative strategies such as numerical

approximations. Numerical approaches have the advantage that approximate solutions for very complex

models can be obtained to, in theory, an arbitrary level of precision. There are nevertheless practical

limitations to what can be computed. While significant advances in computing machinery have been

1

made since the development of the electronic computer in the twentieth century, many calculations will

remain intractable for the foreseeable future due to the sheer number of operations that they require.

In light of these facts, the model that one selects for a particular application should balance the accu-

racy of the model under the conditions specified by the targeted application with the resources available

to construct solutions (numerical or otherwise) to the equations that are used to describe the model. Nu-

merical simulations of some systems can be performed relatively easily: there exist simple models that

are sufficiently descriptive for the intended application and numerical methods are known that produce

adequate approximations to the equations describing these models with a small computational cost.

However, many complex systems display radically different behavior in different situations, and captur-

ing the diverse array of phenomena that these systems exhibit using a single model and a single uniform

numerical approximation is often prohibitively expensive.

This has led to the development of a variety of approaches for combining different models or differ-

ent numerical methods in the hope that these combinations will be able to more efficiently describe the

behavior of systems that exhibit complex behavior. These approaches are often characterized using qual-

ifiers such as hybrid, multiscale, or adaptive. In many cases these methods are constructed in roughly

three steps: (i) divide the problem into different components that isolate each of the important regimes,

(ii) apply different models or numerical approximations to each component that are tailored towards

their respective regimes, and (iii) devise a mechanism for interfacing between the different models or

numerical representations that are used for each component. There are challenges associated with each

of these steps that depend on the particular application and methods under consideration.

Combining methods that are tailored strongly towards disparate regimes is perhaps counterproduc-

tive when intermediate regimes are also present. While the accuracy of the methods that are used may

be acceptable when conditions lie firmly within their respective regimes, accuracy can degrade rapidly

as conditions move away from those regimes. Because of this, a significant loss of accuracy can occur

when combined approaches are applied to problems that contain conditions between those that are the

focus of the combined approximation.

There are two primary motivations for the present work. The first is to investigate extensions of

a collision-based hybrid method for time-dependent radiation transport simulations that was recently

described in [97]. This hybrid approach is based on the first-collision source technique, which divides the

solution into streaming and collisional components to which different models and numerical approxi-

2

mations can be applied (c.f., section 1.2). The general idea is to apply a method such as ray tracing to the

streaming component and something closer to a fluid approximation to the collisional component. This

work extends that of [97] by considering hybrid methods using arbitrarily high-order discretizations in

space and time and arbitrarily accurate angular quadratures of different types. Additionally, these meth-

ods are implemented in a high-performance C++ research code that incorporates robust solver tech-

niques such as GMRES [168].

The second motivation for this work is to investigate the use of defect correction methods to increase

the effectiveness of combined or hybrid approaches. Defect correction methods are based on the idea

that an approximate solution for a problem can be iteratively refined towards a solution with a higher

level of accuracy by using a method that is less accurate, but also less expensive. Our goal is to con-

sider applying the principle of defect correction to the models that are used to describe the behavior of a

system in addition to the numerical approximations that are applied to these models. In this case the de-

sired high-accuracy solution would be described using a more detailed model of the underlying system,

and the lower-accuracy solution that is used would be composed of a combination of models that are

applicable to one or more limiting regimes present in the system (e.g., the streaming/collisional decom-

position mentioned previously). The hybrid methods considered in this work are constructed based on a

combination of high- and low-resolution numerical approximations that are applied to complementary

models of radiation transport. While the impact of full physical model reduction is not explicitly explored

in this work, it should be noted that the separation between reduced physical models and very coarse

numerical approximations is often not fully well-defined. As such, we believe that this work provides a

strong first step in the exploration of these ideas in the context of radiation transport applications. Fu-

ture extensions of this work will seek to examine the impact of further reductions in model complexity

in the context of defect correction methods.

This thesis provides roughly three contributions to the study of defect correction methods and hybrid

methods for radiation transport, each of which corresponds to one of chapters 2 to 4. First, low-storage

integral deferred correction (LS-IDC) methods are presented in chapter 2 that require approximately

half the storage required by traditional integral deferred correction (IDC) methods. The importance of

this contribution is related to the significant memory requirements imposed by the high-dimensional

nature of certain models, including kinetic models of radiation transport. In chapter 3, extensions of a

collision-based hybrid method for time dependent radiation transport simulations are discussed, with a

3

focus on the use of arbitrarily high-order numerical approximations including, in particular, high-order

time-integration schemes. In chapter 4, two mechanisms for increasing the effectiveness of the hybrid

methods are presented. The first is a reconstruction procedure for mapping between arbitrary discrete

ordinates quadratures within the context of these hybrid methods. The second is a variation of the previ-

ous hybrid methods using IDC integrators in which the hybrid approach is written as a two-grid iterative

method in angle that is combined with an IDC time-integration scheme. It is demonstrated that the

resulting methods are able to iteratively reduce the error due to the application of discrete ordinates

quadratures of different resolutions to the components of the hybrid method.

1.2 Radiation transport

1.2.1 Introduction

Radiation transport models are used to describe the behavior of charge-neutral particles as they prop-

agate through a medium. These models are commonly applied to a diverse range of particle species

including neutrons [62, 135], neutrinos [141], photons [160], and phonons [180]. In many applications,

these components are responsible for a significant portion of the transfer of energy that occurs between

different regions or constituent parts of the system.

Radiation transport models are typically composed of two main parts: particle streaming and parti-

cle collisions. These models assume that a particle traveling in some direction through a medium will

continue to travel in that direction (particle streaming) until it collides with the material it is traveling

through (particle collision). When a collision event occurs, energy is exchanged between the particle and

the background material in a manner that is determined by the physical properties of the system under

consideration. Interactions between the radiation particles themselves may also be considered, but in

many applications these types of interactions are neglected, either because they have a relatively small

impact on the overall behavior of the system or because including these effects significantly increases

the complexity of the model. In many cases the importance of these effects depends on the scales of

interest.

Kinetic descriptions of radiation transport are commonly expressed in terms of a first-order integro-

differential equation of the form

∂t f(cid:161)(cid:126)x,(cid:126)Ω,E, t(cid:162)+ (cid:126)Ω·∇(cid:126)x f(cid:161)(cid:126)x,(cid:126)Ω,E, t(cid:162) =C(cid:163)f(cid:164),

1
v

(1.1)

4

where the distribution function f depends on as many as seven variables: time t > 0, energy E > 0, po-
sition (cid:126)x ∈ (cid:82)3, and a unit vector (cid:126)Ω ∈ (cid:83)2 specifying direction of travel. Here v > 0 is the particle speed,

which may depend on various factors such as the type and energy E of the particle. The terms on the
left of (1.1) express particle streaming and the operator C on the right of (1.1) is the collision operator

that is used to express the behavior of collision events; i.e., radiation-material interactions, interactions

between radiation particles, etc.

There are a number of factors that make solving equations of the form (1.1) difficult. In the absence

of the collision term, the solution of (1.1) can be written explicitly for each direction (cid:126)Ω and energy E in-

dependently following a simple ray-tracing approach. However, this approach no longer works when the

collision term is included because the collision operator typically couples the particle directions and en-

ergies. Additionally, many problems of practical interest require dynamic interaction between the radi-

ation components and the background material. For example, in radiation hydrodynamics applications
the collision operator C couples the transport equation (1.1) to a hydrodynamic model for the evolution

of the background material [143, 160]. The coupling between the radiation and background material is

often highly nonlinear in nature, resulting in a significant degree of numerical stiffness when the radia-

tion and background material evolve on different timescales. This combination of high-dimensionality

and numerical stiffness renders radiation transport problems very computationally expensive to solve in

practice.

In this work we consider a simple linear, mono-energetic, time-dependent transport equation of the

form

∂t ψ+ (cid:126)Ω·∇(cid:126)x ψ+ σtψ = σs
4π

〈ψ〉+ q,

〈·〉 =(cid:90)

(·)d(cid:126)Ω,

(cid:83)2

(1.2)

where ψ = ψ((cid:126)x,(cid:126)Ω, t) is flux of radiation at time t ≥ 0 and position (cid:126)x ∈ X ⊂ (cid:82)3 in the direction (cid:126)Ω ∈ (cid:83)2. The
parameters σs, σa, and σt = σs + σa are the scattering, absorption, and total cross sections of the ma-
terial, respectively, and q = q((cid:126)x,(cid:126)Ω, t) is a source of radiation that is assumed to be known for all times
t ≥ 0. The equation (1.2) is assumed to be normalized such that particles travel with unit speed, and the

background material is assumed to be fixed in time: specifically, the cross sections of the material are as-

sumed to be non-negative functions of position only. The model (1.2) accounts for at most three forms of

interaction between the radiation and background material: scattering, absorption, and emission. The

5

collision operator C can be written in the following form that separates the contributions of these effects:

(cid:181) 1

4π

(cid:182)− σaψ+ q.

C[ψ] = σs

〈ψ〉− ψ

(1.3)

Note that the scattering of radiation off of the background material is assumed to be isotropic.

The equation (1.2) has a relatively simple form compared to that of more advanced models, but it

retains many of the same characteristics that make solving systems of this type difficult. Even though

energy-dependence is excluded, this system still contains a total of six dimensions. Hence storing nu-

merical representations of solutions can quickly become prohibitively expensive as the resolution of the

numerical approximations that are applied is increased. In this work only fully implicit time discretiza-

tions will be considered. This is motivated by the fact that in many applications where coupling between

the radiation and background material is present the radiation components evolve on timescales that

are much faster than that of the background material: Restricting global timestep sizes based on the

evolution of the radiation components is usually prohibitively expensive in such situations. However the

use of implicit methods poses a challenge due to the high dimensionality of the system and the need to

solve the equations that define the updated solution at each timestep. The simplicity of (1.2) does elim-

inate some of the difficulties present in more advanced models. Most notably, (1.2) is linear and lacks

the degree of stiffness commonly encountered when nonlinear material coupling is present. It remains

to be determined whether the methods introduced in this work are sufficiently robust for cases where

material coupling is included.

The remainder of this section is organized as follows: Section 1.2.2 examines some mathematical

properties of solutions of the transport equation (1.2) that are intimately tied to the hybrid approach used

in later chapters. Section 1.2.3 provides a very brief overview of common numerical methods applied to

the angular variable (cid:126)Ω of (1.2). Section 1.2.4 discusses some hybrid methods that share similarities with

the approach that is used in this work.

1.2.2 Structure of solutions

The purpose of this section is to outline some important properties of solutions of (1.2). Two character-

istics in particular are discussed: the diffusion limit for strong scattering and vanishing mean-free-path,

and a series form of the solution known as the multiple collision expansion.

6

The diffusion limit

As mentioned previously, the kinetic equation (1.1) is composed of two parts: particle streaming on the

left and the collision term on the right. The streaming terms on the left have the form of a simple hy-

perbolic conservation law, while the collision term on the right serves to relax the solution of the system

towards a local equilibrium state. Systems of this form can be more generally described under the um-

brella of hyperbolic conservation laws with relaxation terms [41]. The strength of the relaxation term

determines the extent to which the solution of the system is able to relax towards an equilibrium state

over a given length of time.

Knowledge of the equilibrium states is essential to understanding the overall behavior of systems

containing relaxation terms. One application of this knowledge is the derivation of reduced models for

scenarios where the scales of interest are much larger than the relaxation time – the time over which the

system is able to relax towards an equilibrium state. In kinetic theory, the relaxation time is related to

the particle mean-free-path or mean-free-time, and a hydrodynamic model is obtained in the limit as

the mean-free-path approaches zero. For example, hydrodynamic models such as the Euler or Navier-

Stokes equations can be derived from Boltzmann’s equation for dilute gases in this manner [38]. The

standard reduced model for the transport equation (1.2) is a diffusion equation [90, 120], the derivation

of which we now briefly recall.

The diffusion limit of (1.2) is used to describe the long-time behavior of systems that are dominated

by particle scattering. These situations occur when the total cross section is large relative to the times of

interest, and the contributions of particle absorption and emission are small. This is described formally

using the following scaling:

σt (cid:55)→ σt

ε

,

σa (cid:55)→ εσa,

t (cid:55)→ t

ε

.

(1.4)

q (cid:55)→ εq,
(cid:179) σt

(cid:180)

Under the asymptotic scaling (1.4), the transport equation (1.2) becomes
φ+ εq,

ε∂t ψ+ (cid:126)Ω·∇(cid:126)x ψ+ σt

− εσa

(1.5)
where φ = 〈ψ〉 and all quantities in (1.5) are assumed to be O(1) with respect to ε. We have the following

ε

ε

ψ = 1
4π

result for the solution of the scaled equation (1.5):

Proposition 1. If ψ solves (1.5) then φ satisfies the diffusion equation

·(cid:181) 1

3σt

(cid:182)+ σaφ = 〈q〉

∇(cid:126)x φ

∂t φ−∇(cid:126)x

7

(1.6)

up to an O(ε) error.

Proof. We do not consider contributions from initial or boundary layers: For more complete discussions

including these components see [90, 120]. To simplify notation slightly, we let(cid:101)ψ = ψ− 1

4π φ and (cid:126)J = 〈(cid:126)Ωψ〉.

The limiting behavior can be derived by substituting a Hilbert expansion in powers of ε

ψ = ψ(0) + εψ(1) + ε2ψ(2) +···

into (1.5) and equating terms for each power of ε. This gives,
at O(ε−1):

at O(1):

and at O(ε):

ψ(0) = 1
4π

φ(0),

(cid:126)Ω·∇(cid:126)x ψ(0) + σt(cid:101)ψ(1) = 0,
∂t ψ(0) + (cid:126)Ω·∇(cid:126)x ψ(1) + σt(cid:101)ψ(2) + σa

4π

φ(0) = q.

Multiplying (1.9) by (cid:126)Ω and integrating over (cid:83)2 yields, after some algebraic manipulation,

∇(cid:126)x φ(0) +(cid:126)J (1) = 0.

1
3σt

(1.7)

(1.8)

(1.9)

(1.10)

(1.11)

Integrating (1.10) over (cid:83)2 and using (1.11) to replace the resulting (cid:126)J (1) term yields (1.6) after some sim-
■

plification.

The importance of the diffusion limit is derived from the fact that many radiation transport appli-

cations contain collision scales that are much smaller than the timescales of interest. In such cases, the

solution of the transport equation (1.2) will exhibit behavior that is similar to the solution of the diffu-

sion equation (1.6). When numerical approximations are applied to (1.2), it is often important to con-

sider how the numerical methods that are used behave under the asymptotic scaling (1.4) in the discrete

setting. It would be desirable for a consistent discretization of (1.5) to produce a consistent discretiza-
tion of (1.6) in the limit as ε → 0. Many methods yield an inconsistent discretization in the limit, and

methods that do yield a consistent discretization may succumb to order reduction or onerous timestep

restrictions in limiting regimes [28]. Methods that do not yield a consistent discretization of the limiting

8

equation (1.6) as ε → 0 are often severely inaccurate for problems containing highly-collisional regimes

[3, 27, 164].

One of the advantages of implicit time discretizations is that they are generally more well-behaved

in limiting regimes than explicit methods [121]. For example, applying an implicit Euler discretization

to (1.5) and examining the behavior of the scheme for small ε yields an implicit Euler approximation of

the limiting equation (1.6) up to an O(ε) error (the proof of this is essentially the same as that of propo-

sition 1). This is one of the reasons (in addition to the unconditional stability of the method) that the

implicit Euler method remains the preferred time discretization for many radiation transport applica-

tions.

The multiple collision expansion

The multiple collision expansion is a series form of the solution of the transport equation that has useful

physical and mathematical interpretations. In the multiple collision expansion, the angular flux distri-

bution is written as a series

ψ(cid:161)(cid:126)x,(cid:126)Ω, t(cid:162) =

(cid:161)(cid:126)x,(cid:126)Ω, t(cid:162)

ψn

∞(cid:88)

n=0

where the distribution functions ψn satisfy

∂t ψ0 + (cid:126)Ω·∇(cid:126)x ψ0 + σtψ0 = q,
∂t ψn + (cid:126)Ω·∇(cid:126)x ψn + σtψn = σs
4π

(cid:173)ψn−1

(cid:174),

(n ≥ 1).

(1.12)

(1.13a)

(1.13b)

Much of the attractiveness of this form of the solution is derived from the fact that each of the ψn can be

interpreted physically as the distribution of particles that have experienced exactly n collision events.

Early appearances of this form of the solution, such as in [161, Section 22] and [14], were motivated

by physical interpretations in terms of collision counts. Generalizations [175] and analysis [176, 177]

were later provided by Syros, who showed rigorously for the case of slab geometry that the series (1.12) is

absolutely convergent and equal to the solution of (1.2) whenever the solution exists. Mathematically, the

series (1.12) corresponds to the Liouville-Neumann series solution of the integral form of the transport

equation [69].

One of the common applications of the multiple collision approach is the derivation of analytic or

semi-analytic benchmark solutions that can be used for code verification. The multiple collision ap-

proach was first used by Kholin to obtain analytic solutions to time-dependent transport problems with

9

anisotropic scattering [114]. Ganapol later derived a closed-form expression of the solution and applied

numerical techniques to the resulting expressions to explicitly obtain benchmark solutions [83]. A num-

ber of other benchmark solutions were later derived using similar techniques [80, 81, 84, 154], including

some for charged-particle systems [78, 82] and Brownian motion [79].

1.2.3 Standard methods

A wide variety of numerical methods have been devised for computing approximate solutions for the

transport equation (1.2). The purpose of this section is to provide a brief overview overview of some of

the more common numerical methods that are used to treat the angular or directional dependence of

the solution. This discussion is intentionally brief and by no means complete, though the references

given in this section should provide a reasonable starting point for those new to the field.

One of the simplest classes of methods are diffusion approximations – see, e.g., [160, Chapter III] or

[134, 149, 152] and references therein for some examples. Diffusion approximations effectively reduce

the system to one that does not depend on the angular direction, and hence do not permit arbitrary

refinement like other methods. The simplicity of most diffusion approximations often makes them much

less expensive than other methods. These approximations work well for highly collisional or optically

thick regimes where the solution is nearly isotropic anyway, but are ill-suited to cases where the behavior

of the solution is not dominated by particle scattering.

The two most common approaches when higher fidelity approximations are required are Monte

Carlo and discrete ordinates methods. Monte Carlo methods use a stochastic approach based on a

sampling of particle histories – see, e.g., [135, Chapter 7] or [52, 87, 139, 186] and references therein

for more thorough discussions. Most Monte Carlo methods in use are based on variations of the implicit

Monte Carlo (IMC) approach introduced by Fleck and Cummings [75]. One of the strongest advantages

of Monte Carlo methods is the ability to handle continuous treatments of the problem. This permits

a greater degree of fidelity for treating complicated geometries and energy-dependence of cross sec-

tion values than is possible with deterministic approaches. However, because Monte Carlo methods are

based on a stochastic approach, solutions contain some degree of noise and convergence of the methods

is generally slow (i.e., order N

−1/2 for N particle histories).

The discrete ordinates method is a collocation approximation for the angular variable of the trans-

10

port equation [135, Chapters 3 & 4]. This yields a coupled system of differential equations, each of which

depends only on time and space. In a physical sense, discrete ordinates methods restrict particle trajec-

tories to a finite number of directions corresponding to the nodes of the quadrature that is used. The use

of a finite number of directions naturally leads to what is called the ray-effects phenomenon [126, 127]

wherein particle trajectories can separate, producing visible lines or rays, for example, emanating from

sharp material interfaces. The popularity of discrete ordinates methods is driven by a number of fac-

tors, including the robustness of the methods, ease of enforcing boundary conditions, and the devel-

opment of extremely effective iterative techniques for solving the systems that result from implicit time

discretizations [4, 123]. One of the primary disadvantages of discrete ordinates methods, which the hy-

brid methods in this work seek to improve upon, is that the cost of computing accurate solutions for

problems containing disparate collisionality regimes can be exceptionally high.

Other approaches such as moment methods are generally less common, though recent efforts have

improved upon many of the difficulties inherent to these approaches. Moment methods approximate

the angular variation in the solution of the transport equation using a spectral approach [138, 148, 160].

This is done by: (i) writing the solution ψ of (1.2) in series form relative to some basis of L2((cid:83)2) (the most

common choice being spherical harmonics), (ii) using this representation in the original transport equa-

tion to produce an infinite, coupled system of equations, and (iii) applying a closure relation to reduce to

the resulting system to a finite-dimensional approximate form. The properties of moment approxima-

tions are determined primarily by the characteristics of the closure that is applied. The simplest closure
is a simple Galerkin projection in L2((cid:83)2), which yields the standard PN method. Other closures and
the addition of filtering strategies have been investigated in an attempt to improve the accuracy and ro-

bustness of moment-based approaches – see [95, 96, 119, 137, 163, 190] and references therein for some

recent examples. Additionally, a good overview and comparison of different closures can be found in

[85]. The most prominent disadvantage of moment methods is that ensuring positivity in the solution

is generally exceptionally difficult, particularly for problems that lack a significant degree of regularity.

While recent work has significantly improved the robustness of moment-based approaches, other as-

pects impacting the practical application of such methods, such as the development of efficient solver

and preconditioning techniques, still lags behind other methods.

Finite element methods have garnered some increased attention in recent years – see [20, 37, 99, 109,

117, 185] for some examples. One of the greatest strengths of finite element methods are their flexibility

11

in handling approaches such as adaptive refinement [117] and sparse grids [185] in a reasonably straight-

forward manner. However, the development of effective solver techniques for the systems of coefficients

resulting from implicit time discretization remains is an issue. Since the number of angular flux vectors

that can be stored is often significantly limited by available memory, advanced solver techniques such

as Krylov subspace methods can be prohibitively expensive when a high level of angular fidelity is re-

quired. The flexibility afforded by finite element methods may nevertheless permit better scalability on

very large distributed-memory systems, an area where methods such as the discrete ordinates method

have faced significant challenges [5, 21].

1.2.4 Hybrid methods

All of the numerical methods mentioned in section 1.2.3 yield approximations that require some trade-

offs in accuracy or efficiency. For example, discrete ordinates methods are generally quite robust for

a diverse array of problems, but they produce ray effects in streaming-dominated regimes [126, 127]

and become ill-conditioned for problems that are very optically thick [4]. Monte Carlo methods do not

produce ray effects like discrete ordinates methods, though they do become very computationally ex-

pensive for optically thick problems [86]. In contrast, most moment-based approaches are better-suited

to highly-collisional problems, but often produce negative values and poorly conditioned systems in

streaming dominated regimes [138].

For problems that contain collision probabilities that are relatively consistent everywhere, one can

select a numerical approach that is tailored towards the regime of interest. However, for problems that

contain disparate collisionality regimes, no one method consistently out-performs the others. As a re-

sult, a wide variety of hybrid methods have been developed that seek to combine different numerical

approaches in ways that allow a diverse array of problems to be solved more accurately and efficiently

than would otherwise be possible with any one method alone. Because the effectiveness of most numeri-

cal methods depends on the degree of collisionality present in a given problem, variations of the multiple

collision decomposition (1.12) and (1.13) are often used as the foundation on which hybrid methods are

constructed. Loosely speaking, this is done by applying a higher-fidelity approximation to terms near

the beginning of the series and a lower-fidelity, more fluid-like approximation to terms in the tail of the

series.

12

A crucial component of this approach is determining the point at which one should switch from a

higher-fidelity approximation to a lower-fidelity one. The most common approach is the first-collision

source technique [8, 9], in which only the first term of the series – the uncollided flux – is treated using a

higher-fidelity approximation and the remaining terms are grouped into a single component – the col-

lided flux – that is treated using a lower-fidelity approximation. Other approaches in which the solution

is split after two [88] or an arbitrarily-specified number of collision events [65, 158, 159] have also been

examined. One could also go one step further and consider decomposing the angular flux into more

than two components to which progressively coarser approximations are applied [181].

The other aspect that must be considered is the determination of which methods should be applied

to each component. In many cases methods such as Monte Carlo [8] or high-resolution discrete ordi-

nates approximations (or variations thereof) [11, 146, 147, 188] are selected for the higher-resolution

approximation. Analytic representations may also be used for the uncollided flux when available [9]. In

many cases the reduced approximation applied to the collisional component is numerical in nature: For

example, a low-resolution discrete ordinates method [8, 9, 88, 188] or a moment method [146, 147] may

be used. Other times a reduced physics model such as a point-kinetics [158, 159] or diffusion approx-

imation [23, 24, 136, 142] is used in place of the kinetic model for the collisional component. Within

the Monte Carlo community, hybrid methods leveraging a stochastic diffusion approximation such as

Implicit Monte Carlo Diffusion [86] or Discrete Diffusion Monte Carlo (DDMC) [65, 66] after a specified

number of collision events have been used with satisfactory results [1, 187].

It should be noted that methods motivated by the same considerations as the first-collision source

technique have been used in isolated instances under different names. Within the phonon transport

community, a similar technique is used that is often referred to as the ballistic-diffusive model [11, 39,

40, 102, 103, 146, 147, 188], where the ballistic and diffusive parts correspond to the uncollided and col-

lided components, respectively. A similar technique called the Isotropic Diffusion Source Approxima-

tion (IDSA) is used within the supernova neturino transport community [23, 24, 136, 142, 153]. The IDSA

splits the flux distribution into trapped and streaming components to which time-dependent diffusion

and steady-state ray-tracing approximations are applied, respectively. It is interesting to note that the

hybrid methods introduced in [97] which form the basis of this work share a great deal of similarity with

the IDSA. The key ingredient in both methods is the introduction of a mechanism by which the collided

distribution can flow into the uncollided distribution. Traditional first-collision source methods lack this

13

capability: the uncollided distribution flows into the collided distribution but not vice versa. In the IDSA,

this mechanism is derived on the continuum level using physical arguments to introduce source terms

into the split equations. In contrast, the feedback mechanism of the hybrid methods of [97] and this work

(i.e., the relabeling or reconstruction procedure) is constructed on the discrete level and is coupled with

the time integration scheme. Further, while this work focuses on hybrid methods using discrete ordi-

nates approximations for both the collided and uncollided components, this choice need not be strictly

enforced, since the approach is compatible with other types of discretizations.

1.3 Defect correction

1.3.1 Introduction

The terms defect correction and deferred correction are used to refer to a broad class of methods in which

the accuracy of an approximate solution to a problem is improved, often in an iterative fashion. One

of the simplest examples of such a method is that of iterative refinement for the solution of a system of

linear equations [100, Chapter 12]. The iterative refinement algorithm can be constructed as follows.

Suppose that a linear system

Ax = b

(1.14)

is to be solved, where A and b are given and x is the unknown solution. Suppose further that a matrix B is
−1 in some reasonable sense. In most applications of iterative refinement, B cor-
known such that B ≈ A
responds to the matrix obtained from a linear solution algorithm (e.g., Gaussian Elimination) executed
−1, it can be used to define an initial approximate

in finite-precision arithmetic. Since B approximates A

solution to (1.14):

(cid:101)x(0) = Bb.

(1.15)

The difference between this approximate solution and the exact solution x will depend on how well B

−1.

approximates A

Now suppose that some approximate solution (cid:101)x(k) has been computed after k applications of the
iterative process to be defined, with the initial value (cid:101)x(0) given by (1.15). Then the error e(k) and the

14

residual r (k) in(cid:101)x(k) are defined by

If the error e(k) were known, then the exact solution x could be obtained through

(1.16a)

(1.16b)

(1.17)

(1.19)

e(k) = x −(cid:101)x(k),
r (k) = b − A(cid:101)x(k).
x =(cid:101)x(k) + e(k).

(cid:101)e(k) = Br (k).
(cid:101)x(k+1) =(cid:101)x(k) +(cid:101)e(k).

However, the problem of determining e(k) is just as difficult as computing x to begin with: The error e(k)

satisfies the equation

Ae(k) = Ax − A(cid:101)x(k) = b − A(cid:101)x(k) = r (k),

(1.18)

which has the same form as the original problem (1.14), but with a different right-hand side. In an at-

tempt to improve the approximation solution, an approximation(cid:101)e(k) of e(k) can be computed using B

as before:

This approximate error can then be used to define an “improved” solution:

(1.20)
−1, this process will increase the accuracy of the
approximate solution at each iteration and the approximate solutions (cid:101)x(k) will converge to the exact

So long as B is a sufficiently good approximation of A

solution x.

The iterative refinement algorithm contains most of the crucial components of many defect correc-

tion methods: a residual is computed for a given approximate solution, which is then used to compute

erative refinement, the difference between A

an error approximation that is used to update the solution in an iterative fashion. In applications of it-
−1 and B is generally due to the stability of the linear solver
algorithm and the accumulated effects of round-off error [106, 169], but sometimes also includes mixed-

precision computations [34]. In a broader context considering other types of defect correction methods,

the difference between the “exact” and approximate solutions can vary widely depending on the algo-

rithm – see the reviews [25, 26] for an assortment of defect correction methods.

Defect correction methods for differential and integral equations have an overall structure that is

similar to that of the iterative refinement algorithm, but there are significant differences due to the con-

tinuum nature of these types of operator equations. Numerical approaches for solving such operator

15

equations typically follow a two-step procedure in which the problem is first discretized into a system of

linear or nonlinear equations that are then solved using standard techniques. The method that is applied

during the discretization step (e.g., a finite difference or finite element method) is typically accompanied

by some form of approximation error. Methods with smaller approximation errors are generally much

more complicated and thus much more computationally expensive to apply. In this context, one goal of

defect correction methods is to reduce the computational cost associated with more accurate numerical

techniques by combining them with less accurate – and less expensive – approximations.

For integral equations, two-grid methods combine high- and low-resolution quadratures to con-

struct an iterative procedure for computing the high-resolution solution [17]. These methods are often

nearly identical in form to the iterative refinement algorithm, but the iterative procedure specifically

targets the numerical error of the low-resolution approximation. In relation to the iterative refinement

algorithm, the matrices A and B for two-grid methods are the matrix corresponding to a high-resolution

quadrature and the inverse of the matrix corresponding to a low-resolution quadrature, respectively.

The intent is not that the low-resolution approximation will yield the inverse of the high-resolution ap-

proximation in exact arithmetic, but instead the two-grid method can be considered more as a form of

preconditioning for the high-resolution system.

Defect correction methods for differential equations typically target increasing the order of the trun-

cation error of the numerical scheme. These approaches are most commonly used in combination with

finite difference methods for boundary value problems [35, 156] and initial value problems [18, 19, 70,

76, 77], though recent work has also extended this idea to discontinuous Galerkin methods applied to the

spatial variables of models of fluid dynamics [72, 73]. These methods hold the numerical grid fixed and

apply a combination of high- and low-order approximations (e.g., finite difference schemes) on that grid.

As such, these methods can be viewed as preconditioned iterative procedures for computing the high-

order numerical solution. However, a curious phenomenon occurs: When properly formulated, these

methods increase the order of accuracy of the solution at each iteration up to the order of the high-order

approximation. The result is that the defect correction iteration yields the same order of accuracy as

the high-order approximation in a finite number of iterations. These methods are therefore commonly

not iterated until convergence, but only up to the point when the maximum order of accuracy has been

attained.

16

1.3.2 Integral deferred correction

Integral deferred correction (IDC) methods are a class of numerical methods for solving initial value

problems in which high-order accuracy is attained through iterative application of a low-order scheme

within a defect correction framework. IDC methods are characterized by the use of a discrete Picard inte-

gral representation of the error based on high-order numerical quadrature. The first IDC methods were

the spectral deferred correction (SDC) methods of [70], in which Euler methods (both implicit and ex-

plicit) were used to compute provisional solutions and Gauss-Legendre quadrature used to approximate

the Picard integral representation of the error. The study of IDC methods has since been expanded to in-

clude additional quadratures [130], various methods for computing provisional solutions and correction

iterations [45, 47, 51], time parallelism [46, 48, 50], Krylov subspace iteration [104, 105], and multilevel

formulations in the style of multigrid algorithms [173, 174], among others.

When it is convergent1, the iterative procedure on which IDC methods are based converges to the

collocation solution associated with the high-order quadrature used to approximate the Picard integral

representation of the error [94]. Because of this, the accuracy and stability properties of IDC methods are

similar to those of collocation methods, but the coupling between internal stages is significantly reduced.

This reduction in coupling can be leveraged to develop IDC methods that incorporate additional features

such as high-order operator splitting [44, 49] and implicit-explicit [144, 145] forms, while maintaining the

accuracy of the underlying collocation approximation. However this comes at the expense of increased

storage requirements. Typical IDC methods require storing the equivalent of two copies of each stage

vector: twice that of the associated collocation scheme. This can render IDC methods impractical for

problems involving high-dimensional partial differential equations.

1The iterative procedure on which IDC methods are based can be divergent in certain situations,
particularly in the strongly stiff limit when large numbers of non-uniformly spaced quadrature nodes
are used [162].

17

LOW-STORAGE INTEGRAL DEFERRED CORRECTION METHODS FOR SCIENTIFIC COMPUTING

CHAPTER 2

2.1 Introduction

In this chapter a modification of the traditional IDC approach that significantly reduces the storage

requirements of the methods is presented. These methods, which we call low-storage IDC or LS-IDC

methods, require storing only one copy of each stage vector throughout the iteration process. This re-

duction is achieved by splitting the spectral integration matrix into upper-triangular and strictly lower-

triangular parts, and applying the lower-triangular portion to the most recent set of stages in the residual

calculation instead of the previous set. This splitting is similar in spirit to a Gauss-Seidel iteration in the

sense that updated stages are used in the residual calculation as soon as they are computed. This allows

previous stage vectors to be overwritten by their corresponding updates immediately after the substep

in which the update is computed.

Much of the existing literature on low-storage integrators focuses on explicit methods, where the pri-

mary concern is the number of stage vectors that must be stored at any given time (see, e.g., [111, 112,

113] and references therein). This work focuses on fully-implicit methods, where additional considera-

tions must be made for the storage requirements of solving the system(s) that define the stage values of

the methods. Collocation integrators often contain fewer stages for a given order of accuracy than other

methods, but because these stages are fully coupled and must be computed simultaneously, the storage

requirements of the associated implicit solve scale proportionally to the number of stages. Other meth-

ods such as diagonally implicit Runge-Kutta methods require more stages for a given order of accuracy,

but these stages are only weakly coupled and can be computed sequentially. The result is that the stor-

age requirements for the implicit solves in these methods depends primarily on the size of a single stage

vector and remains relatively constant as the number of stages of the method increases. The advantage

of LS-IDC methods is derived from the fact that, relative to other fully-implicit schemes, LS-IDC meth-

ods minimize both the number of stages that must be stored simultaneously and the additional storage

required for solving the systems that define the stage values.

This chapter is organized as follows. In section 2.2 a review of collocation and IDC integrators is

18

provided to establish notation, and LS-IDC methods are introduced. In section 2.3 it is shown that LS-

IDC methods converge with the same order of accuracy as traditional IDC methods for linear, constant-

coefficient problems, and the linear stability of some LS-IDC methods is examined numerically. In sec-

tion 2.4 LS-IDC methods are applied to a nonlinear ODE and a linear transport equation to compare

their accuracy and memory efficiency with other fully-implicit methods.

2.2 Method descriptions

We consider the numerical solution of an initial value problem (IVP) of the form

y

(cid:48) = f (t, y)
y(t0) = y0

(2.1)

where y ∈ (cid:82)m. We assume that the function f : (cid:82)+ × (cid:82)m → (cid:82)m and initial condition y0 ∈ (cid:82)m are known.

2.2.1 Collocation

Integrating (2.1) with respect to t yields the Picard integral equation

y(t) = y0 +(cid:90) t

t0

f (τ, y(τ))d τ,

(2.2)

the solution of which is equivalent to that of (2.1). Let ∆t denote the desired timestep size: the equation

(2.2) can be written as

(cid:90) (s−t0)/∆t

0

f(cid:161)t0 + s∆t, y(t0 + s∆t)(cid:162)d s

y(t) = y0 + ∆t

(2.3)

through a change of variable. Let {cn}N

n=1 be a set of distinct points in the closed interval [0,1], and let

(cid:169)θn(s)(cid:170)N

n=1 defined by

denote the quadrature weights associated with the points {cn} for integrating over the interval [0, s],

where the functions (cid:96)n(ζ) are the Lagrange basis polynomials for the points {cn} satisfying

(cid:96)n(ζ)d ζ

0

θn(s) =(cid:90) s
1, n = k,

0, n (cid:54)= k.

(cid:96)k (cn) =

19

(2.4)

(2.5)

Replacing the integral in (2.3) by the above quadrature yields the collocation approximation

Y (t) = y0 + ∆t

N(cid:88)

n=1

θn

(cid:181) t − t0

(cid:182)

∆t

f(cid:161)tn,Yn

(cid:162),

(2.6)

with Yn = Y (tn) an approximation of y(tn) and tn = t0 + cn∆t for n = 1,..., N . The values Yn are defined
implicitly through the nonlinear system

Yn = y0 + ∆t

θn,k f(cid:161)tk ,Yk

(cid:162),

N(cid:88)

k=1

(n = 1,..., N ),

(2.7)

where θn,k = θk (cn). The result of the timestep update is given by Y (t0 + ∆t), which is obtained from
(2.6) once the Yn are known.

In subsequent sections it will be useful to write the system (2.7) using the vector notation

Y = y0 + ∆t(Θ⊗ I )f,
(cid:164)T ,

f =(cid:163)f (Y1),..., f (YN )(cid:164)T ,

Y =(cid:163)Y1,...,YN

(2.8)

(2.9)

where

and Θ =(cid:161)θn,k

(cid:162)N

n,k=1 is the matrix of quadrature weights.

2.2.2 Integral deferred correction

We consider only the case of integral deferred correction (IDC) methods constructed using implicit Euler

as the low-order method. The following notations will be used in this and subsequent sections: Latin

letters will be used for exact quantities and Greek letters will be used for approximate quantities. The

correction level or index of the correction iteration corresponding to a solution will be denoted using

a superscript enclosed in brackets: square brackets will be used for individual point values and round

brackets will be used to denote polynomial interpolants of point values.

Given a timestep size ∆t and a set of points {cn}N

n=1

set of provisional solutions(cid:169)ξ[0]

n=1 using the implicit Euler method:

⊂ [0,1] the IDC method begins by computing a

(cid:170)N
+ hn∆t f

n

(cid:179)

(cid:180)

n = ξ[0]
ξ[0]
n−1

where

t0 + cn∆t, ξ[0]

n

,

(n = 1,..., N ),

hn = cn − cn−1

for

n = 1,..., N

20

(2.10)

(2.11)

denote the substep sizes. The superscript [p] is used to denote the correction level, with p = 0 the first

set of provisional solutions computed by the implicit Euler method.

The IDC correction procedure can be constructed as follows. Let ξ(p−1)(t) denote the polynomial
(cid:170)N
interpolant (in time) of a set of provisional solutions(cid:169)ξ[p−1]
n=1 at correction level p − 1. The solution

n

error is given by

and the residual is given by

e(p−1)(t) = y(t)− ξ(p−1)(t)

r (p−1)(t) = d

d t

ξ(p−1)(t)− f

(cid:179)

(cid:180)

.

t, ξ(p−1)(t)

The derivative of the error satisfies

(2.12)

(2.13)

(2.14a)

(2.14b)

(2.14c)

d
d t

d t

e(p−1)(t) = d
y(t)− d
(cid:179)
d t
= f (t, y)− f
(cid:179)
= f

(cid:179)

ξ(p−1)(t),
t, ξ(p−1)(cid:180)− r (p−1)(t),
t, ξ(p−1) + e(p−1)(cid:180)− f
(cid:180)− hn∆t f

t, ξ(p−1)(cid:180)− d
(cid:179)

tn, ξ[p−1]

+ 
[p−1]

d t

n

n

(cid:90) t

t0

r (p−1)(s)d s.
(cid:180)−(cid:90) tn

r (p−1)(s)d s,

(cid:179)

n

tn, ξ[p−1]
(cid:170)N

Applying implicit Euler approximations to the time derivatives in (2.14c) yields


[p−1]

n

= 
[p−1]
n−1

+ hn∆t f

from which approximations(cid:169)
[p−1]
n=1 of the error in the set of provisional solutions can be obtained
using the ansatz that the error is initially zero; i.e., e(p−1)(t0) = 0 for all p ≥ 1. These approximations of

n

tn−1

(2.15)

the error can be used to update the provisional solutions and define a new set of provisional solutions:

n = ξ[p−1]
ξ[p]

n

+ 
[p−1]

n

,

(n = 1,..., N ).

(2.16)

Observing that the integral of an interpolant is equivalent to quadrature and applying the relations (2.13)

and (2.16) to (2.15) yields

n = ξ[p]
ξ[p]
n−1

+ hn∆t f

(cid:179)

where

tn, ξ[p]
n

tn, ξ[p−1]

n

(cid:180)+ ∆t

N(cid:88)

k=1

(cid:179)

tk , ξ[p−1]

k

(cid:180)

,

γn,k f

(cid:179)

(cid:180)− hn∆t f
γn,k =(cid:90) cn

(cid:96)k (s)d s

cn−1

21

(2.17)

(2.18)

are the normalized quadrature weights associated with the nodes {ck }N
vals [cn−1,cn] for n = 1,..., N .
In practice, the values ξ[p]

n are computed sequentially using (2.17). Nevertheless, (2.17) defines a

k=1 for integrating over the inter-

system of N nonlinear equations that can be written using the vector notation

where

and

ξ[p] = y0 + ∆t (H ⊗ I )f[p] + ∆t ((Θ− H)⊗ I )f[p−1],
(cid:162)(cid:180)T
ξ[p] =(cid:179)

(cid:162),..., f(cid:161)ξ[p]

f[p] =(cid:179)

f(cid:161)ξ[p]

(cid:180)T

,

ξ[p]
1 ,..., ξ[p]

N

N

1



h1

h1 h2
...
...

h1 h2

.

...
··· hN

H =

(2.19)

,

(2.20)

(2.21)

This representation of the correction procedure will be useful for the analysis in subsequent sections.

(cid:170)N

Once sufficiently many correction iterations have been performed, say P correction iterations, an
approximation of the solution of (2.1) at the time t0 + ∆t is constructed from the provisional solutions
(cid:169)ξ[P]
n=1 from the last correction iteration. For the numerical tests in section 2.4 we consider only
quadrature sets with cN = 1, in which case it is sufficient to use ξ[P]
N as the resulting approximation for
the timestep.

n

2.2.3 Low-storage integral deferred correction

An examination of the correction equation (2.17) reveals that each of the provisional solutions on correc-
tion level p is dependent on all of the provisional solutions on correction level p − 1. Because of this, the

(cid:170)N
n=1 and(cid:169)ξ[p]

traditional IDC iteration requires storing the equivalent of two sets of provisional solutions in memory at
n=1. This storage requirement can be prohibitively

any given time; specifically1,(cid:169)ξ[p−1]
1In practice, one commonly stores the function evaluations(cid:169)f(cid:161)tn, ξ[p−1]
(cid:170)N
(cid:169)ξ[p−1]

n=1 instead of the stages
n=1 in order to avoid repeating the function evaluations at each substep. Note that this changes
only the computational requirements of the implementation: the storage requirements of these ap-
proaches are identical.

(cid:162)(cid:170)N

(cid:170)N

n

n

n

n

22

expensive when high-order IDC methods are applied to large, complex systems of partial differential

equations.

The low-storage IDC (LS-IDC) methods considered here are a simple modification of traditional IDC

methods in which the updated provisional solutions are used in the computation of the residual as they

become available. Traditional IDC methods evaluate the residual using the same set of provisional solu-

tions

(cid:169)ξ[p−1]

(cid:170)N

k=1

(2.22)
for each substep n = 1,..., N . In contrast, LS-IDC methods compute the residual for the nth substep

k

(2.23)

(2.24)

which yields a different stencil for each substep. With this idea, (2.17) can be modified as:

using

(cid:169)ξ[p]

k

k=1

(cid:170)n−1
(cid:179)

and

k

k=n,

(cid:170)N

(cid:169)ξ[p−1]
(cid:180)− hn∆t f
(cid:179)
(cid:180)+ ∆t
tk , ξ[p−1]
n has been computed ξ[p−1]

tn, ξ[p−1]
N(cid:88)

γn,k f

k=n

(cid:180)

(cid:179)

n

k

n

(cid:180)

.

n = ξ[p]
ξ[p]
n−1

+ hn∆t f
n−1(cid:88)
+ ∆t

k=1

tn, ξ[p]
n

(cid:179)

γn,k f

tk , ξ[p]
k

The advantage of this approach is that once ξ[p]

is no longer needed:

can be overwritten by ξ[p]

n immediately after the nth substep. The result is that LS-IDC

hence ξ[p−1]

n

methods only require storing the equivalent of one correction level at any given time: half the memory

requirement of traditional IDC methods.

For the analysis in subsequent sections, we note that (2.24) defines a system of K nonlinear equations

of the form

ξ[p] = y0 + ∆t((H + ΘL)⊗ I )f[p] + ∆t((ΘU − H)⊗ I )f[p−1]

(2.25)

(cid:170)N

where ΘL is the strictly lower-triangular portion of Θ and ΘU is the upper-triangular portion of Θ. Con-
structing an approximation of the solution of (2.1) at the time t0 + ∆t from the set of provisional solutions
(cid:169)ξ[P]
n=1 from the last correction iteration proceeds in the same manner as traditional IDC methods.
For the numerical tests in section 2.4 we consider only quadrature sets with cN = 1, in which case it is
sufficient to use ξ[P]

N as the resulting approximation for the timestep.

n

23

2.3 Convergence and stability

We now examine some properties of the LS-IDC methods, focusing on the case of linear, constant

coefficient systems.

2.3.1 Convergence in the limit of small timestep size

For sufficiently small ∆t, it is known that IDC methods converge to the collocation solution associated
n=1 as the number of correction iterations p → ∞ [94, 104, 105]. We now provide
with the points {cn}N

a proof that this holds also for LS-IDC methods in the case of linear, constant-coefficient ODEs. The

following proposition will be used:

Proposition 2. Let A and B be arbitrary square matrices, and let ˜A = I − ∆t A with ∆t > 0. For ∆t suffi-
ciently small ˜A

−1 exists, and

−1

K→∞ ˜A
lim

−1(cid:180)k

∆tB ˜A

K(cid:88)

(cid:179)
(cid:161) ˜A− ∆tB(cid:162)−1 .

k=0

(2.26)

(2.27)

(2.28)

(2.29)

(2.30)

exists and is equal to

Proof. For ∆t < 1/ρ(A) (where ρ(A) denotes the spectral radius of the matrix A) we have

ρ (∆t A) = ∆t ρ (A) < 1.

It follows that I − ∆t A is invertible and it’s inverse is given by the Neumann series expansion

∞(cid:88)
k=0
(c.f., [140, Section 7.10]). Given an operator norm (cid:107)·(cid:107), for

(I − ∆t A)

−1 =

(∆t A)k

∆t <

1

(cid:107)A(cid:107)+(cid:107)B(cid:107)

24

we have, using the absolute convergence of the Neumann series,

ρ(cid:161)∆tB ˜A

−1(cid:162) ≤ ∆t(cid:107)B(cid:107)(cid:107) ˜A

(cid:176)(cid:176)(cid:176)(cid:176)(cid:176),

(∆t A)k

∆t k(cid:107)A(cid:107)k ,

(cid:176)(cid:176)(cid:176)(cid:176)(cid:176) K(cid:88)
K(cid:88)

k=0

k=0

−1(cid:107),
= ∆t(cid:107)B(cid:107) lim
K→∞
≤ ∆t(cid:107)B(cid:107) lim
K→∞
= ∆t(cid:107)B(cid:107)
1− ∆t(cid:107)A(cid:107) ,
< 1.

Hence the limit in (2.26) exists and

K(cid:88)

k=0

˜A

−1 lim
K→∞

(cid:179)

−1(cid:180)k = ˜A

−1(cid:179)

∆tB ˜A

I − ∆tB ˜A

−1(cid:180)−1 =(cid:161) ˜A− ∆tB(cid:162)−1 .

(2.31a)

(2.31b)

(2.31c)

(2.31d)

(2.31e)

(2.32)

■

In the case of linear, constant-coefficient problems, we have f (t, y) = F y for some matrix F . Then the
n=1 can be written

system (2.8) for the stages of the collocation method associated with the points {cn}N

as

Y = y0 + ∆t(Θ⊗ F )Y

(2.33)
Proposition 3. Let f (t, y) = F y in (2.1). Then for ∆t sufficiently small the iterates ξ[p] of the IDC method

so that

Y =(cid:163)I− ∆t(Θ⊗ F )(cid:164)−1y0.

(2.19) satisfy

lim

p→∞ ξ[p] = Y, and
(cid:179)
∆t p+1(cid:180)

(i)
(ii) (cid:107)Y− ξ[p](cid:107) = O
Proof. For f (t, y) = F y the system (2.19) takes the form

.

ξ[p] = y0 + ∆t(H ⊗ F )ξ[p] + ∆t((Θ− H)⊗ F )ξ[p−1].

Letting A = H ⊗ F and B = (Θ− H)⊗ F , and noting that

equation (2.34) implies that

ξ[0] =(cid:163)I− ∆t(H ⊗ F )(cid:164)−1y0,
−1(cid:180)k

p(cid:88)

−1

(cid:179)

ξ[p] = ˜A

∆tB ˜A

,

k=0

25

(2.34)

(2.35)

(2.36)

where ˜A = I− ∆tA. Applying proposition 2 to (2.36) under the assumption that ∆t is sufficiently small,

and applying some algebraic manipulation to the result yields

p→∞ ξ[p] =(cid:161) ˜A− ∆tB(cid:162)−1 y0,

lim

The first result implies the second upon observing that

=(cid:163)I− ∆t(H ⊗ F )− ∆t((Θ− H)⊗ F )(cid:164)−1y0,
=(cid:163)I− ∆t(Θ⊗ F )(cid:164)−1y0 = Y.
−1(cid:180)k

∞(cid:88)

−1

(cid:179)

∆tB ˜A

y0

Y− ξ[p] = ˜A

k=p+1

(2.37a)

(2.37b)

(2.37c)

(2.38)

(2.39)

(2.40)

(2.41)

(2.42a)

(2.42b)

(2.42c)

and applying relevant norms with the absolute convergence of the Neumann series.
Proposition 4. Let f (t, y) = F y in (2.1). Then for ∆t sufficiently small the iterates ξ[p] of the LS-IDC

■

method (2.25) satisfy

lim

p→∞ ξ[p] = Y, and
∆t p+1(cid:180)
(cid:179)

(i)
(ii) (cid:107)Y− ξ[p](cid:107) = O
Proof. The proof is similar to that of proposition 3. For f (t, y) = F y the system (2.25) takes the form

.

ξ[p] = y0 + ∆t((H + ΘL)⊗ F )ξ[p] + ∆t((ΘU − H)⊗ F )ξ[p−1].

Letting A = (H + ΘL)⊗ F and B = (ΘU − H)⊗ F , equation (2.39) implies that
(cid:180)p
−1B

−1(cid:180)k

ξ[p] = ˜A

p−1(cid:88)

∆tB ˜A

∆t ˜A

−1

(cid:179)

ξ[0],

where ˜A = I− ∆tA. Assuming that ∆t is sufficiently small, using proposition 2 and the fact that

y0 +(cid:179)
(cid:180)p
−1B

k=0

(cid:179)

p→∞
lim

∆t ˜A

ξ[0] = 0

in (2.40) implies that

p→∞ ξ[p] =(cid:161) ˜A− ∆tB(cid:162)−1 y0,

lim

=(cid:163)I− ∆t((H + ΘL)⊗ F )− ∆t((ΘU − H)⊗ F )(cid:164)−1y0,
=(cid:163)I− ∆t(Θ⊗ F )(cid:164)−1y0 = Y.

26

Now,

Y− ξ[p] =(cid:179)
=(cid:179)

Since

it follows that

(cid:180)p(cid:179)
−1B
(cid:180)p(cid:104)(cid:179)

−1B

˜A

−1y0 − ξ[0](cid:180)+ ˜A
(cid:180)+(cid:179)
−1y0 − Y
˜A

∆t ˜A

∆t ˜A

−1
−1y0 − Y = ˜A
˜A

∆tB ˜A

y0,

−1(cid:180)k
(cid:179)
∞(cid:88)

∆tB ˜A

k=p+1

−1

k=p+1

(cid:179)
∞(cid:88)
Y− ξ[0](cid:180)(cid:105)+ ˜A
−1(cid:180)−1
(cid:179)
∞(cid:88)

∆tB ˜A

−1

k=1

y0,

(cid:107) ˜A

−1y0 − Y(cid:107) = O(cid:161)∆t(cid:162).
(cid:107)Y− ξ[0](cid:107) = O(cid:161)∆t 2(cid:162).

−1(cid:180)k

(2.43a)

y0.

(2.43b)

(2.44)

(2.45)

(2.46)

Additionally, since ξ[0] is computed using the implicit Euler method, we have

Applying relevant norms to (2.43b), and using (2.45), (2.46), and the absolute convergence of the Neu-
■

mann series yields the second result.

2.3.2 Convergence for arbitrary timestep sizes

Consider the standard linear test problemy

(cid:48) = λy,
y(t0) = 1.

λ ∈ (cid:67),

(2.47)

When an IDC method is applied to (2.47) each correction iteration can be written in the form of a matrix-

vector multiplication in which the iteration matrix of the method G(z) is applied to the vector of previous
stage values, where z = λ∆t and ∆t is the timestep size. The contraction factor of the method is defined
as the spectral radius of the iteration matrix ρ (G(z)). For traditional IDC methods the iteration matrix

takes the form

G(z) = z (I − zH)

−1 (Θ− H),

and the iteration matrix for LS-IDC methods takes the form

G(z) = z (I − z (H + ΘL))

−1 (ΘU − H).

27

(2.48)

(2.49)

The analysis discussed in the preceding section establishes convergence for IDC and LS-IDC meth-

ods when the timestep size used is sufficiently small. The arguments rely on the fact that the operator

applied at each correction iteration is a contraction when the timestep size is sufficiently small. However,

what constitutes “small enough” is determined by the spectrum of the iteration matrix: this spectrum is

governed by the operator F of the system to be solved, and, via the matrices H and Θ, the quadrature
nodes {cn}N
n=1. Upon considering the problem (2.47), it becomes clear that the results of section 2.3.1
can be portrayed as describing the behavior of the methods in the limit |z| → 0 instead of ∆t → 0. While
these two characterizations are mathematically equivalent, depicting this behavior in terms of |z| → 0

more explicitly acknowledges the dependence on the properties of the actual system being solved.

Many problems of practical interest possess eigenvalues of sufficiently large magnitude that the be-
havior of methods when |z| is small is of little useful importance due to the exceptionally small step sizes

required to capture this limiting behavior. Instead, it is more desirable that a method is well-behaved for
|z| large. For defect correction methods, this translates into the requirement that that the contraction
factor of the iteration matrix remain less than one, and ideally much less than one, even for |z| large.

The contraction factor associated with a given IDC method is strongly dependent on both the num-

ber and type of quadrature nodes used. For IDC methods constructed with equispaced nodes, the con-
traction factor has the desirable property that ρ(G(z)) → 0 as z → −∞ with z ∈ (cid:82) [76, 184]. However,

schemes using equispaced nodes are not as well behaved near the imaginary axis as other schemes

(they are not A-stable) and can be negatively impacted by the Runge phenomenon when large num-

bers of quadrature nodes are used. The situation is quite different for Gauss-Radau, Gauss-Lobatto, and

Gauss-Legendre nodes, the use of which is desirable due to their spectral accuracy and superior stability
properties. IDC methods constructed using these types of quadrature nodes have limz→−∞ ρ(G(z)) > 0,
and, when large numbers of quadrature nodes are used, ρ(G(z)) > 1 away from the origin [162]. Because

of this, order reduction can be observed when these methods are applied to very stiff problems.

The contraction factors of IDC and LS-IDC methods along the negative real axis are shown in fig-

ure 2.1 and the contraction factors in the strongly stiff limit are shown in table 2.1. For most quadrature

nodes, the low-storage modification negatively impacts the contraction factors of the methods. This

effect is most pronounced for right-biased quadratures: that is, quadratures that include the right end-

point of the timestep interval and not the left. The contraction factors of right-biased quadratures in-

crease rapidly in the stiff limit as the number of nodes is increased. This effect is most pronounced

28

Nodes

Gauss-Radau
LS-IDC
IDC

Chebyshev-Radau
IDC

LS-IDC

Equal Right

IDC

LS-IDC

2
3
4
5
6
7
8

0.250
0.434
0.618
0.737
0.816
0.873
0.915

0.546
0.998
1.498
2.078
2.761
3.568
4.523

0.333
0.463
0.642
0.760
0.839
0.895
0.936

0.488
0.607
1.027
1.612
2.279
3.113
4.060

0.000
0.000
0.000
0.001
0.003
0.008
0.013

1.618
4.235
3.111
6.9E+1
5.4E+2
5.4E+4
3.3E+6

(a) Contraction factors for quadrature sets including the right endpoint of the timestep interval and not the left.

Nodes

Gauss-Legendre
LS-IDC
IDC

Gauss-Lobatto
LS-IDC
IDC

Chebyshev

Equal Both

IDC

LS-IDC

IDC

LS-IDC

2
3
4
5
6
7
8

0.317
0.421
0.561
0.665
0.742
0.800
0.845

0.493
0.642
0.954
1.305
1.710
2.182
2.735

0.500
0.500
0.592
0.684
0.758
0.815
0.860

0.500
0.556
0.581
0.591
0.596
0.857
1.280

0.396
0.460
0.582
0.679
0.753
0.810
0.854

0.479
0.554
0.566
0.595
0.866
1.211
1.629

0.500
0.500
0.553
0.616
0.682
0.748
0.813

0.500
0.556
0.546
0.856
1.812
3.079
4.169

(b) Contraction factors for quadrature sets containing both endpoints or neither endpoint of the timestep interval.

Table 2.1: Contraction factors for IDC and LS-IDC methods in the stiff limit using different numbers and
types of quadrature nodes with implicit Euler substeps.

for equally-spaced right-biased quadratures, and, in stark contrast to the behavior of traditional IDC

methods, all LS-IDC methods constructed with equally-spaced right-biased quadratures yield contrac-

tion factors greater than one in the stiff limit.

LS-IDC methods constructed with Gauss-Lobatto quadratures display notably different behavior

than methods using other quadratures: When four, five, or six quadrature nodes are used, LS-IDC meth-

ods with Gauss-Lobatto quadratures produce contraction factors in the stiff limit that are smaller than

those of traditional IDC methods using the same quadratures. Additionally, the contraction factors for

IDC and LS-IDC methods constructed using two or three Gauss-Lobatto nodes are comparable. These

results suggest that LS-IDC methods can be a competitive replacement for IDC methods when moder-

ately high-order accuracy is desired, but may exacerbate the difficulties faced by IDC methods for very

stiff problems.

29

2.3.3 Linear stability

We now examine the linear stability properties of LS-IDC methods. We begin by briefly reviewing some

basic terminology related to linear stability theory of numerical methods for the solution of initial-value

problems for ODEs before returning to the discussion of LS-IDC methods: see, e.g., [15, 92] for more

detailed discussions of linear stability theory.

A timestep update with step size ∆t of any one-step method applied to the test problem (2.47) can
be written as yn+1 = R(λ∆t)yn. The function R(z) is called the stability function of the method and
the quantity |R(z)| the amplification factor. The region of absolute stability of the method is the set
{z ∈ (cid:67) : |R(z)| ≤ 1}. A method is called A-stable if its region of absolute stability contains the entire left
half-plane, and L-stable if it is A-stable and |R(z)| → 0 as z → ∞.

The stability of standard IDC methods using implicit Euler as the base method was examined nu-

merically in [61]: it was found that it is possible to obtain IDC methods that are L-stable to within the

limits of double-precision arithmetic. The construction of such methods is based on three observations:
(i) any A-stable IDC method with c1 (cid:54)= 0 is necessarily L-stable [130, Theorem 3.1]; (ii) Radau-IIA colloca-
tion schemes are L-stable (and hence also A-stable) at all orders of accuracy [93]; and (iii) the provisional

solutions of IDC methods converge to the stage values of the collocation method corresponding to the
points {cn}N
n=1 used by the IDC method as the number of correction iterations increases. L-stable IDC
methods were constructed in [61] by choosing the points {cn}N
n=1 to correspond to a set of (right-biased)
Gauss-Radau quadrature nodes and applying sufficiently many correction iterations until the desired

stability properties were obtained. Note that these additional correction iterations do not increase the

overall accuracy of the method2, and serve only to refine the method’s stability properties.

The boundaries of the regions of absolute stability of IDC and LS-IDC methods constructed using two

and three Gauss-Radau quadrature nodes are shown in figure 2.2. The boundaries of the regions of ab-

solute stability of IDC and LS-IDC methods constructed using four and five Gauss-Lobatto quadrature

nodes are shown in figures 2.3 and 2.4, respectively. For Gauss-Radau quadratures, it is clear that the

2Additional correction iterations do not increase the overall accuracy of the IDC method once the
maximum order determined by the collocation scheme is attained. For sufficiently small ∆t and well-
behaved problems, this maximum order of accuracy can be attained using, e.g., 2N − 2 correction iter-
ations for N Radau quadrature nodes. Additional correction iterations may help reduce the effects of
order reduction observed when IDC methods are applied to very stiff problems or very large timesteps
are used [104, 130, 144].

30

1.0

0.8

0.6

0.4

0.2

0.0
−1E+ 4

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0
−1E+ 4

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

)
)
z
(

G

(
ρ

)
)
z
(

G

(
ρ

)
)
z
(

G

(
ρ

Gauss-Legendre

2pt IDC
3pt IDC
4pt IDC
2pt LS-IDC
3pt LS-IDC
4pt LS-IDC

0.6

0.5

0.4

0.3

0.2

0.1

)
)
z
(

G

(
ρ

Chebyshev

2pt IDC
3pt IDC
4pt IDC
2pt LS-IDC
3pt LS-IDC
4pt LS-IDC

−1E+ 2

−1E+ 0

z

Gauss-Lobatto

−1E− 2

0.0
−1E+ 4

−1E+ 2

−1E+ 0

−1E− 2

z

Equispaced Both

−1E+ 2

−1E+ 0

z

Gauss-Radau

3pt IDC
4pt IDC
5pt IDC
3pt LS-IDC
4pt LS-IDC
5pt LS-IDC

0.8

0.6

0.4

0.2

)
)
z
(

G

(
ρ

3pt IDC
4pt IDC
5pt IDC
3pt LS-IDC
4pt LS-IDC
5pt LS-IDC

−1E− 2

0.0
−1E+ 4

−1E+ 2

−1E+ 0

−1E− 2

z

Equispaced Right

2pt IDC
3pt IDC
4pt IDC
2pt LS-IDC
3pt LS-IDC
4pt LS-IDC

2pt IDC
3pt IDC
4pt IDC
2pt LS-IDC
3pt LS-IDC
4pt LS-IDC

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

)
)
z
(

G

(
ρ

0.0
−1E+ 4

−1E+ 2

−1E+ 0

z

−1E− 2

0.0
−1E+ 4

−1E+ 2

−1E+ 0

−1E− 2

z

Figure 2.1: Contraction factors for IDC and LS-IDC methods along the negative real axis using different
numbers and types of quadrature nodes with implicit Euler substeps.

31

low-storage modification negatively impacts the stability properties of the methods, and that recover-

ing L-stability with LS-IDC methods becomes much more difficult as the order of the method increases.

When two Gauss-Radau quadrature nodes are used the stability results for the LS-IDC method are similar

to that of the traditional IDC method. However, the stability of the LS-IDC method using three Gauss-

Radau quadrature nodes is considerably worse than the corresponding IDC method. In this case the

LS-IDC method is unable to recover L-stability in the same number of correction iterations as the IDC

method. For Gauss-Lobatto quadratures, the stability regions of IDC and LS-IDC methods behave sim-

ilarly near the origin, but the small oscillations that appear in the boundary of the stability regions of

LS-IDC methods protrude further away from the imaginary axis than the corresponding IDC methods.

From a broader viewpoint, the stability of IDC and LS-IDC methods using Gauss-Lobatto quadratures is

rather different. The stability regions of IDC methods using Gauss-Lobatto quadratures are unbounded

when the number of correction iterations applied is small, but appear to be bounded when larger num-

bers of correction iterations are used. The bounded stability regions are quite large in all cases, and seem

to increase in size once additional correction iterations are performed. In contrast, the stability regions

of LS-IDC methods using Gauss-Lobatto quadratures are unbounded for all numbers of correction itera-

tions considered. These results suggest that IDC and LS-IDC methods using Gauss-Lobatto quadratures

are best suited to different types of problems. For problems where stability for values near the imaginary

axis is more important, traditional IDC methods are likely a better choice from the perspective of sta-

bility. For problems where stability for values away from the imaginary axis is more important, LS-IDC

methods seem to more consistently yield better stability when Gauss-Lobatto quadratures are used.

2.4 Numerical tests

We now consider the application of LS-IDC methods to two test problems: a nonlinear ODE for which

an exact solution is known is used to study the accuracy of the methods and a linear transport equation

solved by the discrete ordinates method is used to illustrate the memory efficiency of the methods.

32

IDC with 2 Gauss-Radau Nodes

LS-IDC with 2 Gauss-Radau Nodes

0.8

)
z
(
m

I

0.4

0.6

0.4

0.2

)
z
(
m

I

0 corrections
1 correction
2 corrections
3 corrections
−8.0E− 5

0.0

−1.2E− 4

−4.0E− 5
Re(z)

0.0E+ 0

4.0E− 5

0.0
−1.2E− 3

1.6

1.2

)
z
(
m

I

0.8

0.4

IDC with 3 Gauss-Radau Nodes

0 corrections
1 correction
2 corrections
3 corrections
4 corrections
5 corrections

3.2

2.4

)
z
(
m

I

1.6

0.8

0 corrections
1 correction
2 corrections
3 corrections
−8.0E− 4

−4.0E− 4
Re(z)

0.0E+ 0

4.0E− 4

LS-IDC with 3 Gauss-Radau Nodes

0 corrections
1 correction
2 corrections
3 corrections
4 corrections
5 corrections
6 corrections
7 corrections

0.0
−8.0E− 4 −6.0E− 4 −4.0E− 4 −2.0E− 4

0.0E+ 0

2.0E− 4

0.0
−4.8E− 2 −3.6E− 2 −2.4E− 2 −1.2E− 2

0.0E+ 0

1.2E− 2

Re(z)

Re(z)

Figure 2.2: Boundaries of the regions of absolute stability for IDC and LS-IDC methods using implicit
Euler substeps on (right biased) Gauss-Radau quadrature nodes with various numbers of correction it-
erations. Regions of absolute stability lie to the left of the origin.

33

LS-IDC with 4 Gauss-Lobatto Nodes

5 corrections
6 corrections
7 corrections
8 corrections

3.2

2.8

2.4

2.0

1.6

1.2

0.8

0.4

IDC with 4 Gauss-Lobatto Nodes

5 corrections
6 corrections
7 corrections
8 corrections

18.0

16.0

14.0

12.0

10.0

8.0

6.0

4.0

2.0

)
z
(
m

I

0.0
−2.0E− 3

−1.5E− 3

−1.0E− 3
Re(z)

−5.0E− 4

0.0E+ 0

0.0
−1.2E− 1

−8.0E− 2

−4.0E− 2

0.0E+ 0

Re(z)

LS-IDC with 4 Gauss-Lobatto Nodes

3 corrections
4 corrections
5 corrections
6 corrections
7 corrections
8 corrections
9 corrections

IDC with 4 Gauss-Lobatto Nodes

3 corrections
4 corrections
5 corrections
6 corrections
7 corrections
8 corrections
9 corrections

3E+ 3

2E+ 3

1E+ 3

3E+ 3

2E+ 3

1E+ 3

)
z
(
m

I

)
z
(
m

I

)
z
(
m

I

0E+ 0

−1E+ 3

−5E+ 2

0E+ 0
Re(z)

5E+ 2

1E+ 3

0E+ 0

−1E+ 3

−5E+ 2

0E+ 0
Re(z)

5E+ 2

1E+ 3

Figure 2.3: Boundaries of the regions of absolute stability for IDC and LS-IDC methods using implicit
Euler substeps on four Gauss-Lobatto quadrature nodes with various numbers of correction iterations.
Regions of absolute stability lie to the left of the origin.

34

IDC with 5 Gauss-Lobatto Nodes

4.0

3.2

2.4

1.6

0.8

7 corrections
8 corrections
9 corrections
10 corrections

0.0
−2.0E− 3

−1.5E− 3

−1.0E− 3
Re(z)

−5.0E− 4

0.0E+ 0

IDC with 5 Gauss-Lobatto Nodes

3 corrections
4 corrections
5 corrections
6 corrections
7 corrections
8 corrections
9 corrections
10 corrections

3E+ 3

2E+ 3

1E+ 3

)
z
(
m

I

)
z
(
m

I

30.0

25.0

20.0

)
z
(
m

I

15.0

10.0

5.0

0.0

3E+ 3

2E+ 3

1E+ 3

)
z
(
m

I

LS-IDC with 5 Gauss-Lobatto Nodes

7 corrections
8 corrections
9 corrections
10 corrections

−1.6E− 1 −1.2E− 1 −8.0E− 2 −4.0E− 2

0.0E+ 0

Re(z)

LS-IDC with 5 Gauss-Lobatto Nodes

3 corrections
4 corrections
5 corrections
6 corrections
7 corrections
8 corrections
9 corrections
10 corrections

0E+ 0

−1E+ 3

−5E+ 2

0E+ 0
Re(z)

5E+ 2

1E+ 3

0E+ 0

−1E+ 3

−5E+ 2

0E+ 0
Re(z)

5E+ 2

1E+ 3

Figure 2.4: Boundaries of the regions of absolute stability for IDC and LS-IDC methods using implicit
Euler substeps on five Gauss-Lobatto quadrature nodes with various numbers of correction iterations.
Regions of absolute stability lie to the left of the origin.

35

2.4.1 A non-stiff, non-linear ODE

The first test problem is a simple nonlinear ODE from [18]:

(cid:179)
(cid:48)
1− y2
= −y2 + y1
(cid:179)
1
1
(cid:48)
= y1 + 3y2
1− y2
y
1
2
y (0) = (1,0)T .
The solution of (2.50) is given by y(t) = (cos t,sin t)T .

y



(cid:180)
− y2
(cid:180)
2
− y2
2

,

,

(2.50)

Convergence studies of IDC and LS-IDC methods applied to the IVP (2.50) are shown in figure 2.5.

Results for the collocation method corresponding to the quadrature used are included in each case

Gauss-Lobatto nodes, and five right-biased equispaced nodes(cid:169)cn = n/5(cid:170)5

for completeness. Three quadratures are considered for each method: three Gauss-Radau nodes, four
n=1. The methods are of order
five, six, and five for the specified quadratures, respectively. The error in each solution is measured with
respect to the ∞-norm at time t = 10 using the noted number of timesteps.

For an IDC or LS-IDC method with a maximum order of accuracy of M, the expected order of accu-
racy of the method is min{M,P + 1} after P correction iterations when implicit Euler is used as the low-
order method. Thus the methods are expected to attain their maximum order of accuracy after M − 1

correction iterations have been performed. For each IDC and LS-IDC method, results are shown using
M − 1 and M correction iterations.

There are two primary observations to be made from these results: (i) the anticipated rates of conver-

gence are attained by all methods, and (ii) the errors in the solutions produced by the LS-IDC methods

are consistently larger than those of the corresponding IDC methods. We do note that LS-IDC methods

with M correction iterations appear to produce convergence rates slightly larger than those expected as

the timestep size becomes very small. We believe that this effect can be attributed to a combination of:

(i) the second observation above, (ii) the fact that the difference between the LS-IDC and collocation

solutions can decrease at a faster rate than the overall orders of the methods (c.f. proposition 4), and

(iii) the results in section 2.3.2 demonstrating that the contraction factors of LS-IDC methods decrease

rapidly for small timestep sizes. Hence the LS-IDC methods yield increasingly better approximations of

the collocation solution as the timestep size becomes small, producing the observed behavior.

36

0
1
=

t

t
a
n
o
i
t
i
s
o
p
n

i

r
o
r
r
E

−4

10

−6

10

−8

10

−10

10

−12

10

−14

10

−16

10

25

3 Gauss-Radau Nodes

4 Gauss-Lobatto Nodes

5 Equispaced Nodes

IDC(4)

IDC(5)

LS-IDC(4)

LS-IDC(5)

Collocation

IDC(5)

IDC(6)

LS-IDC(5)

LS-IDC(6)

Collocation

IDC(4)

IDC(5)

LS-IDC(4)

LS-IDC(5)

Collocation

27

26
29
Number of timesteps

28

210

25

27

26
29
Number of timesteps

28

210

25

27

26
29
Number of timesteps

28

210

Figure 2.5: Convergence of IDC, LS-IDC, and collocation integrators using three Gauss-Radau nodes,
four Gauss-Lobatto nodes, and five right-biased equispaced nodes for the IVP (2.50). Errors are com-
puted with respect to the ∞-norm at time t = 10. The number of correction iterations used with each
IDC and LS-IDC method is indicated in the key of each plot.

2.4.2 Linear transport equation

The second test problem is the linear transport equation (1.2). For the time being, we simply state that

the system (1.2) is solved using a method of lines approach with a discontinuous Galerkin (DG) spatial

discretization and a discrete ordinates angular discretization. Detailed descriptions of these methods

and their application to (1.2) can be found in subsequent chapters of this thesis.

We consider a problem in two-dimensional x y-geometry with parameters outlined in figure 2.6 that

we refer to as the lattice problem. A plot of the scalar flux distribution

φ((cid:126)x, t) =(cid:90)

ψ((cid:126)x,(cid:126)Ω, t)d(cid:126)Ω

(cid:83)2

(2.51)

is included in figure 2.6c for context. The goal of the tests conducted here is to assess the memory ef-

ficiency (i.e., error versus memory footprint) of different time integration schemes. This is done by ap-

plying each method to the lattice test problem using a fixed discrete ordinates approximation and a DG

spatial discretization of the same order as the time integration scheme. Refinement studies are then per-

formed with fixed CFL numbers of 32, 64, and 128. Two sets of studies are considered: one of third-order

methods and one of fifth-order methods. An approximation of the error in each solution is computed by

comparing the scalar flux distribution of each solution with that of a reference solution at the final time

37

0

1

2

3

4

5

6

7

0

3

4

2

1

6
(a) Geometric layout.

5

0

1

2

3

4

5

6

7

0

q

0
0
1

Region σt σs σa
0
white
10
gray
red
0

1
10
1

1
0
1

(b) Parameter values.

7

3

2

1
6
(c) Sample solution.

5

4

0

-1

-2

-3

-4

-5

-6

7

Figure 2.6: Layout and sample solution for the lattice test problem for the transport equation (1.2). Color
scale of sample solution is log10.

using the L2(d(cid:126)x) norm

e(t) =(cid:181)(cid:90)

X

(cid:179)

(cid:180)2
φ((cid:126)x, t)− φref((cid:126)x, t)

(cid:182)1/2

,

d(cid:126)x

(2.52)

where φref is the chosen reference solution. All solutions use a discrete ordinates approximation with a

TN quadrature [179] of order 16.

Four third-order time integration schemes are considered: an IDC method using two Gauss-Radau

nodes and two correction iterations, which we refer to as IDC3; an LS-IDC method using two Gauss-

Radau nodes and two correction iterations, which we refer to as LS-IDC3; the L-stable DIRK(3,3) method

of [10], which we refer to as SDIRK3 (tableau given in figure 3.1b); and a linear DG time integration

scheme that we refer to as STDG3. Five fifth-order schemes are considered: an IDC method using three

Gauss-Radau nodes and four correction iterations, which we refer to as IDC5; an LS-IDC method using

three Gauss-Radau nodes and four correction iterations, which we refer to as LS-IDC5; the L-stable,

7-stage DIRK method of [118], which we refer to as Kværnø7 (tableau given in figure 3.2a); the L-stable,

8-stage DIRK method of [110], which we refer to as ARK8 (tableau given in figure 3.2b); and a quadratic

DG time integration scheme that we refer to as STDG5. It is useful to note that the time integration

components of the STDG3 and STDG5 methods are equivalent to Radau collocation schemes using two

and three quadrature nodes, respectively [125].

The tests were run on up to 16 nodes of the Laconia cluster at Michigan State University’s Institute for

Cyber-Enabled Research. Each Laconia node is equipped with two Intel Xeon E5-2690v4 processors and

at least 128GB of RAM. Third-order methods were tested using between 56 and 896 mesh cells and fifth-

order methods were tested using between 56 and 448 mesh cells, with the number of cells increasing by

a factor of two in each refinement.

38

Method Order (odd):

Initial Condition
Source
Integrator Stages
Solver

Total

STDG

N

1
1

(N + 1)/2
(N + 1)/2
N + 3

3

1
1
2
2

6

5

1
1
3
3

8

IDC
5

N

1
1
1
1
6 N + 1
1
1
9 N + 4

3

1
1
4
1

7

LS-IDC
N

1
1

(N + 1)/2

1

(N + 1)/2+ 3

3

1
1
2
1

5

5

1
1
3
1

6

DIRK
5
3

1
1
3
1

6

1
1
6
1

9

Table 2.2: Number of angular flux vectors required by each class of time integration scheme for solv-
ing the transport equation (1.2) using the discrete ordinates method. Here N represents the order of
accuracy of the method, which may differ from the number of quadrature nodes or stages used by the
method.

The memory requirements of solving (1.2) using the discrete ordinates method are determined pri-

marily by the number of angular flux vectors that must be stored during each timestep update. The

number of angular flux vectors required by each time integrator are summarized in table 2.2. There are

four contributions to the number of angular flux vectors: (i) the initial condition for the timestep; (ii) a

vector containing the source term q, which is assumed to be constant in time; (iii) the internal stages of

the time integrator; and (iv) additional vectors required by the solver for the implicit systems. Each of

the time integrators considered require one angular flux vector for the initial condition and one for the

source term. The differences in memory requirements between the time integrators are due to differ-

ences in stage counts and storage requirements of the solvers. The STDG and LS-IDC integrators require

storing the fewest number of stages simultaneously, and the stage counts for these methods are roughly

half that of both the IDC and DIRK integrators. The number of angular flux vectors required by the solver

for the implicit systems depends on the number of stages that are coupled into each solve: The solver

for the IDC, LS-IDC, and DIRK integrators requires only one additional angular flux vector, irrespective

of the order of the methods, and the number of additional angular flux vectors required by the solver

for the STDG integrator is equal to the number of stages. The LS-IDC methods consistently produce the

smallest memory footprint amongst all of the methods considered by minimizing both the number of

stages that must be stored simultaneously and the number of additional angular flux vectors required by

the implicit solver.

Figures 2.7a and 2.7b plot the total memory requirement of each solve for the third- and fifth-order

methods, respectively, against the estimated L2(d(cid:126)x) error in the scalar flux distribution of each solution

as measured relative to a refined reference solution. We note that the global convergence rates of the

39

methods in these tests are limited by the lack of regularity in the spatial variables. We observe that the

reduced memory footprint of the LS-IDC methods does not appear to negatively impact the accuracy of

the methods for this problem. For the third-order methods, the LS-IDC method appears to have a smaller

error for a given timestep size that either the STDG or IDC method of the same order. It is not clear why

this is the case, but we conjecture that the observed behavior is due to changes in the dispersive and

dissipative properties of the methods when the low-storage modification is applied.

40

CFL = 32

CFL = 64

CFL = 128

SDIRK3
STDG3
IDC3
LS-IDC3

SDIRK3
STDG3
IDC3
LS-IDC3

SDIRK3
STDG3
IDC3
LS-IDC3

−1 20

2

21

22

23

24

25

26

−1 20

2

21

22

23

24

25

26

−1 20

2

21

22

23

24

25

26

Memory usage (GB)

Memory usage (GB)

Memory usage (GB)

(a) Third-order methods.

CFL = 32

CFL = 64

CFL = 128

LS-IDC5
IDC5
STDG5
ARK8
Kværnø7

LS-IDC5
IDC5
STDG5
ARK8
Kværnø7

LS-IDC5
IDC5
STDG5
ARK8
Kværnø7

−1

10

−2

10

−3

10

−4

10

−1

10

−2

10

−3

10

−4

10

r
o
r
r
e
)
(cid:126)x
d
(

2
L

r
o
r
r
e
)
(cid:126)x
d
(

2
L

20

22

23

21
25
Memory usage (GB)

24

26

20

23

22

21
25
Memory usage (GB)

24

26

20

23

22

21
25
Memory usage (GB)

24

26

Figure 2.7: Memory usage vs. L2(d(cid:126)x) error in the scalar flux distribution for the lattice test problem.

(b) Fifth-order methods.

41

CHAPTER 3

HIGH-ORDER HYBRID METHODS FOR RADIATION TRANSPORT

3.1 The transport equation and discrete ordinates methods

3.1.1 The transport equation

We consider the numerical solution of a mono-energetic, time-dependent, linear transport equation of
the form (1.2). It is assumed that an inflow boundary condition ψB is known for all times t ≥ 0 on the set

(cid:110)(cid:161)(cid:126)x,(cid:126)Ω(cid:162) ∈ ∂X × (cid:83)2 : (cid:126)n((cid:126)x)· (cid:126)Ω < 0

(cid:111)

,

(3.1)

where (cid:126)n((cid:126)x) is the unit outward normal vector on the boundary ∂X of the spatial domain. In order to avoid

complications due to problem geometry, the spatial domain X is assumed to be a convex polyhedron
that is fixed for all times t ≥ 0.

The operator notations used in this and subsequent chapters can be outlined as follows. Continuum

operators will be written using non-bold, majuscule letters in a script font. For example, the equation

(1.2) can be written in the form

∂t ψ = −(L−SP) ψ+ q,

where

L = (cid:126)Ω·∇(cid:126)x

+ σt,

S = σs
4π

,

and

P (·) =(cid:90)

(·) d(cid:126)Ω(cid:48)

.

(cid:83)2

(3.2)

(3.3)

As in (3.2), the continuum quantities to which these operators are applied will be written using non-bold,

miniscule Greek or Latin letters. Approximations of operators and quantities with respect to the direc-

tional variable (cid:126)Ω will be written using bold letters to denote vectors of components of the discretization

that is applied; e.g., see the discrete ordinates system (3.5).

In the few cases where it is necessary to indicate that a spatial discretization has been applied, the

discrete operators will be written using a Gothic font and the quantities to which these operators are

applied will be written using majuscule Greek or Latin letters. Any such quantities may be written in

bold or non-bold faces to indicate the discretization state of the directional variable (cid:126)Ω. Because Gothic

typefaces are generally difficult to read and the focus of this work is the approximations with respect to

42

time and angle, the use of this notation will be avoided whenever possible. Instead, many discussions

will focus on equations in semi-discrete form (i.e., without a spatial discretization having been applied)

with the understanding that a spatial discretization would be applied before the corresponding systems

are solved numerically.

3.1.2 A first-order discrete ordinates method

Given a suitable quadrature(cid:169)(cid:126)Ωk , ωk

(cid:170)K

k=1

⊂ (cid:83)2 × (cid:82), a discrete ordinates approximation for (1.2) satisfies

the system of equations

K(cid:88)

∂t ψk + (cid:126)Ωk ·∇(cid:126)x ψk + σtψk = σs
4π

ω(cid:96)ψ(cid:96) + qk ,

where qk = q(cid:161)(cid:126)x,(cid:126)Ωk , t(cid:162) and ψk ((cid:126)x, t) approximates ψ(cid:161)(cid:126)x,(cid:126)Ωk , t(cid:162). The system (3.4) can be written in the

(cid:96)=1

(k = 1,...,K ),

(3.4)

compact form

where

the operator L has the form

∂t ψ = −(L−SP) ψ+ q,

q =(cid:163)q1,..., qK

(cid:164)T ,

(cid:164)T ,
ψ =(cid:163)ψ1,..., ψK
(cid:162)
L = Diag(cid:161)L1,...,LK
(cid:105)T
S =(cid:104) σs

,...,

σs
4π

4π

with

and

+ σt

Lk = (cid:126)Ωk ·∇(cid:126)x
P =(cid:163)ω1,..., ωK

(cid:164).

for k = 1,...,K , and the operators S and P are defined by

(3.5)

(3.6)

(3.7)

(3.8)

Once a spatial discretization has been applied, (3.5) is transformed into the following initial value

problem:

∂t Ψ = −(L− SP) Ψ+ Q.

(3.9)

The quantities in (3.9) denote representations of discrete approximations to the quantities in (3.5); e.g.,

expansion coefficients with respect to some finite element basis for the spatial variable. The operators

are derived from the original operators, and their action on the expansion coefficients is inherited from

the action of the original operators.

43

We note that the operators L and L are not linear, but affine: Inflow of radiation at the boundary of

the domain results in a source term that forms the nonlinear component of the operator. The notations

L(·) = (cid:101)L(·)− ψ

L(·) =(cid:101)L(·)− ΨB

and

B

will be used to denote the decomposition of the operators L and L into their linear components (cid:101)L and
(cid:101)L, and boundary components ψ

B and ΨB are determined
from the incoming flux of radiation ψB on the boundary of the domain. The distinction between the

B and ΨB, respectively, where the values of ψ

(3.10)

linear and boundary components of these operators in not of critical importance in the present context,

but will be necessary for correctly formulating the defect correction methods described later.

Applying an implicit Euler approximation to the time derivative in (3.9) with step size ∆t yields a

system of the form

(cid:161)(cid:98)L− SP(cid:162) Ψn =(cid:98)Qn

that must be solved at each timestep, where

(cid:98)L =(cid:101)L+ 1

I,

∆t

(cid:98)Qn = 1

∆t

Ψn−1 + Qn + Ψn
B ,

(3.11)

(3.12)

and I is the identity operator. The linear system (3.11) is most commonly solved using one of several

iterative methods built around inverting the streaming operator (cid:98)L at each iteration. The traditional

approach to solving (3.11) is a fixed-point method known as source-iteration [4]: given an initial guess

Ψ(0) for Ψn, iterate

Ψ(k) =(cid:98)L

−1(cid:179)

SPΨ(k−1) +(cid:98)Qn(cid:180)

until the difference between successive iterates is below a given tolerance in some norm.

A more modern approach, which we employ for all numerical results in this work, is to use a Krylov

subspace method. One of the primary advantages of Krylov subspace methods over traditional fixed-

point methods such as source iteration is a reduction in the number of iterations required for conver-

gence [183], particularly in strongly-scattering regimes. Using one of several matrix inversion identities

[98], the solution Ψn of (3.11) can be written as

(3.13)

(3.14)

(3.15)

where PΨn can be obtained by solving

Ψn =(cid:98)L

(cid:179)

I− P(cid:98)L

−1S

−1(cid:161)(cid:98)Qn + S(cid:161)PΨn(cid:162)(cid:162),
(cid:180)(cid:161)PΨn(cid:162) = P(cid:98)L

−1(cid:98)Qn.

44

The system (3.15) is solved using a matrix-free GMRES iteration to find PΨn+1, which then yields Ψn+1
through (3.14). The benefit of this approach is that the dimension of the system (3.15) to which the

GMRES algorithm is applied is much smaller than the dimension of the original system (3.11). Hence

the additional costs associated with the GMRES procedure, such as storage of previous iterates and con-

struction of associated least-squares solutions, are significantly reduced.

Under some mild restrictions on the boundary conditions and the spatial discretization that is ap-

plied, direct inversion of (cid:98)L can be accomplished via so-called transport sweeps. The term transport
sweep is used to refer to a form of block forward substitution used to invert (cid:98)L in which the solution in

each mesh element is updated by traversing the spatial mesh in the characteristic direction determined

by each ordinate direction, with the inflow boundary condition used to begin the process for each or-

dinate. Transport sweeps occupy the vast majority of the solve time required by each of the iterative

methods described above, and a considerable amount of work has focused on efficient implementations

of transport sweeps using various numerical approaches on a wide range of computer architectures –

see, e.g., [5, 21, 60, 63, 101, 115, 150, 155, 157, 182, 189] and references therein for some examples.

The overall cost of these iterative solution procedures is determined by: (i) the cost of each iteration,

which is dominated by the cost of the transport sweep, and (ii) the number of iterations required for

the solver to converge within the desired tolerance. The interaction between these two factors is highly

problem-dependent. Problems that are streaming dominated require many ordinates to mitigate the

well-known “ray-effects” phenomenon [126, 127], but the ordinates are only weakly coupled and the

resulting linear systems can be solved in relatively few iterations. For increasingly scattering-dominated

problems the ordinates become more strongly coupled and more iterations are required. However, the

increased coupling has the effect of smoothing out ray effects, so that fewer ordinates are necessary to

resolve the bulk effects of the radiation. To obtain accurate solutions for problems with both streaming-

and scattering-dominated regions, a uniform treatment of the angular flux requires large numbers of

strongly coupled ordinates, producing systems that are very expensive to solve.

3.1.3 First-order hybrid discrete ordinates methods

We consider hybrid methods constructed following the first-collision source technique [8, 9], in which

the angular flux ψ is decomposed into a collided component ψc and an uncollided component ψu with

45

ψ = ψu + ψc such that ψu and ψc satisfy

∂t ψu + (cid:126)Ω·∇(cid:126)x ψu + σtψu = q,
∂t ψc + (cid:126)Ω·∇(cid:126)x ψc + σtψc = σs
4π

(cid:163)〈ψu〉+〈ψc〉(cid:164).

(3.16a)

(3.16b)

In theory, there are multiple approaches that one could consider for dividing the initial and boundary

conditions of the original system (1.2) between the collided and uncollided components of (3.16). Here

we consider only the case where the uncollided system (3.16a) inherits the initial and boundary con-

ditions of the original system (1.2), and the collided system (3.16b) is endowed with vacuum boundary

conditions and an initial condition of zero.

The advantage of the splitting (3.16) is that it separates the limiting collisionality regimes of the orig-

inal equation (1.2) in a self-consistent manner, with the free-streaming limit captured by the uncollided

component and the diffusion limit captured by the collided component. Because the collision operator

in this case is linear, this splitting is exact on the continuum level; that is, if ψu and ψc solve (3.16), then
ψ = ψu + ψc solves (1.2). However, if different numerical approximations are applied to the collided and
uncollided components, then an effect that is similar to a splitting error will result.

Our intent is to utilize angular discretizations for the collided and uncollided components that are

tailored to the limiting collisionality regimes captured by each component. Because solutions to prob-

lems near the free-streaming limit often contain a great deal of variation in the angular dimension, a

higher-resolution angular approximation is used in the uncollided component in order to more accu-

rately capture this behavior. The increasingly high collision rates found in problems that lie closer to-

wards the diffusion limit have the effect of smoothing and diminishing the angular variation of the so-

lution. Hence a lower-resolution angular approximation is used for the collided component to reduce

computational cost.

ponents. Given two quadrature sets(cid:169)(cid:126)Ωu,k , ωu,k

In this work, discrete ordinates approximations are used for both the collided and uncollided com-
⊂ (cid:83)2 × (cid:82), a discrete

⊂ (cid:83)2 × (cid:82) and(cid:169)(cid:126)Ωc,k , ωc,k

(cid:170)Ku

k=1

(cid:170)Kc

k=1

ordinates approximation for (3.16) satisfies

∂t ψu = −Luψu + q,
∂t ψc = −(cid:161)Lc −ScPc

(cid:162)ψc +ScPuψu,

(3.17a)

(3.17b)

46

where

(cid:164)T ,

ψu =(cid:163)ψu,1,..., ψu,Ku

(cid:164)T .
(cid:161)(cid:126)x,(cid:126)Ωc,k , t(cid:162), and
Here ψu,k = ψu,k ((cid:126)x, t) approximates ψu
qk ((cid:126)x, t) = q(cid:161)(cid:126)x,(cid:126)Ωu,k , t(cid:162). The operators Lu, Lc, Sc, Pu, and Pc are defined analogously to (3.7) and (3.8)

q =(cid:163)q1,..., qKu
(cid:161)(cid:126)x,(cid:126)Ωu,k , t(cid:162), ψc,k = ψc,k ((cid:126)x, t) approximates ψc

ψc =(cid:163)ψc,1,..., ψc,Kc

(cid:164)T ,

and

(3.18)

with subscripts used to denote the quadrature associated with each operator.

The split system (3.17) is discretized using a strategy similar to the one used for the standard discrete

ordinates system (3.5). Applying an implicit Euler approximation to the time derivatives in (3.17) with

step size ∆t yields

(cid:181)

(cid:181)
Lu + 1
∆t

Lu + 1
∆t
−ScPc

(cid:182)
(cid:182)

u = 1
ψn
∆t
c = 1
ψn
∆t

ψn−1
u + qn,
ψn−1

c

+ScPuψn
u.

(3.19a)

(3.19b)

Once a spatial discretization has been applied to (3.19), the resulting systems are solved sequentially

using standard methods. Because (3.19a) does not contain a scattering source, the uncollided flux ψn
u

can be computed using a single transport sweep. The collided flux ψn

c is obtained by solving (3.19b)

using one of the iterative methods discussed in section 3.1.2. The cost of this solve is reduced by using

fewer ordinates in the approximation of the collided flux. Because the number of ordinates Kc scales with

the square of the quadrature order in two- and three-dimensional geometries, reducing the precision of

the collided quadrature by small amounts can significantly decrease the cost of this solve.

In the absence of any external source (i.e., q ≡ 0 and ψB ≡ 0), the uncollided component ψu will
decay exponentially in time at a rate determined by the total cross section σt. This will have the effect

of reducing the accuracy of the split method (3.19) to that of the angular quadrature used in the collided

component as the system is evolved in time. The approach introduced in [97] to mitigate this effect is

to periodically “relabel” the collided flux as uncollided to reduce the proportion of the angular flux that

is designated as collided at any given time. We consider the following generalization of this approach.

Under the assumption that the initial flux distribution for each timestep is composed entirely of an un-

collided component ψn−1∗

and the collided flux is initially zero, (3.19) can be written as

(cid:181)

(cid:181)
Lc + 1
∆t

Lu + 1
∆t
−ScPc

(cid:182)
(cid:182)

ψn−1∗ + qn,

u = 1
ψn
∆t
c =ScPuψn
ψn
u.

47

(3.20a)

(3.20b)

Once the systems in (3.20) have been solved a single approximation of the resulting angular flux distri-
bution at the time tn, denoted by ψn∗, is constructed from the components ψn
c . In order to
maintain the accuracy of the solution, ψn∗ is constructed on the higher-resolution quadrature set used

u and ψn

for the uncollided component. Because the approximations of the collided and uncollided components

are computed using different angular quadratures, these components use different numerical represen-
tations: hence computing ψn∗ requires a reconstruction procedure to map between these representa-

tions.

The idea introduced in [97] is to apply a reconstruction operator R that maps the collided flux from

the collided quadrature set to the uncollided quadrature set. The resulting flux distribution for the

timestep is then defined explicitly by

(3.21)
A suitable definition of R generally depends on the problem geometry and the choice of angular quadra-

ψn∗ = ψn

u +Rψn
c .

tures for the collided and uncollided components. The reconstruction operators used in this work are

discussed in section 3.3.2.

Each timestep of the hybrid discrete ordinates method follows a three-step cadence: First solve the

uncollided system, then solve the collided system, and finally apply a reconstruction to obtain the solu-

tion for the timestep. This process is outlined in detail in algorithm 3.1.

3.2 High-order time discretizations

Hybrid methods using four classes of high-order time integrators are now described. We consider

diagonally-implicit Runge-Kutta (DIRK) methods, space-time discontinuous Galerkin (STDG) methods,

integral deferred correction (IDC) methods, and the low-storage integral deferred correction (LS-IDC)

methods introduced in chapter 2. Each time integration scheme is combined with a standard discrete

ordinates approximation (we refer to these methods as nonhybrid methods) and several types of hybrid

approximations.

A type specifier has been assigned to each hybrid method in order to allow the different hybrid ap-

proaches to be discussed in a reasonable manner. The hybrid methods for each integrator are grouped

into at most two distinct types, each denoted by an upper-case Roman numeral. The type designation is

used to characterize how the splitting is applied to the system. The types considered here are as follows:

48

Algorithm 3.1 Hybrid Euler timestep.

input: Initial condition ψn−1∗
1: Compute ψn

u by solving (3.20a):

.

(cid:182)

(cid:181)
Lu + 1
∆t

2: Compute ψn

c by solving (3.20b):

(cid:181)
Lc + 1
∆t

3: Apply reconstruction: ψn∗ ← ψn
u

+Rψn
c .

return: ψn∗

ψn−1∗

+ qn.

u = 1
ψn
∆t
(cid:182)

−ScPc

c =ScPuψn
ψn
u .

I. Hybrid-I methods are constructed by solving the continuum split system (3.17) using a standard

method-of-lines approach and applying a reconstruction procedure between timesteps.

II. Hybrid-II methods follow a different construction depending on the type of the time integration

scheme. For more traditional time integrators such as DIRK methods, the hybrid-II methods are

constructed by applying the time integrator before applying a splitting analogous to (3.16) at the

semi-discrete level. For IDC and LS-IDC methods the hybrid-II splitting is unique in the sense

that the splitting is wrapped inside of the defect correction iteration in a way such that the defect

correction procedure is able to iteratively correct for the hybrid splitting that is employed. We will

defer discussion of the hybrid-II IDC and LS-IDC methods to chapter 4.

The hybrid methods are further characterized by at most three subtypes, each denoted by a lower-case

Latin letter. The subtype designation is used to characterize the reconstruction procedure that is applied.

We consider only subtypes a and b in this chapter, which are characterized as follows:

(a) Hybrid-a methods apply a reconstruction procedure at the end of each timestep that is based on a
reconstruction operator R maps from the collided quadrature set to the uncollided quadrature set

directly.

(b) Hybrid-b methods are specific to IDC and LS-IDC integrators. These methods apply a reconstruc-
tion procedure after each substep that is based on a reconstruction operator R maps from the

collided quadrature set to the uncollided quadrature set directly.

49

The subtype c is used in chapter 4 to designate that the method employs a Nyström reconstruction pro-

cedure.

We will not consider all possible combinations of all types of hybrid methods with all subtypes for

each class of time integrator. Dirk methods of types Ia and IIa are described in section 3.2.1. For STDG

integrators, the hybrid methods of type I and II are equivalent. We therefore drop the type designation

for hybrid STDG methods and characterize them only by their subtype. The construction of the hybrid-a

STDG methods is described in section 3.2.2 using the approach of the hybrid-I methods. Hybrid-I IDC

and LS-IDC methods are described in sections 3.2.3 and 3.2.4, respectively, for subtypes a and b. As

stated previously, discussions of hybrid-II IDC and LS-IDC methods is deferred to chapter 4.

3.2.1 Diagonally implicit Runge-Kutta methods

Given an initial value problem (IVP) of the form

(cid:48) = f (t, y),

y

y(t0) = y0,

(3.22)

a general s-stage Runge-Kutta (RK) method evolves an approximate solution of (3.22) in time through an

algebraic system of equations of the form

yn = yn−1 + ∆t
Y (i ) = yn−1 + ∆t

s(cid:88)
s(cid:88)

i=1

j=1

(cid:179)

tn−1 + ci ∆t,Y (i )(cid:180)
(cid:179)
tn−1 + c j ∆t,Y (j )(cid:180)

,

bi f

ai j f

,

(i = 1,..., s),

(3.23a)

(3.23b)

where yn approximates y(tn), tn = tn−1 + ∆t, and ∆t denotes the timestep size. Each RK method is
defined by the parameters A = (ai j )s
i=1, which may be chosen to satisfy
various order and stability criteria (c.f., [15, 91, 92]). Diagonally-implicit RK (DIRK) methods are char-

i=1, and c = (ci )s

i ,j=1, b = (bi )s

acterized by the fact that the coefficient matrix A is lower triangular; i.e., ai j = 0 whenever j > i . This
allows the stages Y (i ) to be computed sequentially in a manner analogous to forward substitution.

50

Algorithm 3.2 Nonhybrid DIRK timestep.

input: Initial condition ψn−1.
1: for i = 1,..., s do
(cid:181)
2:
L+ 1
∆t ai i

−SP

Compute ψ(i ) by solving (3.24b):

(cid:182)

ψn−1 − i−1(cid:88)
j=1

ai j
ai i

(cid:104)

(L−SP) ψ(j ) − q(j )(cid:105)+ q(i )

ψ(i ) = 1

∆t ai i

3: end for

4: if the DIRK scheme is stiffly accurate then
5:
6: else

ψn ← ψ(s).
ψn is defined by (3.24a): ψn ← ψn−1 − ∆t

7:

8: end if

return: ψn

Nonhybrid DIRK method

(cid:104)
(L−SP) ψ(i ) − q(i )(cid:105)

.

s(cid:88)
i=1

bi

Applying a DIRK scheme (3.23) to the standard discrete ordinates system (3.5) yields

where q(i ) = q(cid:161)tn−1 + ci ∆t(cid:162) for i = 1,..., s. The nonhybrid DIRK timestep algorithm is outlined in algo-

rithm 3.2.

Hybrid-I DIRK method

The derivation of the hybrid-I DIRK methods begins by applying a general DIRK scheme (3.23) to the

split discrete ordinates system (3.17) with the collided flux initially zero and the initial condition ψn−1∗

51

ψn = ψn−1 − ∆t

ψ(i ) = ψn−1 − ∆t

(cid:104)

(L−SP) ψ(i ) − q(i )(cid:105)
(L−SP) ψ(j ) − q(j )(cid:105)
(cid:104)

,

,

bi

ai j

s(cid:88)
i(cid:88)

i=1

j=1

(i = 1,..., s),

(3.24a)

(3.24b)

Algorithm 3.3 Hybrid-Ia DIRK timestep.

input: Initial condition ψn−1∗
1: for i = 1,..., s do
Compute ψ(i )
2:

.

u by solving (3.25c):

(cid:182)

(cid:181)
Lu + 1
∆t ai i

ψ(i )
u

= 1

∆t ai i

ψn−1∗

− i−1(cid:88)
j=1

ai j
ai i

(cid:183)
Luψ(j )
u

(cid:184)+ q(i ).

− q(j )

3:

Compute ψ(i )

c by solving (3.25d):

(cid:181)
Lc + 1
∆t ai i

(cid:182)

−ScPc

ψ(i )
c

= − i−1(cid:88)
j=1

ai j
ai i

(cid:183)(cid:161)Lc −ScPc

(cid:162)ψ(j )

c

(cid:184)+ScPuψ(i )

u .

−ScPuψ(j )
u

4: end for

5: if the DIRK scheme is stiffly accurate then
6:

ψn
u
ψn
c

← ψ(s)
u .
← ψ(s)
c .

7:
8: else

9:

10:

ψn

u is defined by (3.25a): ψn
u

ψn

c is defined by (3.25b): ψn
c

11: end if
12: ψn∗ ← ψn
u
return: ψn∗

+Rψn
c .

(cid:104)Luψ(i )
− q(i )(cid:105)
s(cid:88)
(cid:104)(cid:161)Lc −ScPc
(cid:162)ψ(i )
i=1

bi

u

c

.

−ScPuψ(i )
u

(cid:105)

.

← ψn−1∗
− ∆t
s(cid:88)
← −∆t
i=1

bi

,

(cid:105)

(cid:104)Luψ(i )
u − q(i )(cid:105)
(cid:104)(cid:161)Lc −ScPc
(cid:162)ψ(i )
c −ScPuψ(i )
u − q(j )(cid:105)
(cid:104)Luψ(j )
(cid:104)(cid:161)Lc −ScPc
(cid:162)ψ(j )
c −ScPuψ(j )

u

u

,

,

for the uncollided component. This yields

bi

bi

ψn

u = ψn−1∗ − ∆t
− ∆t
c =
ψn
u = ψn−1∗ − ∆t
ψ(i )

i=1

i=1

s(cid:88)
s(cid:88)
i(cid:88)
i(cid:88)
(cid:170)s
i=1 and(cid:169)ψ(i )

− ∆t

j=1

j=1

c

ai j

ai j

c =
ψ(i )
The stages(cid:169)ψ(i )

u

(cid:170)s

i=1 are computed using an alternating cadence between the uncollided

and collided stages; i.e., in the order ψ(1)

u , ψ(1)

c , ψ(2)

u , ψ(2)

c ,....

The hybrid-Ia DIRK method is obtained by applying an operator R to map the collided component

52

(3.25a)

(3.25b)

(3.25c)

(3.25d)

(cid:105)

,

(i = 1,..., s),

(i = 1,..., s).

Algorithm 3.4 Hybrid-IIa DIRK timestep.

.

input: Initial condition ψn−1∗
1: for i = 1,..., s do
Compute ψ(i )
(cid:181)
u by solving (3.26a):
2:
Lu + 1
∆t ai i

ψ(i )
u

(cid:182)

ψn−1∗

= 1

∆t ai i

(cid:183)(cid:161)Lu −SuPu

(cid:162)ψ(j )∗ − q(j )

(cid:184)+ q(i ).

ai j
ai i

− i−1(cid:88)
j=1
(cid:182)

3:

Compute ψ(i )

c by solving (3.26b):(cid:181)

Lc + 1
∆t ai i

−ScPc

ψ(i )
c

=ScPuψ(i )
u .

ψ(i )∗ ← ψ(i )
u

4:
5: end for

+Rψ(i )
c .

6: if the DIRK scheme is stiffly accurate then
7:
8: else

ψn∗ ← ψ(s)∗ .
ψn∗ is defined by (3.28): ψn∗ ← ψn−1∗

9:

− ∆t

10: end if

return: ψn∗

(cid:104)(cid:161)Lu −SuPu

(cid:162)ψ(i )∗ − q(i )(cid:105)

.

s(cid:88)
i=1

bi

u and ψn
at the final time to the uncollided quadrature set to obtain the reconstructed solution. Once ψn
c
are known – either through (3.25a) and (3.25b), or directly from ψ(s)
c if the DIRK scheme is stiffly
accurate – the reconstructed solution ψn∗ is given by (3.21). The hybrid-Ia DIRK timestep algorithm is

u and ψ(s)

outlined in algorithm 3.3.

Hybrid-II DIRK method

The hybrid-II DIRK methods are derived by changing the order in which the collision-based splitting

and the time discretization are applied. In the derivation of the hybrid-I DIRK methods, the splitting

was applied to the continuum transport equation (1.2) to obtain the split system (3.16), which was then

discretized in angle, space, and time. For the hybrid-II DIRK methods, the time discretization is applied

first. A collision-based splitting that is analogous to (3.16) is then applied to the equations that define

the stage values, with the understanding that each such equation has the form of a steady-state trans-

port equation. The resulting systems are then discretized in angle and space in a manner similar to the

methods described earlier.

53

We omit a detailed derivation of the hybrid-II DIRK methods and simply state the final result. Each

u and ψ(i )

c satisfy

stage of the hybrid-II DIRK method is composed of three parts: an uncollided component ψ(i )
component ψ(i )
ψ(i )

u and ψ(i )

c , and a reconstructed value ψ(i )∗ that is computed from ψ(i )
(cid:181)
Lu + 1
∆t ai i
−ScPc

u = 1
ψ(i )
∆t ai i
c =ScPuψ(i )
ψ(i )
u .

(cid:104)(cid:161)Lu −SuPu

ψn−1∗ − i−1(cid:88)

(cid:162)ψ(j )∗ − q(j )(cid:105)+ q(i ),

ai j
ai i

j=1

(cid:182)
(cid:182)

(cid:181)
Lc + 1
∆t ai i

u , a collided

c . The components

(3.26a)

(3.26b)

Since the collided and uncollided components of each stage depend on the reconstructed values of the

previous stages, the reconstruction procedure must be applied between stages. This is in contrast to the

hybrid-Ia DIRK method, where the reconstruction procedure was applied only to the approximations at

the final time after all of the stages had been computed.

The hybrid-IIa DIRK method is obtained by applying an operator R to map the collided component

for each stage to the uncollided quadrature set to obtain the reconstructed stage values. Once the com-
ponents ψ(i )

c for a stage are known, the reconstructed stage value ψ(i )∗ is given by

u and ψ(i )

The solution for the timestep is computed using the following modified form of (3.24a)

ψ(i )∗ = ψ(i )
s(cid:88)

u +Rψ(i )
c .
(cid:104)(cid:161)Lu −SuPu

bi

i=1

(cid:162)ψ(i )∗ − q(i )(cid:105)

ψn∗ = ψn−1∗ − ∆t

(3.27)

,

(3.28)

or is obtained directly from ψ(s)∗ if the DIRK scheme is stiffly accurate. The hybrid-IIa DIRK timestep

algorithm is outlined in algorithm 3.4.

3.2.2 Space-time discontinuous Galerkin methods

We now consider traditional and hybrid discrete ordinates methods using space-time discontinuous

Galerkin (STDG) methods of arbitrary order. Assume the spatial domain X is a convex polyhedron and
let Th be a partition of X into non-overlapping polyhedral cells K of diameter less than h. Given a time
tn for which an approximate solution ψn is known and a timestep size ∆t, we consider advancement of
the solution to the time tn+1 = tn + ∆t using an orthogonal space-time mesh obtained by extruding the
spatial mesh Th through the time interval In =(cid:163)tn, tn+1
(cid:164). This yields a space-time mesh (cid:101)Th,n that parti-
tions the space-time slab X × In into polytopal cells of the form (cid:101)K =K× In for each K ∈Th. Let Vh,n be

54

a finite-dimensional space of functions of both space and time such that the restriction of each v ∈ Vh,n
to each cell (cid:101)K ∈(cid:101)Th,n is continuous.

Nonhybrid STDG method

For each ordinate (cid:126)Ωk , the boundary ∂X of the spatial domain can be decomposed into two components
∂X = ∂X

∪ ∂X

−
k

+
k where

−
k

∂X

=(cid:169)(cid:126)x ∈ ∂X : (cid:126)n((cid:126)x)· (cid:126)Ωk < 0(cid:170)

=(cid:169)(cid:126)x ∈ ∂X : (cid:126)n((cid:126)x)· (cid:126)Ωk ≥ 0(cid:170)

+
k

∂X

(3.29)
are the inflow and outflow components of the boundary for the kth ordinate, respectively. Let Eh denote

and

the set of interior edges of the partition Th with(cid:161)(cid:83)

e(cid:162)∩ ∂X = (cid:59), and to each edge e ∈Eh associate a

unit vector (cid:126)ne normal to the edge whose orientation may be chosen arbitrarily. On each edge e ∈ ∂X
the associated unit normal vector (cid:126)ne is specified to be the outward normal.

e∈Eh

±
k

For each ordinate, the STDG approximation of the solution ψk of the discrete ordinates system (3.5)

(k = 1,...,K ),

(3.30)

(cid:90)
(cid:161)(cid:126)Ωk ·(cid:126)ne

K (vu)

(cid:162) v

(cid:175)(cid:175)(cid:175)t=tn+1

d(cid:126)x

−

−

u

d s d t,

K∈Th

,

k

k

(cid:179)

(cid:90)

(cid:101)K

Bh,n
k

k
Bh,n
k

is given by ψh,n

∈ Vh,n satisfying
(cid:179)
ψh,n

, vh,n(cid:180) = ¯Bh,n(cid:179)

φh,n, vh,n(cid:180)+Ah,n
for every vh,n ∈ Vh,n with the bilinear and linear forms
(cid:162)[v](cid:98)u d s d t + (cid:88)

vh,n(cid:180)
(u, v) = − (cid:88)
(cid:161)∂t v + (cid:126)Ωk ·∇(cid:126)x v + σtv(cid:162)u d(cid:126)x d t + (cid:88)
(cid:101)K∈(cid:101)Th,n
(cid:90)
− (cid:88)
(cid:161)(cid:126)Ωk ·(cid:126)ne
e×In
(cid:90)
e∈Eh
(cid:88)
(cid:101)K σsvu d(cid:126)x d t,
(cid:101)K∈(cid:101)Th,n
(cid:90)
(cid:101)K qk v d(cid:126)x d t − (cid:88)
(v) = (cid:88)
(cid:101)K∈(cid:101)Th,n
(cid:90)
+ (cid:88)
k v(cid:161)(cid:126)x, t n(cid:162) d(cid:126)x,
K ψn
K∈Tn
ψh,n =(cid:104)

¯Bh,n (u, v) = 1
4π

(cid:161)(cid:126)Ωk ·(cid:126)ne

(cid:90)
e×In

Ah,n
k

(cid:162) ψ

e×In

e∈∂X

e∈∂X

(cid:105)T

+
k

−
k

(cid:90)

where

B,k = ψB

ψ

(cid:161)(cid:126)x,(cid:126)Ωk , t(cid:162) is the value of the inflow boundary condition specified on ∂X

,..., ψh,n

ψh,n

K

1

,

φh,n =Pψh,n,

−

d s d t

B,k v

±

v

((cid:126)x, t) = lim

ε→0+ v ((cid:126)x ± ε(cid:126)ne , t),

[v] = v

+ − v

−

,

55

(3.31a)

(3.31b)

(3.31c)

(3.32)

−
k for the ordinate (cid:126)Ωk ,

(3.33)

and(cid:98)v is a numerical flux specified in terms of v
tion flow imposed by the continuous operator L, we use the upwind flux
(cid:98)v ((cid:126)x, t) = lim

ε→0+ v(cid:161)(cid:126)x − ε(cid:126)Ωk , t(cid:162).

and v

−

+

. In order to maintain the direction of informa-

(3.34)

Ψh,n

k

, then

(3.35)

(3.36)

(3.37)

(3.38)

(3.39)

In practice (3.30) is written as a linear system relative to some basis of Vh,n. If Bh,n is a vector whose

elements form a basis of Vh,n and Ψh,n
k
(3.30) can be written as

is the vector of coefficients such that ψh,n

k

= BT
h,n

where (cid:179)

with Qh,n

k

Ψh,n(cid:180)T =(cid:183)(cid:179)
(cid:161)Bh,n

=Ah,n
k

Ψh,n

1

Lh,nΨh,n = Sh,nPh,nΨh,n + Qh,n,
(cid:179)

Qh,n(cid:180)T =(cid:183)(cid:179)

(cid:180)T
(cid:179)
(cid:162); the operator Lh,n has the form

(cid:180)T(cid:184)

Ψh,n

,...,

and

K

(cid:179)

(cid:180)

(cid:180)T

,...,

(cid:179)

Qh,n

K

(cid:180)T(cid:184)

Qh,n

1

(cid:179)

=Bh,n
k

BT
h,n,Bh,n

(cid:180)

Lh,n = Diag

Lh,n
k
for k = 1,...,K ; and the operators Sh,n and Ph,n are defined by

,...,Lh,n

Lh,n
1

with

K

Sh,n = 1K ⊗ ¯Sh,n

and Ph,nΨh,n = K(cid:88)

ωk Ψh,n
k

,

k=1

where

¯Sh,n = ¯Bh,n(cid:179)

(cid:180)

BT
h,n,Bh,n

and 1K is the K -dimensional column vector of ones.

As was the case with the implicit time integration schemes in the previous sections, the STDG dis-

cretization yields a linear system (3.35) that must be solved at each timestep. The system (3.35) can be

solved using the same iterative techniques described in section 3.1.2 where the streaming operator Lh,n
is inverted through transport sweeps. The approximate solution at the time tn+1 is given by evaluating
ψh,n at tn+1.

Hybrid STDG method

Unlike the DIRK methods discussed in section 3.2.1, STDG integrators produce identical methods irre-

spective of the order in which the collision-based splitting and the time discretization are applied. This

56

is due to the fact that the temporal degrees of freedom in the STDG method are fully coupled, preventing

the reconstruction procedure from being applied at intermediate stages. Here we consider discretiza-

tion of the split system (3.16) in a manner similar to the STDG discrete ordinates method described

previously.

−
k

Most of the previous discussion applies here with minor modifications to notation. The decomposi-

∪ ∂X

+
k becomes ∂X = ∂X

tion of the boundary ∂X = ∂X

+
c,k for the un-
collided and collided ordinates respectively, where the components are defined similarly to (3.29) with
(cid:126)Ωu,k and (cid:126)Ωc,k replacing (cid:126)Ωk . For each ordinate, the STDG approximations of the uncollided and collided
components ψu,k and ψc,k of the split system (3.16) are given by ψh,n

+
u,k and ∂X = ∂X

∈ Vh,n satisfying

∪ ∂X

∪ ∂X

−
u,k

−
c,k

u,k , ψh,n

c,k

Bh,n
u,k
Bh,n
c,k

(cid:179)
(cid:179)

ψh,n

ψh,n

,

u,k

vh,n(cid:180)
u,k , vh,n(cid:180) =Ah,n
(cid:179)
φh,n, vh,n(cid:180)
c,k , vh,n(cid:180) = ¯Bh,n(cid:179)
u,k and Bh,n
(cid:90)

,

for every vh,n ∈ Vh,n. The bilinear forms Bh,n

Bh,n

u,k (u, v) = − (cid:88)
(cid:101)K∈(cid:101)Th,n
(cid:90)
− (cid:88)
e∈Eh

e×In

c,k are defined analogously to (3.31a); i.e.,

(cid:101)K

(cid:161)∂t v + (cid:126)Ωu,k ·∇(cid:126)x v + σtv(cid:162)u d(cid:126)x d t + (cid:88)
(cid:90)
(cid:161)(cid:126)Ωu,k ·(cid:126)ne

(cid:162)[v](cid:98)u d s d t + (cid:88)

K∈Th

(cid:90)
(cid:161)(cid:126)Ωu,k ·(cid:126)ne

K (vu)

(cid:175)(cid:175)(cid:175)t=tn+1
(cid:162)v

−

−

u

d(cid:126)x

(3.41)

d s d t

e∈∂X

+
u,k

e×In

(k = 1,...,Ku),
(k = 1,...,Kc),

(3.40a)

(3.40b)

(cid:90)

e×In

(cid:161)(cid:126)Ωu,k ·(cid:126)ne

(cid:162)ψ

−

B,k v

d s d t

(3.42)

and similarly for Bh,n

Ah,n

u,k is defined by

c,k . The linear form Ah,n
u,k (v) = (cid:88)
(cid:101)K∈(cid:101)Th,n
+ (cid:88)
K∈Tn
= ψn∗(cid:161)(cid:126)x,(cid:126)Ωu,k , t(cid:162),

(cid:90)
(cid:101)K qk v d(cid:126)x d t − (cid:88)
(cid:90)
K ψn∗,k v(cid:161)(cid:126)x, t n(cid:162) d(cid:126)x,
qk =(cid:161)(cid:126)x,(cid:126)Ωu,k , t(cid:162),

e∈∂X

−
u,k

where

ψn∗,k

(3.43)
for k = 1,...,Ku, are the initial condition, source, and boundary condition for the timestep, respectively,
given on the uncollided discrete ordinates set. The bilinear form ¯Bh,n is again defined by (3.31b), but
the scalar flux φh,n is defined here by

and

(cid:161)(cid:126)x,(cid:126)Ωu,k , t(cid:162),

B,k = ψB

ψ

φh,n =Puψh,n

u +Pcψh,n

c

,

(3.44)

57

Algorithm 3.5 Hybrid-a STDG timestep.

input: Initial condition ψn∗.

1: Compute Ψh,n

u

by solving (3.46a):

2: Compute Ψh,n

c

by solving (3.46b):

(cid:179)

= Qh,n.

u Ψh,n
Lh,n
u
(cid:180)

Lh,n

c

− Sh,n

c Ph,n

c

Ψh,n

c

= Sh,n

c Ph,n

c Ψh,n
u .

3: Ψh,n∗ ← Ψh,n
u
return: ψh,n∗

with

.

c

+ RΨh,n

(cid:175)(cid:175)(cid:175)t=tn+1
u =(cid:104)

ψh,n

(cid:105)T

and

ψh,n
u,1 ,..., ψh,n
u,Ku

c =(cid:104)

ψh,n

ψh,n
c,1 ,..., ψh,n
c,Kc

(cid:105)T

.

(3.45)

If Bh,n is a vector whose elements form a basis of Vh,n and Ψh,n

u,k and Ψh,n
c,k , then (3.40) can be written as

Ψh,n

= BT
h,n

such that ψh,n
u,k

= BT
h,n

Ψh,n
u,k and ψh,n

c,k

c,k are vectors of coefficients

(cid:179)

(cid:180)

u = Qh,n,
u Ψh,n
Lh,n
c = Sh,n
Ψh,n

c − Sh,n
Lh,n

c Ph,n

c

c Ph,n

u Ψh,n
u ,

(3.46a)

(3.46b)

and Lh,n

c

are defined

u , Ψh,n

where Ψh,n
analogously to (3.37); and Sh,n

c

, and Qh,n are defined analogously to (3.36); the operators Lh,n

u

, Ph,n

c

, and Ph,n

u

are defined analogously to (3.38) and (3.39). The

c

systems in (3.46) are solved in a manner similar to the previous hybrid methods: The uncollided system

(3.46a) is solved using a single transport sweep, and then the collided system (3.46b) is solved using one

of the iterative methods discussed in section 3.1.2.

Once the components ψh,n

u

and ψh,n

c

have been computed, a reconstruction is performed to obtain

a single approximation ψh,n∗ to the angular flux with

ψh,n∗ =(cid:104)

(cid:105)T

ψh,n∗,1 ,..., ψh,n∗,Ku

,

ψh,n∗,k

= BT
h,n

Ψh,n∗,k ,

for k = 1,...,Ku,

(3.47)

and Ψh,n∗ defined analogously to (3.36). The hybrid-a STDG method is obtained by applying an opera-
tor R to map the collided component to the uncollided quadrature set to obtain the coefficients of the

reconstructed approximation:

Ψh,n∗ = Ψh,n

u + RΨh,n

c

.

(3.48)

58

The approximate solution at the time tn+1 is given by evaluating ψh,n∗
timestep algorithm is outlined in algorithm 3.5.

at tn+1. The hybrid-a STDG

3.2.3 Integral deferred correction methods

A detailed derivation of integral deferred correction (IDC) methods can be found in section 2.2.2. The

notation used in this section deviates from that of section 2.2.2 in two respects. First, the substep index
and correction iteration index are both written as superscripts; i.e., ξn,[0] instead of ξ[0]

n . Second, the

conventions for Latin and Greek letters stated at the beginning of section 2.2.2 are not imposed here.

The remainder of the notation introduced in section 2.2.2 for IDC methods is preserved. In this section

the symbol δn,(cid:96) denotes the Kronecker delta:

δn,(cid:96) =

1,

0,

if n = (cid:96),
if n (cid:54)= (cid:96).

(3.49)

Nonhybrid IDC method

solutions(cid:169)ψn,[0](cid:170)N

Following the discussion in section 2.2.2, the IDC iteration begins by computing a set of provisional
n=1 satisfying an implicit Euler approximation of the discrete ordinates system (3.9),

which can be written as

(cid:104)

(L−SP) ψn,[0] − qn(cid:105)

,

ψn,[0] = ψn−1,[0] − hn∆t

where ψ0 is the initial condition for the timestep, ψ0,[p] = ψ0 for all p ≥ 0, and qn = q(cid:161)tn

(3.50)

(cid:162). Applying

(n = 1,..., N ),

(2.17) to (3.9) yields the correction equation

ψn,[p] = ψn−1,[p] − hn∆t (L−SP) ψn,[p] + hn∆t (L−SP) ψn,[p−1]

(cid:104)
(L−SP) ψ(cid:96),[p−1] − q (cid:96)(cid:105)

,

(3.51)

N(cid:88)

(cid:96)=1

− ∆t

γn,(cid:96)

for n = 1,..., N . The IDC timestep algorithm for the discrete ordinates system (3.9) is outlined in algo-

rithm 3.6.

59

Algorithm 3.6 Nonhybrid IDC timestep.

input: Initial condition ψ0, number of correction iterations P.

1: for p = 0,...,P do
ψ0,[p] ← ψ0.
2:
3: end for
4: for n = 1,..., N do
5:

(cid:181)
Compute ψn,[0] by solving (3.50):
L+ 1
hn ∆t

(cid:182)

−SP

ψn,[0] = 1
hn ∆t

ψn−1,[0] + qn.

(cid:46) Set initial condition.

(cid:46) Prediction steps.

6: end for
7: for p = 1,...,P do
8:
9:

for n = 1,..., N do
(cid:181)
L+ 1
hn ∆t

−SP

(cid:182)

Compute ψn,[p] by solving (3.51):

ψn,[p] = 1
hn ∆t

ψn−1,[p] + N(cid:88)
(cid:96)=1

(cid:46) Correction iterations.

(cid:181)
δn,(cid:96) − γn,(cid:96)
hn

(cid:182)

(L−SP) ψ(cid:96),[p−1] + N(cid:88)

(cid:96)=1

γn,(cid:96)
hn

(cid:96)

.

q

end for

10:
11: end for

return:ψN ,[P]

Hybrid-I IDC methods

Two subtypes of hybrid-I IDC methods are considered in this section: a and b. The construction of the

hybrid-Ia IDC method proceeds as follows. Using the notation of (2.10), an implicit Euler approximation

of (3.17) for the initial set of provisional solutions(cid:169)ψn,[0]

(cid:170)N

u

(cid:170)N
n=1 and(cid:169)ψn,[0]
(cid:104)Luψn,[0] − qn(cid:105)
(cid:104)
,
(Lc −ScPc) ψn,[0]

c

c

−ScPuψn,[0]

u

n=1 takes the form

(cid:105)

,

(3.52a)

(3.52b)

ψn,[0]

u

ψn,[0]

c

= ψn−1,[0]
= ψn−1,[0]

u

c

− hn∆t
− hn∆t

for n = 1,..., N . It is assumed that the solution at the initial time for the step is entirely uncollided and
= 0 for all p ≥ 0, where ψ0∗ is the initial
that the collided flux is initially zero; i.e., ψ0,[p]

= ψ0∗ and ψ0,[p]

u

c

condition for the timestep.

60

Applying (2.17) to (3.17) yields the correction equation

ψn,[p]

u

ψn,[p]

c

u

= ψn−1,[p]
= ψn−1,[p]

c

u

u

− ∆t

− hn∆tLuψn,[p]
(cid:104)(cid:161)Lc −ScPc
− hn∆t
(cid:104)(cid:161)Lc −ScPc
+ hn∆t
N(cid:88)
− ∆t

N(cid:88)
+ hn∆tLuψn,[p−1]
(cid:105)
(cid:162)ψn,[p]
−ScPuψn,[p]
(cid:162)ψn,[p−1]
−ScPuψn,[p−1]
(cid:104)(cid:161)Lc −ScPc
(cid:162)ψ
(cid:96),[p−1]
−ScPuψ
c

(cid:96)=1

γn,(cid:96)

c

c

u

u

(cid:105)

(cid:105)

.

(cid:96),[p−1]
u

(cid:96)=1

(cid:104)Luψ

(cid:96),[p−1]
u

− q (cid:96)(cid:105)

,

γn,(cid:96)

(3.53a)

(3.53b)

The provisional solutions for the prediction level and correction levels are computed using an alternating

cadence between the uncollided and collided components; i.e., in the order

ψ1,[0]

u

, ψ1,[0]

c

, ψ2,[0]

u

, ψ2,[0]

c

,

..., ψ1,[1]

u

, ψ1,[1]

c

, ψ2,[1]

u

, ψ2,[1]

c

,

....

(3.54)

The final approximation for the timestep is obtained by applying a reconstruction operator to map

the approximation of the collided component from the last substep of the final correction iteration to

the uncollided quadrature set:

ψN ,[P]

∗

= ψN ,[P]

u

+RψN ,[P]

c

.

(3.55)

Here P denotes the number of correction iterations performed using (3.53). The hybrid-Ia IDC timestep

algorithm is outlined in algorithm 3.7.

The hybrid-Ib IDC method is a modification of the hybrid-Ia IDC method in which the reconstruction
operator R is applied after each substep of the IDC prediction and correction iterations. This yields
reconstructed approximations ψn,[p]

defined by

∗

ψn,[p]

∗

= ψn,[p]

u

+Rψn,[p]

c

,

(3.56)

for each n = 1,..., N and p = 0,...,P. Equations (3.52) and (3.53) are modified such that the initial con-
ditions of the uncollided and collided fluxes are ψn−1,[p]

and zero, respectively, for each substep. More

∗

specifically, (3.52) becomes

ψn,[0]

u

ψn,[0]

c

(cid:104)Luψn,[0] − qn(cid:105)

,

∗

= ψn−1,[0]
− hn∆t
(cid:104)
= −hn∆t
(Lc −ScPc) ψn,[0]

c

−ScPuψn,[0]

u

(cid:105)

,

(3.57a)

(3.57b)

61

Algorithm 3.7 Hybrid-Ia IDC timestep.

input: Initial condition ψ0∗, number of correction iterations P.

u

1: for p = 0,...,P do
← ψ0∗.
2:
← 0.

ψ0,[p]
ψ0,[p]

3:
c
4: end for
5: for n = 1,..., N do
Compute ψn,[0]
6:
u

7:

Compute ψn,[0]

c

(cid:46) Set initial condition.

(cid:46) Prediction steps.

by solving (3.52a):

(cid:181)
Lu + 1
hn ∆t

(cid:182)

ψn,[0]

u

= 1
hn ∆t

ψn−1,[0]

u

+ qn.

by solving (3.52b):

(cid:181)
Lc + 1
hn ∆t

−ScPc

(cid:182)

ψn,[0]

c

= 1
hn ∆t

ψn−1,[0]

u

+ScPuψn,[0]

u

.

8: end for
9: for p = 1,...,P do
10:

11:

12:

Compute ψn,[p]

for n = 1,..., N do
(cid:181)
u
Lu + 1
hn ∆t

(cid:182)

(cid:46) Correction iterations.

by solving (3.53a):

ψn,[p]

u

= 1
hn ∆t

ψn−1,[p]

u

+ N(cid:88)
(cid:96)=1

(cid:181)
δn,(cid:96) − γn,(cid:96)
hn

(cid:182)

Luψ

(cid:96),[p−1]
u

+ N(cid:88)
(cid:96)=1

γn,(cid:96)
hn

(cid:96)

.

q

Compute ψn,[p]

c

(cid:181)
Lc + 1
hn ∆t

(cid:182)

−ScPc

ψn,[p]

c

by solving (3.53b):

(cid:181)
(cid:182)(cid:183)
δn,(cid:96) − γn,(cid:96)
hn
ψn−1,[p]

= N(cid:88)
(cid:96)=1
+ 1
hn ∆t

c

(Lc −ScPc) ψ
+ScPuψn,[p]

.

u

(cid:96),[p−1]
c

−ScPuψ

(cid:96),[p−1]
u

(cid:184)

end for

13:
14: end for

15: ψN ,[P]

∗

← ψN ,[P]

u
return: ψN ,[P]

∗

+RψN ,[P]

c

.

62

Algorithm 3.8 Hybrid-Ib IDC timestep.

input: Initial condition ψ0∗, number of correction iterations P.

1: for p = 0,...,P do
ψ0,[p]∗ ← ψ0∗.
2:
3: end for
4: for n = 1,..., N do
Compute ψn,[0]
5:
u

6:

Compute ψn,[0]

c

(cid:46) Set initial condition.

(cid:46) Prediction steps.

(cid:182)

ψn,[0]

u

= 1
hn ∆t

ψn−1,[0]
∗

+ qn.

(cid:182)

−ScPc

ψn,[0]

c

=ScPuψn,[0]

u

.

by solving (3.57a):

by solving (3.57b):

(cid:181)
Lu + 1
hn ∆t
(cid:181)
Lc + 1
hn ∆t

+Rψn,[0]

c

.

ψn,[0]∗ ← ψn,[0]

u

7:
8: end for
9: for p = 1,...,P do
10:

11:

12:

Compute ψn,[p]

for n = 1,..., N do
(cid:181)
u
Lu + 1
hn ∆t

(cid:182)

(cid:46) Correction iterations.

by solving (3.58a):

ψn,[p]

u

= 1
hn ∆t

ψn−1,[p]
∗

+ N(cid:88)
(cid:96)=1

(cid:181)
δn,(cid:96) − γn,(cid:96)
hn

(cid:182)

Luψ

(cid:96),[p−1]
u

+ N(cid:88)
(cid:96)=1

γn,(cid:96)
hn

(cid:96)

.

q

Compute ψn,[p]

c

(cid:181)
Lc + 1
hn ∆t

(cid:182)

by solving (3.58b):

−ScPc

ψn,[p]

c

(cid:181)
= N(cid:88)
δn,(cid:96) − γn,(cid:96)
(cid:96)=1
hn
+ScPuψn,[p]

.

u

(cid:182)(cid:183)

(Lc −ScPc) ψ

(cid:96),[p−1]
c

−ScPuψ

(cid:96),[p−1]
u

(cid:184)

∗ ← ψn,[p]
ψn,[p]

u

13:
14:
15: end for

end for

return: ψN ,[P]

∗

+Rψn,[p]

c

.

63

and (3.53) becomes
= ψn−1,[p]

ψn,[p]

u

ψn,[p]

c

u

u

∗

(cid:104)(cid:161)Lc −ScPc
= −hn∆t
(cid:104)(cid:161)Lc −ScPc
+ hn∆t
N(cid:88)
− ∆t

+ hn∆tLuψn,[p−1]
− hn∆tLuψn,[p]
(cid:105)
(cid:162)ψn,[p]
−ScPuψn,[p]
(cid:162)ψn,[p−1]
(cid:104)(cid:161)Lc −ScPc
(cid:162)ψ

−ScPuψn,[p−1]
(cid:96),[p−1]
−ScPuψ
c

γn,(cid:96)

u

u

c

c

− ∆t
(cid:105)

(cid:96),[p−1]
u

(cid:105)

,

(cid:96)=1

(cid:104)Luψ

(cid:96),[p−1]
u

− q (cid:96)(cid:105)

,

γn,(cid:96)

N(cid:88)

(cid:96)=1

(3.58a)

(3.58b)

for n = 1,..., N . In this case, the provisional solutions are computed using a three-step cadence between

the uncollided, collided, and reconstructed components; i.e., in the order

ψ1,[0]

, ψ1,[0]

, ψ1,[0]∗

u

c
..., ψ1,[1]

u

, ψ1,[1]

c

, ψ2,[0]

, ψ2,[0]

u
c
, ψ1,[1]∗
, ψ2,[1]

u

, ψ2,[0]∗

,

...,

, ψ2,[1]

c

, ψ2,[1]∗

,

....

(3.59)

The hybrid-Ib IDC timestep algorithm is outlined in algorithm 3.8.

3.2.4 Low-storage integral deferred correction methods

A detailed derivation of low-storage IDC (LS-IDC) methods can be found in section 2.2.3. The methods

in this section differ from those in the preceding section (section 3.2.3) only in the way that the residuals

are evaluated during the correction iterations. Because of the similarity of the IDC and LS-IDC methods,

we give only very brief descriptions of the LS-IDC methods.

Nonhybrid LS-IDC method

An initial set of provisional solutions(cid:169)ψn,[0](cid:170)N

The prediction step of the nonhybrid LS-IDC method is the same as that of the nonhybrid IDC method:
n=1 is computed using the implicit Euler approximation
(3.50) of the discrete ordinates system (3.9). The LS-IDC correction equation is obtained through a sim-

ple modification of the residual term of (3.51):

ψn,[p] = ψn−1,[p] − hn∆t (L−SP) ψn,[p] + hn∆t (L−SP) ψn,[p−1]

n−1(cid:88)

(cid:96)=1

− ∆t

γn,(cid:96)

(cid:104)
(L−SP) ψ(cid:96),[p] − q (cid:96)(cid:105)− ∆t

N(cid:88)

(cid:96)=n

γn,(cid:96)

(cid:104)
(L−SP) ψ(cid:96),[p−1] − q (cid:96)(cid:105)

(3.60)

,

for n = 1,..., N . The LS-IDC timestep algorithm for the discrete ordinates system (3.9) is outlined in

algorithm 3.9.

64

Hybrid-I LS-IDC methods

As before, two subtypes of hybrid-I LS-IDC methods are considered in this section: a and b. The predic-

tions(cid:169)ψn,[0]

(cid:170)N
n=1 and(cid:169)ψn,[0]

c

(cid:170)N

tion steps are the same as those of the hybrid-Ia and hybrid-Ib IDC methods: The initial provisional solu-
n=1 are computed using (3.52) and (3.57) for the hybrid-Ia and hybrid-Ib
methods, respectively. The correction equations for the hybrid-Ia LS-IDC method are obtained by mod-

u

ifying the residual terms of (3.53):

ψn,[p]

u

= ψn−1,[p]

u

ψn,[p]

c

= ψn−1,[p]

c

u

u

γn,(cid:96)

(cid:96)=n

(cid:96),[p]
u

− hn∆tLuψn,[p]
− ∆t

+ hn∆tLuψn,[p−1]
− q (cid:96)(cid:105)− ∆t
(cid:104)Luψ
N(cid:88)
n−1(cid:88)
(cid:104)(cid:161)Lc −ScPc
(cid:105)
γn,(cid:96)
(cid:162)ψn,[p]
(cid:96)=1
− hn∆t
−ScPuψn,[p]
(cid:104)(cid:161)Lc −ScPc
(cid:162)ψn,[p−1]
−ScPuψn,[p−1]
+ hn∆t
(cid:104)(cid:161)Lc −ScPc
n−1(cid:88)
(cid:162)ψ
− ∆t
(cid:104)(cid:161)Lc −ScPc
N(cid:88)
(cid:162)ψ

(cid:104)Luψ
(cid:105)
(cid:105)

(cid:96),[p]
c
(cid:96),[p−1]
c

−ScPuψ

−ScPuψ

(cid:96),[p]
u

− ∆t

(cid:96)=1

γn,(cid:96)

γn,(cid:96)

u

u

c

c

(cid:96)=n

− q (cid:96)(cid:105)

,

(cid:96),[p−1]
u

(3.61a)

(3.61b)

(cid:105)

.

(cid:96),[p−1]
u

Similarly, the correction equations for the hybrid-Ib LS-IDC method are obtained by modifying the

residual terms of (3.58):

ψn,[p]

u

ψn,[p]

c

− q (cid:96)(cid:105)

,

(cid:96),[p−1]
u

u

u

γn,(cid:96)

(cid:96)=n

∗
− ∆t

= ψn−1,[p]
(cid:104)Luψ
n−1(cid:88)
(cid:104)(cid:161)Lc −ScPc
(cid:96)=1
= −hn∆t
(cid:104)(cid:161)Lc −ScPc
+ hn∆t
n−1(cid:88)
− ∆t
N(cid:88)

+ hn∆tLuψn,[p−1]
− hn∆tLuψn,[p]
(cid:104)Luψ
− q (cid:96)(cid:105)− ∆t
N(cid:88)
(cid:96),[p]
(cid:105)
γn,(cid:96)
(cid:162)ψn,[p]
u
−ScPuψn,[p]
(cid:105)
(cid:162)ψn,[p−1]
(cid:104)(cid:161)Lc −ScPc
(cid:105)
(cid:162)ψ
(cid:104)(cid:161)Lc −ScPc
(cid:162)ψ

−ScPuψn,[p−1]

(cid:96),[p]
c
(cid:96),[p−1]
c

−ScPuψ

−ScPuψ

(cid:96),[p]
u

− ∆t

(cid:96)=1

γn,(cid:96)

γn,(cid:96)

u

u

c

c

(cid:96)=n

(cid:105)

.

(cid:96),[p−1]
u

(3.62a)

(3.62b)

The provisional solutions for the prediction level and correction levels are computed using the same

alternating cadences as the corresponding IDC methods (i.e., (3.54) and (3.59)), and the reconstructed

solutions are defined similarly as well (i.e., using (3.55) and (3.56)). The hybrid-Ia and hybrid-Ib LS-IDC

timestep algorithms are outlined in algorithms 3.10 and 3.11, respectively.

65

Algorithm 3.9 Nonhybrid LS-IDC timestep.

input: Initial condition ψ0, number of correction iterations P.
for p = 0,...,P do
ψ0,[p] ← ψ0.

end for
for n = 1,..., N do

(cid:181)
Compute ψn,[0] by solving (3.50):
L+ 1
hn ∆t

(cid:182)

−SP

ψn,[0] = 1
hn ∆t

ψn−1,[0] + qn.

(cid:46) Set initial condition.

(cid:46) Prediction steps.

(cid:46) Correction iterations.

for n = 1,..., N do

end for
for p = 1,...,P do
(cid:181)
L+ 1
hn ∆t

Compute ψn,[p] by solving (3.60):

(cid:182)

−SP

ψn,[p] = 1
hn ∆t

ψn−1,[p] − n−1(cid:88)
(cid:182)
(cid:181)
(cid:96)=1
δn,(cid:96) − γn,(cid:96)
hn

+ N(cid:88)
(cid:96)=n

γn,(cid:96)
hn

(L−SP) ψ(cid:96),[p]

(L−SP) ψ(cid:96),[p−1] + N(cid:88)

γn,(cid:96)
hn

(cid:96)

q

(cid:96)=1

end for

end for

return:ψN ,[P]

3.3 Implementation details

This section provides a detailed description of the numerical methods used in our implementations

and the numerical tests in subsequent sections. Solving the kinetic equation (1.2) in full generality re-

quires significant computational resources. In order to reduce the computational demands of the tests

conducted for this work, we consider test cases in the following reduced geometries [135, Section 1-3]:
(i) One spatial dimension (slab geometry): In the case that ∂x ψ = ∂y ψ = 0, (1.2) can be reduced to

(cid:90) 1
−1
where µ ∈ [−1,1] is the z-component of (cid:126)Ω = (ξ, η, µ) ∈ (cid:83)2.

∂t ψ+ µ∂z ψ+ σtψ = σs
2

ψ(cid:161)µ(cid:48)(cid:162) d µ(cid:48) + q

(3.63)

(ii) Two spatial dimensions (x y-geometry): In the case that ∂z ψ = 0 the angular flux is considered as a

function of x y-position only, reducing (1.2) to

∂t ψ+ ξ∂x ψ+ η∂y ψ+ σtψ = σs
4π

66

(cid:90)

(cid:83)2

ψ(cid:161)(cid:126)Ω(cid:48)(cid:162) d(cid:126)Ω(cid:48) + q.

(3.64)

Here the angular space is not reduced as it is for slab geometry. While it is possible to reduce
the angular space to one hemisphere (e.g., µ ≥ 0) due to symmetry in the z-dimension, this re-

duces the number of ordinates by the same constant factor (one half) for all discrete ordinates sets

considered in this work. Hence relative performance comparisons between discrete ordinates ap-

proximations of different resolutions should not depend significantly on whether this reduction is

performed or not. We do not consider this type of reduction so that discussions of the numerical

treatment of the angular variable in this geometry extend naturally to the full problem in three

spatial dimensions.

Throughout this section, Nu and Nc are used to represent positive integers that denote the resolution

of the high- and low-resolution angular quadratures used for the uncollided and collided components,

respectively. The number of ordinates Ku and Kc in each quadrature corresponding to the values of Nu

and Nc will depend on the type of the quadrature that is used. For each of the quadratures considered
here a relationship of the form Ku = f (Nu) can be established where f is a simple algebraic expression.

3.3.1 Specifics of high-order time discretizations

The high-order time discretizations described in section 3.2 are discussed in general forms: each class

requires that certain parameters be specified in order to generate a particular scheme of that type. We

now provide detailed descriptions of the schemes considered for the numerical tests conducted for this

work.

DIRK schemes

We consider four DIRK schemes:

• The L-stable DIRK(2,2) scheme of [10], which we refer to as SDIRK2. The scheme is second order

and stiffly accurate. The Butcher tableau of the scheme is given in figure 3.1, with the parameter

α = 1−

(cid:112)
2
2

.

(3.65)

• The L-stable DIRK(3,3) scheme of [10], which we refer to as SDIRK3. The scheme is third order and

67

Algorithm 3.10 Hybrid-Ia LS-IDC timestep.

input: Initial condition ψ0∗, number of correction iterations P.
for p = 0,...,P do

ψ0,[p]

← ψ0∗; ψ0,[p]

← 0.

c

u
end for
for n = 1,..., N do
Compute ψn,[0]
u

by solving (3.52a):

(cid:181)
Lu + 1
hn ∆t

(cid:182)

ψn,[0]

u

= 1
hn ∆t

ψn−1,[0]

u

+ qn.

(cid:46) Set initial condition.

(cid:46) Prediction steps.

Compute ψn,[0]

c

by solving (3.52b):

(cid:181)
Lc + 1
hn ∆t

−ScPc

(cid:182)

ψn,[0]

c

= 1
hn ∆t

ψn−1,[0]

u

+ScPuψn,[0]

u

.

end for
for p = 1,...,P do

Compute ψn,[p]

for n = 1,..., N do
(cid:182)
(cid:181)
u
Lu + 1
hn ∆t

ψn,[p]

u

by solving (3.61a):

= 1
hn ∆t

+ N(cid:88)
(cid:96)=1

ψn−1,[p]

u

γn,(cid:96)
hn

(cid:96)

.

q

(cid:46) Correction iterations.

− n−1(cid:88)
(cid:96)=1

γn,(cid:96)
hn

Luψ

(cid:96),[p]
u

+ N(cid:88)
(cid:96)=n

(cid:181)
δn,(cid:96) − γn,(cid:96)
hn

(cid:182)

Luψ

(cid:96),[p−1]
u

c

(cid:181)
Compute ψn,[p]
Lc + 1
hn ∆t

−ScPc

(cid:182)

by solving (3.61b):

ψn,[p]

c

= 1
hn ∆t

+ScPuψn,[p]

ψn−1,[p]
(cid:183)
u
(Lc −ScPc) ψ

c
γn,(cid:96)
(cid:181)
hn
δn,(cid:96) − γn,(cid:96)
hn

(cid:182)(cid:183)

(cid:96),[p]
c

(Lc −ScPc) ψ

− n−1(cid:88)
(cid:96)=1
+ N(cid:88)
(cid:96)=n

(cid:184)

−ScPuψ
(cid:96),[p]
u
(cid:96),[p−1]
c

−ScPuψ

(cid:184)

.

(cid:96),[p−1]
u

end for

end for

← ψN ,[P]

ψN ,[P]

∗

u
return: ψN ,[P]

∗

+RψN ,[P]

c

.

68

Algorithm 3.11 Hybrid-Ib LS-IDC timestep.

input: Initial condition ψ0∗, number of correction iterations P.
for p = 0,...,P do
ψ0,[p]∗ ← ψ0∗.

end for
for n = 1,..., N do
Compute ψn,[0]
u

Compute ψn,[0]

c

by solving (3.57a):

by solving (3.57b):

(cid:181)
Lu + 1
hn ∆t
(cid:181)
Lc + 1
hn ∆t

(cid:182)

ψn,[0]

u

= 1
hn ∆t

ψn−1,[0]
∗

+ qn.

(cid:182)

−ScPc

ψn,[0]

c

=ScPuψn,[0]

u

.

(cid:46) Set initial condition.

(cid:46) Prediction steps.

ψn,[0]∗ ← ψn,[0]

u

+Rψn,[0]

c

.

end for
for p = 1,...,P do

Compute ψn,[p]

for n = 1,..., N do
(cid:181)
(cid:182)
u
Lu + 1
hn ∆t

ψn,[p]

u

by solving (3.62a):

= 1
hn ∆t

+ N(cid:88)
(cid:96)=1

ψn−1,[p]
∗

γn,(cid:96)
hn

(cid:96)

.

q

(cid:46) Correction iterations.

− n−1(cid:88)
(cid:96)=1

γn,(cid:96)
hn

Luψ

(cid:96),[p]
u

+ N(cid:88)
(cid:96)=n

(cid:181)
δn,(cid:96) − γn,(cid:96)
hn

(cid:182)

Luψ

(cid:96),[p−1]
u

(cid:183)

(Lc −ScPc) ψ

(cid:182)(cid:183)

γn,(cid:96)
hn
(Lc −ScPc) ψ

(cid:96),[p]
c
(cid:96),[p−1]
c

−ScPuψ

−ScPuψ

(cid:184)

(cid:96),[p]
u
(cid:96),[p−1]
u

(cid:184)

.

by solving (3.62b):

Compute ψn,[p]

c

(cid:181)
Lc + 1
hn ∆t

(cid:182)

−ScPc

ψn,[p]

c

− n−1(cid:88)
(cid:181)
(cid:96)=1
δn,(cid:96) − γn,(cid:96)
hn

=ScPuψn,[p]
+ N(cid:88)
(cid:96)=n

u

∗ ← ψn,[p]
ψn,[p]

u

end for

end for

return: ψN ,[P]

∗

+Rψn,[p]

c

.

69

α
1

α

1− α α
1− α α

α
τ
1

α
τ− α
b1
b1

α
b2 α
b2 α

(a) SDIRK2 with α given by (3.65).

(b) SDIRK3 with parameters given by (3.66).

Figure 3.1: Butcher tableaus of SDIRK2 and SDIRK3 schemes.

stiffly accurate. The Butcher tableau of the scheme is given in figure 3.1, with the parameters

,

6

α = 0.435866521508459,
τ = α+ 1
b1 = − 6α2 − 16α+ 1
b2 = 6α2 − 20α+ 5

4

,

.

4

(3.66a)

(3.66b)

(3.66c)

(3.66d)

• The L-stable, 7-stage DIRK method of [118], which we refer to as Kværnø7. The scheme is fifth

order and stiffly accurate. The Butcher tableau of the scheme is given in figure 3.2.

• The L-stable, 8-stage DIRK method of [110], which we refer to as ARK8. The scheme is fifth order

and stiffly accurate. The Butcher tableau of the scheme is given in figure 3.2.

While many of these schemes are constructed with embedded pairs to provide error estimates for adap-

tive timestepping, we do not make use of these capabilities and hence the embedded pairs are omitted.

IDC and LS-IDC schemes

Each IDC and LS-IDC scheme is generated by specifying the quadrature nodes {cn} which form the

substeps of the method. The substep sizes hn and the quadrature weights for evaluating the residual

γn,k are computed from the positions of the quadrature nodes using (2.11) and (2.18), respectively (see

section 2.2.2 for a thorough description of IDC methods). We consider IDC and LS-IDC methods con-

structed with two quadratures: the two- and three-point Gauss-Radau quadratures. These quadratures

yield IDC and LS-IDC methods of order three and five, respectively, and we refer to these methods as

IDC3 and IDC5, and LS-IDC3 and LS-IDC5, respectively.

70

0.0
0.52
1.230333209967908
0.895765984350076
0.436393609858648
1.0
1.0

0.0
0.26
0.13
0.22371961478320505
0.16648564323248321
0.13855640231268224
0.13659751177640291

0.26
0.84033320996790809
0.47675532319799699
0.10450018841591720
0.0
0.0

0.13659751177640291

0.0

0.26
−0.06470895363112615
0.03631482272098715
−0.04245337201752043
−0.05496908796538376
−0.05496908796538376

0.26
−0.13090704451073998
0.02446657898003141
−0.04118626728321046
−0.04118626728321046

(a) Butcher tableau of the Kværnø7 scheme.

0.26
0.61943039072480676
0.62993304899016403

0.26
0.06962479448202728

0.62993304899016403

0.06962479448202728

0.26

0.26

0.0
41
100

2935347310677
11292855782101
1426016391358
7196633302097

92
100
24
100
3
5
1

0
41
200
41
400

683785636431
9252920307686
3016520224154
10081342136671
218866479029
1489978393911
1020004230633
5715676835656
− 872700587467
9133579230613
− 872700587467
9133579230613

41
200

− 567603406766
11931857230679

0

0

0

0

0

0

41
200

− 110385047103
1367015193373
30586259806659
12414158314087
638256894668
5436446318841
25762820946817
25263940353407

0

0

41
200

− 22760509404356
11113319521817
− 1179710474555
5321154724896
− 2161375909145
9755907335909
22348218063261
9555858737531

22348218063261
9555858737531

(b) ARK8

41
200

− 60928119172
8023461067671
− 211217309593
5846859502534
− 1143369518992
8141816002931
− 1143369518992
8141816002931

41
200

− 4269925059573
7827059040749
− 39379526789629
19018526304540
− 39379526789629
19018526304540

41
200

32727382324388
42900044865799

32727382324388
42900044865799

41
200

41
200

Figure 3.2: Butcher tableaus of Kværnø7 and ARK8 schemes.

71

STDG schemes

Each STDG scheme is generated by specifying the space Vh,n in which the approximate solution lies.

Because the space-time mesh is obtained by extruding a spatial mesh through the timestep interval,

the form of the spatial and temporal components may be specified separately: the resulting space-time

approximation space is composed of the tensor product of the spatial and temporal components.

We consider two STDG schemes, which we refer to as STDG3 and STDG5. The STDG3 scheme uses

a tensor product basis in each mesh element that consists of a polynomial of degree 2 in each spatial

dimension and a polynomial of degree 1 in the time dimension. The STDG5 scheme uses a tensor prod-

uct basis in each mesh element that consists of a polynomial of degree 4 in each spatial dimension and

a polynomial of degree 2 in the time dimension. These methods are formally third- and fifth-order ac-

curate in space and second- and third-order accurate in time, respectively, at all space-time points. Ad-

ditionally, the schemes are third- and fifth-order superconvergent in time, respectively, at the timestep

points (similar superconvergence results can be shown for values along certain boundaries of the spatial

mesh) [122, 124, 125]. Further, the time-integration components of the methods can be shown to be

equivalent to Radau collocation schemes, which are L-stable at all orders of accuracy [64, 125].

The approximation spaces used for the STDG methods are chosen such that the order of accuracy of

the methods aligns with the choices made for DIRK and IDC methods. For each test problem considered

in this work, solution errors are measured at the final simulation time by comparing the scalar flux dis-

tribution of each approximate solution with that of a reference solution using a continuum Lp distance

metric with respect to the spatial variables; e.g., (2.52). The DG spatial discretizations that are used (c.f.

section 3.3.3) are not superconvergent in this norm. Since the timestep sizes are always chosen such that

the final simulation time is an integer multiple of the timestep size, the final simulation time is a point of

superconvergence of the DG time discretizations. Hence using a polynomial basis of lower degree in the

time dimension and higher degree in the spatial dimensions yields STDG methods with the same order

of accuracy in both space and time with respect to the metrics used to evaluate solution errors.

The motivation for this choice is to maintain consistency between the different classes of time inte-

gration schemes, and further, by the relationship between IDC, collocation, and DG methods. The IDC

and STDG methods used are chosen such that the IDC solutions converge to the STDG solutions in the

limit of infinitely many correction iterations. This observation is based on the following facts: (i) IDC

72

methods converge to the collocation solution corresponding to the choice of quadrature nodes that is

used in the limit of infinitely many correction iterations; (ii) the IDC methods considered in this work

are constructed using Gauss-Radau quadratures; and (iii) the DG time integration schemes used here

are equivalent to Radau collocation schemes.

3.3.2 Angular quadratures and relabeling

Slab geometry
In slab geometry the angular variable takes values µ ∈ [−1,1] and the ordinates (and their associated

weights) are obtained from Gauss-Legendre quadrature. As was done in [61, 97], the angular recon-

struction is performed at each spatial point by evaluating the Lagrange interpolant (over the angular

variable) of the collided flux at each of the uncollided ordinates. This method defines a linear map from

the collided quadrature set to the uncollided quadrature set that can be applied using a single matrix

multiplication. This yields a reconstruction procedure requiring 2KuKc operations per spatial point.

x y-geometry
In two-dimensional x y-geometry the angular variable takes values (cid:126)Ω ∈ (cid:83)2.
Gauss-type quadratures over (cid:83)2 do not exist as they do in the one-dimensional case [170]. Because of

It is known that optimal

this, a large number of quadratures have been developed for use in discrete ordinates calculations – c.f.

[2, 6, 7, 32, 33, 71, 116, 132, 167] for some examples. The design of most angular quadratures is motivated

by the desire to optimize one or more particular characteristics of the quadrature. Many quadratures

emphasize the exact integration of as many spherical harmonics as possible [6, 33, 131, 132, 133]. This

restriction is often accompanied by a requirement on the symmetry of the quadrature: Most quadratures

are chosen such that they are symmetric with respect to octahedral reflections and rotations aligned with

the coordinate axes [33, 131, 132, 133], though quadratures using other forms of symmetry such as those

corresponding to icosahedral rotations have been described [6]. Other quadratures are designed around

geometric considerations that permit arbitrary levels of refinement, local refinement, and a distribution

of nodes over the sphere that is reasonably close to uniform [107, 129, 178, 179]

One of the focal points of this work is the use of methods that permit an arbitrary level of accuracy,

and in particular, methods of arbitrarily high order. As such, the use of angular quadratures that per-

73

mit an arbitrary level of refinement is desirable. Unfortunately, many quadratures designed for discrete

ordinates calculations have limitations that either make them ill-suited to the construction of arbitrar-

ily accurate solvers or to the construction of high-order reconstruction procedures: For example, some

quadratures require negative weights as the number of nodes is increased [128]. These limitations are

a result of the fact that certain features of the quadrature must be sacrificed in order to optimize other

features. We do not seek to conduct an exhaustive (or even particularly thorough) comparison of angular

quadratures in this work. However, we consider two kinds of angular quadratures for multidimensional

geometries for the sake of variety: CL tensor-product quadratures and the TN quadratures of [178, 179].

Both quadrature families permit an arbitrary level of refinement in the quadrature, but the choices that

are made in the design of these quadratures emphasize different characteristics in each case.

Chebyshev-Legendre (CL) quadratures

The Chebyshev-Legendre (CL) quadratures are constructed using a tensor product approach in which an

N -point Gauss-Legendre quadrature is applied to the cosine of the polar angle of the sphere (i.e., the z

Cartesian variable) and a 2N -point composite midpoint rule is applied to the azimuthal angle at each

latitude of the sphere corresponding to one of the Gauss-Legendre nodes. The CL quadrature of order N

thus contains a total of 2N 2 quadrature nodes. A detailed definition of the quadrature is given in defini-

tion 1. The advantages of CL quadratures are that: (i) they are relatively easy to construct; (ii) they possess

strictly positive quadrature weights for all orders of accuracy; and (iii) the number of spherical harmon-

ics that are integrated exactly by the quadrature increases proportionally to the order of the quadrature:

More specifically, the CL quadrature rule of order N on (cid:83)2 integrates all spherical polynomials of degree

less than 2N exactly [16, Theorem 5.4].

Definition 1 (Chebyshev-Legendre (CL) quadrature of order N ). Let {µ1,..., µN } and {w1,..., wN } denote
the N th set of Gauss-Legendre nodes and weights, respectively, over the interval [−1,1], and let

Then the Chebyshev-Legendre quadrature rule of order N on (cid:83)2 is given by

: k = 1,...,2N and (cid:96) = 1,..., N

(cid:111)

(k = 1,...,2N ).

,

2N

φk = (2k − 1)π
(cid:110)(cid:179)(cid:126)Ωk,(cid:96), ωk,(cid:96)

(cid:180)

(3.67)

(3.68)

74

where

(cid:126)Ωk,(cid:96) =(cid:179)

cos φk

(cid:113)
1− µ2

(cid:96), sin φk

(cid:113)
1− µ2

(cid:96), µ(cid:96)

(cid:180)

and

ωk,(cid:96) = π

N

w(cid:96).

(3.69)

In contrast to the slab-geometry case, a single, straightforward interpolation approach cannot be

used to define reconstruction procedures for hybrid methods in higher-dimensional settings. One of

the difficulties in this regard is selecting the interpolation space: the choice of interpolation space on

(cid:83)2 depends strongly on the quadrature that is used, and in most cases an appropriate choice is not

obvious. This is further complicated by the fact that solutions to interpolation problems on higher-

dimensional manifolds such as (cid:83)2 do not always exist [170], and the interpolants that do exist are often

too ill-conditioned to be of practical use. For example, CL quadratures are constructed using a tensor

product of one-dimensional Gauss-Legendre and composite midpoint quadratures. This construction

can be used to define interpolants for CL nodes that is based on a product of one-dimensional polyno-

mial and trigonometric interpolations. While the conditioning of both one-dimensional interpolations

is satisfactory in most cases, we have found that the two-dimensional product is often inaccurate due to

the commonly encountered “pole problem” [29] on the sphere.

One alternative is a technique called hyperinterpolation wherein a continuous reconstruction is per-

formed using half as many spherical harmonics as nodes in the CL quadrature. The spherical harmonics

are eigenfunctions of the Laplace-Beltrami operator on (cid:83)2, and form an orthonormal basis of spherical

polynomials in L2((cid:83)2). Following [16], the (real) spherical harmonics can be written as

Yn,1(µ, φ) = cnPn(µ)
Yn,2m(µ, φ) = cn,mPm
Yn,2m+1(µ, φ) = cn,mPm

n (µ)cos(mφ)

n (µ)sin(mφ)

n ≥ 0
n ≥ 1; m = 1,...,n
n ≥ 1; m = 1,...,n

(3.70a)

(3.70b)

(3.70c)

where Pm

n is the associated Legendre function and the normalization constants cn and cn,m are given

by

cn =

(cid:114) 2n + 1

4π

(cid:115)

,

cn,m =

2n + 1
2π

(n − m)!
(n + m)!

.

(3.71)

A reconstruction procedure for hybrid methods using CL quadratures can be defined as follows:
Given a collided flux solution defined on a set of CL nodes of order Nc, compute a discrete L2 projec-

tion onto the basis of spherical harmonics of degree less than Nc using the quadrature rule (this is the

75

hyperinterpolation step). The values of the reconstructed collided flux solution are obtained by evaluat-

ing the expansion of spherical harmonics generated by the hyperinterpolation procedure at each node

of the higher-resolution quadrature. Since the CL quadrature of order N contains 2N 2 nodes and there

are only N 2 spherical harmonics of degree less than N , this hyperinterpolation method does not in gen-

eral interpolate the nodal values. However, hyperinterpolation on CL nodes is stable in the uniform

norm provided that the solution is sufficiently smooth [171]. Additionally, it is straightforward to show

that this reconstruction procedure preserves the angular moments of the collided flux that are used as

part of the hyperinterpolation step; that is, the angular moments of degree less than Nc in the recon-

structed collided flux distribution are the same as those of the original collided flux distribution on the

low-resolution quadrature.

As in the one-dimensional case, this reconstruction procedure for CL quadratures based on hyper-

interpolation defines a linear map from the collided quadrature set to the uncollided quadrature set.

This map can be decomposed into two components: one that maps from the nodal values on the col-

lided quadrature set to the coefficients of the spherical harmonics expansion, and one that maps from

the coefficients of the spherical harmonics expansion to the nodal values on the uncollided quadrature

set. Because the dimension of the vector containing the coefficients of the spherical harmonics expan-

sion is less than the dimensions of the vectors containing the nodal values on either quadrature set, this

reconstruction is most efficiently applied using a sequence of two matrix multiplications, one for each

component. This results in a reconstruction procedure requiring 4N 2

c + 4K 4

uN 2

c operations1.

The tensor-product construction of CL quadratures makes them easy to construct and their prop-

erties straightforward to describe, but also limits their efficiency. Since each latitudinal component of

these quadratures contains the same number of nodes around the circumference of the sphere, these

quadratures produce a significant degree of clustering near the poles of the sphere. This clustering ren-

ders them rather inefficient relative to other quadratures in streaming-dominated regimes, where the

degree of uniformity in the distribution of quadrature nodes is one of the primary determining factors

of the accuracy of the discrete ordinates method. This characteristic motivates our second choice of

quadratures: the TN quadratures [178, 179].

(cid:179)

(cid:180)

1If the reconstruction procedure is applied as a single matrix multiplication, then 8N 2

uN 2

c operations

are required: 4N 2
c

u − N 2
N 2

c

operations more than the two-matrix approach.

76

TN quadratures

The TN quadratures are based on tessellations applied to the faces of an octahedron inscribed within the

sphere that are projected onto the sphere’s surface. The result is essentially a generalized midpoint rule

in which the quadrature nodes consist of the projections of the centroids of the triangles onto the sphere

and the quadrature weights are computed as the surface areas of the projected triangles on the sphere.

Each TN quadrature can be defined by considering a single octant and then reflecting the quadrature

nodes from that octant across each of the coordinate planes to the other octants. Here, as in [178], we

consider only the first octant. The tessellation is performed in the context of a reference triangle called

the basal triangle. This triangle is embedded within a two-dimensional plane called the basal plane in

which coordinates are denoted using an ordered pair (u, v): The vertices of the basal triangle lie at the
coordinates (−1/2,0), (1/2,0), and (0,1).

A uniform tessellation is applied to the basal triangle according to the order N of the quadrature. This
tessellation consists of N rows of triangles indexed n = 0,..., N − 1, where the nth row contains 2n + 1
triangles that are indexed by k = 0,...,2n. The coordinates of the vertices of the kth triangle in the nth
row are given by (cid:181) k − n
(cid:181) k − n

(cid:181) k − n + 1
(cid:181) k − n + 1

(cid:181) k − n − 1

(cid:181) k − n − 1

N − n
N

N − n − 1

N − n − 1

for k even and

(cid:182)

,

(3.72a)

,

2N

,

(cid:182)

(cid:182)

(cid:182)

,

2N

,

2N

N

(cid:182)

,

N

N − n − 1
(cid:182)

N − n
N

,

N − n
N

,

,

2N

2N

(3.72b)

,

2N

N

, v

∗

∗

for k odd. The coordinates (u

) of the centroid of each triangle in the tessellation of the basal triangle

are given by

(cid:161)u1 + u2 + u3

(cid:162),

∗ = 1
3

u

(cid:161)v1 + v2 + v3

(cid:162),

∗ = 1
3

v

(3.73)

where {(ui , vi )}3

i=1 are the coordinates of the vertices of the triangle.

The projection from the basal triangle to the surface of the sphere follows a two-step process. The
first step maps each ordered pair (u, v) ∈ (cid:82)2 in the basal plane to an ordered triple (x, y, z) ∈ (cid:82)3 for the cor-
responding point on the face of the octahedron centered at the origin with vertices at (±1,0,0), (0,±1,0),
and (0,0,±1). This map is defined explicitly by

x = 1
2

− u − 1
2 v,

y = 1
2

+ u − 1
2 v,

z = v.

(3.74)

77

The second step projects each point (x, y, z) ∈ (cid:82)3 on the octahedron to a point (ξ, η, µ) ∈ (cid:83)2. This projec-

tion is defined explicitly by

ξ =

x(cid:113)
x2 + y2 + z2

,

η =

y(cid:113)
x2 + y2 + z2

,

µ =

z(cid:113)
x2 + y2 + z2

.

(3.75)

The quadrature weights are computed from the surface area on the sphere of the projection of each

triangle in the tessellation, which can be written explicitly using results from elementary trigonometry.
Let (cid:126)α, (cid:126)β,(cid:126)γ ∈ (cid:83)2 denote the vectors containing the coordinates of the vertices of a projected triangle. The

arc lengths of each edge of the projected triangle are given by

−1(cid:161)(cid:126)β·(cid:126)γ(cid:162),

a = cos

−1(cid:161)(cid:126)α·(cid:126)γ(cid:162),

b = cos

−1(cid:161)(cid:126)α· (cid:126)β(cid:162),

c = cos

and the semi-perimeter of the projected triangle is given by

s = 1
2

(a + b + c).

The surface area of the triangle is given by

(cid:181)

(cid:182)

,

tanr
sin(s − a)

A = 2tan

−1

where

and

A+ B +C − π
(cid:181)

−1

tanr
sin(s − b)

(cid:182)

,

C = 2tan

−1

(cid:181)

(cid:182)

,

tanr
sin(s − c)

B = 2tan
(cid:115)

tanr =

sin(s − a)sin(s − b)sin(s − c)

.

sin s

(3.76)

(3.77)

(3.78)

(3.79)

(3.80)

A reconstruction procedure for hybrid methods using TN quadratures can be defined based on a

piecewise-constant interpolation over the tessellation used to construct the low-resolution quadrature.

Evaluating this interpolant at each of the high-resolution quadrature nodes yields the reconstructed val-

ues of the collided flux. One advantage of this reconstruction procedure is that it is very inexpensive

to apply: a simple copy operation is all that is required once the mapping between nodes has been de-

termined. When the order Nu of the high-resolution quadrature is an integer multiple of the order Nc

of the low-resolution quadrature, the tessellation on which the high-resolution quadrature is based is

a refinement of the tessellation on which the low-resolution quadrature is based. In this case, deter-

mining the mapping between nodes is reasonably straightforward, and, due to the way the quadrature

78

weights are computed, the resulting reconstruction procedure is guaranteed to preserve the scalar flux of

the solution. In this work we consider only hybrid methods using TN quadratures that differ by integer

multiples.

At this point, we would like to emphasize that the properties of the reconstruction procedures de-

fined for the CL and TN quadratures are quite different. Whereas the reconstruction for CL quadratures

preserves as many angular moments as are used in the hyperinterpolation step, the reconstruction for

TN quadratures preserves only the zeroth angular moment of the solution exactly. In particular, the TN

reconstruction does not preserve the solution’s current (first angular moments), which may impact the

accuracy of solutions near material interfaces [74]. Because of the nonlinearities found in thermal ra-

diative transfer problems, it is often necessary to enforce positivity in the scalar flux distribution. While

the TN reconstruction will preserve positivity in reconstructed solutions, the CL reconstruction can in-

troduce negative values due to oscillations in the spherical harmonics expansion, even if the original

solution is non-negative at each low-resolution quadrature node.

3.3.3 Spatial discretization

Each high-order time discretization is paired with an arbitrarily high-order discontinuous Galerkin spa-

tial discretization, which is constructed in essentially the same manner as the STDG methods described

in section 3.2.2, but without the components in the time dimension. The discontinuous Galerkin (DG)

method was first proposed by Reed and Hill for approximation of spatial variables in steady-state neutron

transport problems [166]. Formal convergence results were later established by Lesaint and Raviart [125]

and Johnson and Pitkäranta [108]. The method has subsequently been generalized to time-dependent

first-order systems of hyperbolic conservation laws by Cockburn and Shu, et al. [53, 55, 57, 58, 59]. Ex-

tensions to second-order elliptic and parabolic equations [12, 13, 22, 30, 36, 54, 56, 68] and other higher-

order equations [42] have also been investigated.

One of the primary advantages of the DG method is the lack of strict continuity enforcement across

boundaries of the finite element mesh. This permits the construction of finite element bases in which

the support of each basis element is contained in a single cell. Combining such localized bases with

“upwind” numerical fluxes (which preserve the characteristic dependence of the continuous operator
L) allows efficient sweep algorithms of arbitrarily high order to be constructed.

79

For all of the Runge-Kutta integrators (i.e., the Euler method, the DIRK methods, and the IDC and LS-

IDC methods) solving the discrete time-dependent system for each stage value is equivalent to solving

a steady-state system with a specific form. Hence we consider a steady-state version of the discrete

ordinates system (3.5), which an be written in the form

(L−SP) ψ = q.

(3.81)

Assume the spatial domain X is a convex polyhedron and let Th be a partition of X into nonover-
lapping polyhedral cells K of diameter less than h. For each ordinate (cid:126)Ωk , the boundary of the spatial
domain can be decomposed into two components ∂X = ∂X

=(cid:169)(cid:126)x ∈ ∂X : (cid:126)n((cid:126)x)· (cid:126)Ωk < 0(cid:170)

−
k

∂X

−
k

∪ ∂X
+
k

+
k where

=(cid:169)(cid:126)x ∈ ∂X : (cid:126)n((cid:126)x)· (cid:126)Ωk ≥ 0(cid:170)

and

∂X

(3.82)

are the inflow and outflow components of the boundary for the kth ordinate, respectively. Let Eh denote
the set of interior edges of the partition Th with
unit vector (cid:126)ne normal to the edge whose orientation may be chosen arbitrarily. On each edge e ∈ ∂X
the associated unit normal vector (cid:126)ne is specified to be the outward normal.

(cid:180)∩ ∂X = (cid:59), and to each edge e ∈Eh associate a

(cid:179)(cid:83)

e∈Eh

±
k

e

For each ordinate, the DG approximation of the solution ψk of the steady-state discrete ordinates

(cid:179)

ψh

∈ Vh satisfying
(cid:179)

k , vh(cid:180) = ¯Bh(cid:179)
(cid:90)

system (3.81) is given by ψh
k
Bh
k

vh(cid:180)
φh, vh(cid:180)+Ah
for every vh ∈ Vh with the bilinear and linear forms
(cid:161)(cid:126)Ωk ·∇(cid:126)x v + σtv(cid:162)u d(cid:126)x − (cid:88)
(cid:162) v
(cid:161)(cid:126)Ωk ·(cid:126)ne

Bh

d s,

−

−

k

,

e∈Eh

e

(k = 1,...,K )
(cid:90)

(cid:161)(cid:126)Ωk ·(cid:126)ne

(cid:162)[v](cid:98)u d s

(3.83)

(3.84a)

(3.84b)

(3.84c)

(3.85)

u

e

K

+
k

e∈∂X

k (u, v) = − (cid:88)
(cid:90)
+ (cid:88)
K∈Th
(cid:90)
(cid:88)
(cid:90)
K qk v d(cid:126)x − (cid:88)
K∈Th
(cid:105)T
ψh =(cid:104)

k (v) = (cid:88)

K σsvu d(cid:126)x,

K∈Th

e∈∂X

Ah

¯Bh(u, v) = 1
4π

−
k

ψh
1 ,..., ψh
K

where

(cid:90)

e

,

(cid:161)(cid:126)Ωk ·(cid:126)ne

(cid:162) ψ

−

d s,

B,k v

φh =Pψh,

80

(cid:161)(cid:126)x,(cid:126)Ωk

B,k = ψB

ψ

(cid:162) is the value of the inflow condition specified on ∂X
−
k for the ordinate (cid:126)Ωk ,
+ − v

((cid:126)x) = lim

[v] = v

±

−

v

,

ε→0+ v((cid:126)x ± ε(cid:126)ne ),

and(cid:98)v is a numerical flux specified in terms of v
tion flow imposed by the continuous operator L, we use the upwind flux
ε→0+ v(cid:161)(cid:126)x − ε(cid:126)Ωk
(cid:98)v = lim

(cid:162).

and v

−

+

. In order to maintain the direction of informa-

In practice (3.83) is written as a linear system relative to some basis of Vh. If Bh is a vector whose
k , then (3.83) can

k be the vector of coefficients such that ψh
k

= BT
h

Ψh

elements form a basis for Vh and ψh
be written as

(3.86)

(3.87)

(3.88)

(3.89)

(3.90)

(3.91)

(3.92)

where

with Qh
k

=Ah
k

(cid:179)

(cid:180)T

LhΨh = ShPhΨh + Qh,
Ψh(cid:180)T =(cid:183)(cid:179)
(cid:180)T(cid:184)
(cid:179)
(cid:162): the operator Lh has the form
(cid:161)Bh
(cid:180)

Ψh
K

Ψh
1

,...,

and

(cid:179)

(cid:179)

Qh(cid:180)T =(cid:183)(cid:179)

Lh = Diag

Lh
1 ,...,Lh
K

with

=Bh
k

Lh
k

for k = 1,...,K ; and the operators Sh and Ph are defined by

Sh = 1K ⊗ ¯Sh

and PhΨh = K(cid:88)
¯Sh = ¯Bh(cid:179)

(cid:180)

BT
h ,Bh

k=1

where

(cid:180)T

(cid:180)T(cid:184)

(cid:179)

Qh
K

,...,

Qh
1

(cid:180)

(cid:179)

BT
h ,Bh

ωk Ψh
k ,

and 1K is the K -dimensional column vector of ones.

For most of the numerical tests considered in this work, each time integrator is paired with a DG spa-

tial discretization of the same order of accuracy with respect to space as the order of the time integrator:

All exceptions are explicitly noted. We consider only uniform Cartesian meshes where each spatial cell

is assigned a tensor product basis of polynomials of degree at most one less than the order of the time

integration scheme.

81

3.4 Numerical tests

Two sets of test problems are considered to assess the accuracy and efficiency of the hybrid methods.

The first set consists of a sequence of convergence tests for smooth problems in one-dimensional slab

geometry under the diffusive scaling (1.4). The aim of the second set of tests is to compare the com-

putational efficiency (i.e., error versus solve time) of hybrid and standard discrete ordinates methods in

two-dimensional x y-geometry. Throughout this section SN will be used to denote a standard discrete

ordinates approximation using an angular quadrature of order N with K total ordinates, and SNu,Nc will
be used to denote a hybrid discrete ordinates approximation using angular quadratures of order Nu and

Nc with Ku and Kc total ordinates in the uncollided and collided components, respectively.

3.4.1 One spatial dimension

We first examine the convergence behavior of the hybrid discrete ordinates methods in one-dimensional
slab geometry using Gauss-Legendre quadratures. With σs = σt, q = 0, and under the asymptotic scaling
(1.4), the slab-geometry form of the transport equation (1.2) can be written as

ε∂t ψ+ µ∂z ψ+ σt

ε

ψ = σt
2ε

〈ψ〉.

(3.93)

A sequence of convergence tests are performed for decreasing ε on a spatial domain of z ∈ [−4,4] with
vacuum boundary conditions (zero inflow) and σt ≡ 1 for all z. Convergence tests are conducted for
ε = 1E+0, 1E−1, 1E−2, and 1E−3. The simulations are run to a final time of t = 0.5 and refinement of the
space-time mesh is performed by successively doubling the number of spatial cells from 27 to 217 under
a constant CFL of 8. The initial condition is set from the isotropic distribution ψ(z, µ,0) = ηr (z)/2, where
ηr is a compactly supported C

function defined by

∞

(cid:33)−1(cid:35)

− 1

(cid:34)(cid:195)|z|2

r 2

C exp

0,

ηr (z) =

|z|2 < r 2

,

otherwise.

(3.94)

Here r is the radius of the support of ηr and C > 0 is a constant such that ||ηr ||
L1 = 1. Each convergence
test is performed with a radius of r = 1. The error in the scalar flux of each solution is approximated by
comparing to a reference solution using 218 spatial cells and an S32 standard discrete ordinates angular

approximation.

82

The results of the convergence tests for DIRK integrators are shown in figures 3.3 and 3.4, for STDG

integrators in figure 3.5, and for IDC5 and LS-IDC5 integrators in figure 3.6. It is clear from these tests

that the use of very coarse angular approximations in the collided component of the hybrid methods can

significantly degrade the overall accuracy of the methods. We observe two distinct phenomena that con-

tribute to the error profiles of the hybrid methods: convergence saturation and order reduction. These

effects appear to be governed by a combination of (i) the resolution of the angular approximation that is

used in the collided component, (ii) the timestep size of the integrator, and (iii) the particle mean-free-

time.

When the particle mean-free-time is smaller than the timestep size, the convergence of the hybrid

methods saturates at the error level of the low-resolution quadrature that is used in the collided flux.

This is a result of the fact that the collided component dominates in highly collisional regimes where the

vast majority of particles are collisional within each timestep. As the mesh is refined the timestep size

eventually becomes small enough to resolve the particle mean-free-time and the solution begins to shift

from the collided component into the uncollided component. This shift causes the hybrid methods to re-

sume converging to the reference solution, but at a rate corresponding to the order of the low-resolution

angular quadrature. Order reduction is therefore observed when this rate is less than the convergence

rate of the time-integration scheme. This order reduction is unexpected, and a rigorous study of this

phenomenon is currently underway. Preliminary results suggest that this phenomenon is a result of a

truncation error with respect to the length of time between applications of the reconstruction procedure

for mapping the collided flux into the uncollided flux, which in this case corresponds to the timestep

size.

We note that there are two reasons for convergence saturation of the hybrid methods in these tests:

(i) the conditioning of the linear systems that must be solved at each stage of the time integrators, and
(ii) the accuracy of the angular approximation used in the collided component. As ε → 0 the linear sys-

tems that must be solved at each stage become increasingly ill-conditioned, and hence the minimum

error that one can expect to realize when solving such systems increases proportionally. Roughly speak-

ing, the loss of accuracy in these tests due to the conditioning of the linear systems that must be solved

is generally within one order of magnitude of 1/ε. This yields minimum effective tolerances of approxi-
mately 1E−13, 1E−12, 1E−11, and 1E−10, depending on the time integrator, for the tests considered here
in order of decreasing ε. We emphasize that any behavior observed in these convergence tests below

83

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

−2

10

−4

10

−6

10

−8

10

−10

10

−12

2
L

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

−14

10

−2

10

−4

10

−6

10

−8

10

−10

10

−14

10

−12

2
L

10

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 IIa
S32,4 IIa
S32,2 IIa

Order 2

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

(a) SDIRK2

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 IIa
S32,4 IIa
S32,2 IIa

Order 2
Order 3

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) SDIRK3

Figure 3.3: L2 convergence studies of hybrid-Ia and hybrid-IIa methods based on SDIRK2 and SDIRK3
integrators for a sequence of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial do-
main z ∈ [−4,4], final time 0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar
flux of each solution is approximated by comparing to a reference solution using 218 spatial cells and an
S32 discrete ordinates approximation in angle. Point marks for hybrid solutions whose convergence is
unhindered lie within the black boxes.

84

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

−2

10

−4

10

−6

10

−8

10

−10

10

−12

2
L

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

−14

10

−2

10

−4

10

−6

10

−8

10

−10

10

−14

10

−12

2
L

10

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 IIa
S32,4 IIa
S32,2 IIa

Order 2
Order 4
Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

(a) Kværnø7

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 IIa
S32,4 IIa
S32,2 IIa

Order 2
Order 4
Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) ARK8

Figure 3.4: L2 convergence studies of hybrid-Ia and hybrid-IIa methods based on Kværnø7 and ARK8
integrators for a sequence of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial do-
main z ∈ [−4,4], final time 0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar
flux of each solution is approximated by comparing to a reference solution using 218 spatial cells and an
S32 discrete ordinates approximation in angle. Point marks for hybrid solutions whose convergence is
unhindered lie within the black boxes.

85

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

−2

10

−4

10

−6

10

−8

10

−10

10

−12

2
L

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

S8
S4
S2
S32

S32,8 a
S32,4 a
S32,2 a

Order 2
Order 3

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

(a) STDG3

S8
S4
S2
S32

S32,8 a
S32,4 a
S32,2 a

Order 2
Order 4
Order 5

−14

10

−2

10

−4

10

−6

10

−8

10

−10

10

−14

10

−12

2
L

10

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) STDG5

Figure 3.5: L2 convergence studies of hybrid-a methods based on STDG integrators for a sequence of
smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time 0.5,
and a CFL of 8 for various values of ε approaching 0. The error in the scalar flux of each solution is
approximated by comparing to a reference solution using 218 spatial cells and an S32 discrete ordinates
approximation in angle. Point marks for hybrid solutions whose convergence is unhindered lie within
the black boxes.

86

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

−2

10

−4

10

−6

10

−8

10

−10

10

−12

2
L

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

−14

10

−2

10

−4

10

−6

10

−8

10

−10

10

−14

10

−12

2
L

10

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 Ib
S32,4 Ib
S32,2 Ib

Order 2
Order 4
Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

(a) IDC5

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 Ib
S32,4 Ib
S32,2 Ib

Order 2
Order 4
Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) LS-IDC5

Figure 3.6: L2 convergence studies of hybrid-I methods based on IDC5 and LS-IDC5 integrators for a
sequence of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4],
final time 0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar flux of each
solution is approximated by comparing to a reference solution using 218 spatial cells and an S32 discrete
ordinates approximation in angle. Point marks for hybrid solutions whose convergence is unhindered
lie within the black boxes.

87

these tolerances should be considered suspect.

Even though convergence saturation and order reduction can be observed, the accuracy of many of

the hybrid schemes is generally quite good. The S32,2 methods display marked increases in error over

the reference S32 approximations (in the form of saturation, order reduction, or both) in nearly every

case, with more significant deviations in accuracy observed for higher-order integrators and moderate

values of ε. However the S32,4 methods only show significant increases in error for fifth-order integrators
and the case of ε = 1E−1.

The behavior of the S32,2 and S32,4 solutions suggests that the splitting error of the hybrid-IIa DIRK

methods is smaller than that of the hybrid-Ia DIRK methods. However, we make two observations con-

cerning this phenomenon. First, the difference in accuracy appears to depend strongly on the time inte-

gration scheme that is applied: The difference in error between the hybrid-Ia and hybrid-IIa methods is

nearly two orders of magnitude for ARK8 and approximately one order of magnitude for SDIRK2, but the

difference for SDIRK3 is much smaller. Second, this difference in accuracy amounts to a reduction in er-

ror by a constant factor: All of the hybrid DIRK methods display similar levels of convergence saturation

and order reduction below the saturation point, but at marginally different tolerances. At this time it is

not clear what properties of the DIRK schemes might affect this behavior.

A similar difference in the splitting error is observed in the hybrid-Ia and hybrid-Ib methods using

IDC and LS-IDC integrators. In this case the methods display similar saturation behavior and the dif-

ference in error appears only in the order-reduction regime. Additionally, the difference in accuracy

between the hybrid-Ia and hybrid-Ib IDC and LS-IDC methods in each test is very small. The fact that

the hybrid-Ia methods apply the reconstruction procedure only once per timestep suggests that they

may offer an advantage over the hybrid-Ib methods if the reconstruction procedure that is used is very

expensive to apply. On the other hand, the increased frequency with which the hybrid-Ib methods apply

the reconstruction procedure may be more advantageous for problems in multidimensional geometries

that contain complex material interfaces.

3.4.2 Two spatial dimensions

To compare the efficiency of the hybrid discrete ordinates methods described in section 3.2 with classical

discrete ordinates methods we consider the following test problem in two-dimensional x y-geometry,

88

which we refer to as the hohlraum problem. The problem is a simplified mock-up of standard hohlraum

configurations used in nuclear fusion devices. The configuration used here is a modification of that

originally introduced in [31], where models of thermal radiative transfer were considered that couple the

transport equation (1.2) to an energy equation for the background material. Since this type of coupling

is not considered in this work, we use a modified configuration in which particle scattering is used to

emulate the effects of thermal absorption and re-emission of particles. The layout of the problem is

shown in figure 3.7. We note that the cross-section values of the material are not rigorously selected to

reproduce the effects of the material energy coupling, but instead are chosen such that the behavior of

the system is qualitatively similar to that of the coupled problem for the time scales selected for these

tests. The problem is simulated to a final time of 2.6 using a CFL of 104 using hybrid and standard

discrete ordinates methods with various angular resolutions. The number of cells in the spatial mesh is

selected based on the order of the method: second-order methods use 312 cells per dimension, third-

order methods use 208 cells per dimension, and fifth-order methods use 104 cells per dimension.

An approximation of the error in each solution is obtained by comparing the scalar flux distribution

of each solution at the final time to a particular reference solution. The reference solution that is used for

each approximate solution is computed using the same space-time discretization as the approximate so-

lution and a highly-refined standard discrete ordinates angular approximation. Since each approximate

solution is compared to a reference solution using the same space-time discretization, the computed er-

ror approximation contains only the angular component of the overall solution error. In particular, this

approach ignores errors in the solution due to approximations in space and time. As a result of this, we

emphasize that these tests cannot be used to make efficiency comparisons between different space-time

discretizations themselves because these components of the error are intentionally suppressed.

We present results for both multidimensional angular quadratures described in section 3.3.2; that

is, we present results for CL quadratures using reconstruction procedures based on hyperinterpolation

and for TN quadratures using reconstruction procedures based on piecewise-constant interpolation.

Reference solutions for methods using CL quadratures are computed using a CL quadrature of order

134 and reference solutions for methods using TN quadratures are computed using a TN quadrature

of order 67. The order of these quadratures is chosen based on prime factors to avoid the possibility of

aliasing in the quadrature directions between approximate and reference solutions from impacting the

error estimates for each approximate solution.

89

0

0.25

1.05

1.3

0

0.45

0.85

1.3

Region

white
black
red
green
blue

σt
0.1
100.0
100.0
100.0
100.0

σs
0.1
0.0
95.0
90.0
50.0

σa
0.0
100.0
5.0
10.0
50.0

0

0.25

1.05

1.3

0

0

-1

-2

-3

-4

0.45

0.85

1.3

(a) Geometric layout.

(b) Parameter values.

(c) Sample solution.

Figure 3.7: Layout and sample solution for the hohlraum test problem. Color scale of solution is log10.

We present results for the hohlraum test problem in three forms. First, we show plots comparing the

scalar flux density and the error distribution between hybrid and standard discrete ordinates methods

using a selection of angular quadrature resolutions. For CL quadratures, standard discrete ordinates

methods using an angular quadrature of order 32 are compared with hybrid discrete ordinates methods

using an angular quadrature of order 64 in the uncollided component and angular quadratures of various

resolutions in the collided component. For TN quadratures, standard discrete ordinates methods using

an angular quadrature of order 16 are compared with hybrid discrete ordinates methods using an angular

quadrature of order 32 in the uncollided component and angular quadratures of various resolutions in

the collided component. Results for DIRK methods are shown in figures 3.8 to 3.15, results for STDG

methods are shown in figures 3.16 and 3.17, results for IDC methods are shown in figures 3.18 to 3.21,

and results for LS-IDC methods are shown in figures 3.22 to 3.25. The solve time (in seconds) and an

approximation of the L

error are included for each solution.

∞

We observe that the hybrid methods require significantly smaller solve times than the standard dis-

crete ordinates methods that they are compared to. The S64,2 and S32,1 approximations require between

one-fourth and one-fifth of the solve time of the S32 and S16 approximations, respectively, and the S64,16

and S32,8 approximations require between one-half and one-third of the solve time of the S32 and S16

approximations, respectively. However this reduction in solve time generally comes at the expense of

accuracy: Using very coarse angular approximations in the collided component of the hybrid methods

increases solution error by as much as an order of magnitude or more, depending on the quadrature.

An examination of the plots of the scalar flux distribution of each solution shows that the increased er-

ror is typically accompanied by an increased presence of ray effects in the solution. The fact that these

90

ray effects are more pronounced in the solutions of higher-order methods is a result of the fact that

each test is run with the same CFL number, so that higher-order methods use a larger timestep size than

lower-order methods. This illustrates the fact that even though the use of a reconstruction procedure be-

tween timesteps to map the collided flux to the higher-resolution quadrature improves the accuracy of

the collision-based splitting [97], increasing the timestep size significantly can decrease the effectiveness

of this approach.

For CL quadratures, the S64,16 hybrid discrete ordinates methods tend to yield an L

∞

error that is

as much as half that of the corresponding S32 standard discrete ordinates method in approximately half

∞

the time. For TN quadratures, the S32,8 hybrid discrete ordinates methods tend to yield an L

error that

is comparable to that of the corresponding S16 standard discrete ordinates method in approximately

half the time. Overall, there is a pronounced qualitative difference between the S32 and S16 discrete

ordinates solutions and the S64,16 and S32,8 hybrid discrete ordinates solutions for each time integra-

tion scheme. The S64,16 and S32,8 methods tend to produce comparable or smaller errors that are more

evenly distributed over the spatial domain. The difference between the hybrid and non-hybrid solutions

is most apparent in regions with low collision probabilities. By using a higher-resolution quadrature in

the uncollided component than the standard discrete ordinates methods to which they are compared,

the hybrid methods that satisfactorily balance the quadrature resolutions that are applied to each com-

ponent tend to produce solutions in which ray effects are less pronounced.

For DIRK integrators, the hybrid-IIa methods produce significantly more accurate solutions than the

hybrid-Ia methods. In many cases the hybrid-IIa methods yield errors that are approximately half that

of the corresponding hybrid-Ia methods. However, similar behavior is observed here as was the case for

the convergence tests in the prior section: The level of improvement offered by the hybrid-IIa methods

over the hybrid-Ia methods is not consistent across the different DIRK schemes. It is interesting to note

that for the ARK8 scheme, the hybrid-IIa S32,4 and S32,8 methods using TN quadratures yield larger
∞

errors than the corresponding hybrid-Ia methods, even though the hybrid-IIa solutions show clear

L

reductions in error over most of the spatial domain. Again, it is not clear at this time what properties of

the methods may impact the expression of this phenomenon.

For IDC and LS-IDC integrators, the hybrid-Ib methods yield solutions with errors that are 20 to 40

percent smaller than the corresponding hybrid-Ia methods while increasing solve time by at most a few

percent. We therefore see no reason that the hybrid-Ib methods should not be preferred over the hybrid-

91

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 3.41E+02
L
error: 1.09E-01

∞

Solve time: 3.60E+02
L
error: 5.35E-02

∞

Solve time: 4.56E+02
L
error: 2.61E-02

∞

Solve time: 8.33E+02
L
error: 1.06E-02

S32

∞

Solve time: 1.68E+03
L
error: 1.81E-02

(a) SDIRK2, Hybrid-Ia, CL

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 3.40E+02
L
error: 6.30E-02

∞

Solve time: 3.71E+02
L
error: 3.10E-02

∞

Solve time: 4.80E+02
L
error: 1.16E-02

∞

Solve time: 8.64E+02
L
error: 6.37E-03

S32

∞

Solve time: 1.68E+03
L
error: 1.81E-02

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) SDIRK2, Hybrid-IIa, CL

Figure 3.8: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid SDIRK2 methods using CL quadratures applied to
error are included for each solution.
the hohlraum test problem. The solve time (in seconds) and the L

∞

92

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 3.36E+02
L
error: 1.11E-01

∞

Solve time: 3.57E+02
L
error: 4.82E-02

∞

Solve time: 4.51E+02
L
error: 1.41E-02

∞

Solve time: 8.43E+02
L
error: 5.75E-03

S16

∞

Solve time: 1.66E+03
L
error: 6.44E-03

(a) SDIRK2, Hybrid-Ia, TN

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 3.47E+02
error: 6.49E-02
L

∞

Solve time: 3.67E+02
error: 2.49E-02
L

∞

Solve time: 4.83E+02
error: 7.18E-03
L

∞

Solve time: 8.47E+02
error: 2.15E-03
L

S16

∞

Solve time: 1.66E+03
error: 6.44E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) SDIRK2, Hybrid-IIa, TN

Figure 3.9: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid SDIRK2 methods using TN quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

93

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 3.49E+02
L
error: 1.32E-01

∞

Solve time: 3.72E+02
L
error: 6.27E-02

∞

Solve time: 4.69E+02
L
error: 3.55E-02

∞

Solve time: 8.30E+02
L
error: 1.43E-02

S32

∞

Solve time: 1.74E+03
L
error: 2.00E-02

(a) SDIRK3, Hybrid-Ia, CL

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 3.74E+02
L
error: 5.57E-02

∞

Solve time: 4.05E+02
L
error: 3.20E-02

∞

Solve time: 4.96E+02
L
error: 1.54E-02

∞

Solve time: 9.02E+02
L
error: 8.13E-03

S32

∞

Solve time: 1.74E+03
L
error: 2.00E-02

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) SDIRK3, Hybrid-IIa, CL

Figure 3.10: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid SDIRK3 methods using CL quadratures applied to
error are included for each solution.
the hohlraum test problem. The solve time (in seconds) and the L

∞

94

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 3.55E+02
L
error: 1.31E-01

∞

Solve time: 3.71E+02
L
error: 5.70E-02

∞

Solve time: 4.78E+02
L
error: 1.65E-02

∞

Solve time: 8.43E+02
L
error: 5.80E-03

S16

∞

Solve time: 1.74E+03
L
error: 8.17E-03

(a) SDIRK3, Hybrid-Ia, TN

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 3.95E+02
error: 5.62E-02
L

∞

Solve time: 3.95E+02
error: 2.77E-02
L

∞

Solve time: 5.20E+02
error: 1.05E-02
L

∞

Solve time: 8.63E+02
error: 3.25E-03
L

S16

∞

Solve time: 1.74E+03
error: 8.17E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) SDIRK3, Hybrid-IIa, TN

Figure 3.11: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid SDIRK3 methods using TN quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

95

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 4.80E+02
L
error: 1.53E-01

∞

Solve time: 5.03E+02
L
error: 7.25E-02

∞

Solve time: 6.06E+02
L
error: 3.59E-02

∞

Solve time: 1.01E+03
L
error: 2.19E-02

S32

∞

Solve time: 2.03E+03
L
error: 2.12E-02

(a) Kværnø7, Hybrid-Ia, CL

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 4.89E+02
error: 2.84E-02
L

∞

Solve time: 5.12E+02
error: 9.84E-03
L

∞

Solve time: 6.11E+02
error: 8.61E-03
L

∞

Solve time: 1.02E+03
error: 8.28E-03
L

S32

∞

Solve time: 2.03E+03
error: 2.12E-02
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) Kværnø7, Hybrid-IIa, CL

Figure 3.12: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid Kværnø7 methods using CL quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

96

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 4.96E+02
L
error: 1.52E-01

∞

Solve time: 5.23E+02
L
error: 8.36E-02

∞

Solve time: 6.17E+02
L
error: 2.53E-02

∞

Solve time: 1.03E+03
L
error: 9.26E-03

S16

∞

Solve time: 2.03E+03
L
error: 8.19E-03

(a) Kværnø7, Hybrid-Ia, TN

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 5.05E+02
error: 2.78E-02
L

∞

Solve time: 5.13E+02
error: 8.97E-03
L

∞

Solve time: 6.12E+02
error: 4.26E-03
L

∞

Solve time: 1.02E+03
error: 2.54E-03
L

S16

∞

Solve time: 2.03E+03
error: 8.19E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) Kværnø7, Hybrid-IIa, TN

Figure 3.13: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid Kværnø7 methods using TN quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

97

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 5.58E+02
L
error: 1.07E-01

∞

Solve time: 5.84E+02
L
error: 5.50E-02

∞

Solve time: 7.18E+02
L
error: 3.01E-02

∞

Solve time: 1.15E+03
L
error: 1.80E-02

S32

∞

Solve time: 2.26E+03
L
error: 2.02E-02

(a) ARK8, Hybrid-Ia, CL

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 5.68E+02
L
error: 1.06E-01

∞

Solve time: 5.94E+02
L
error: 5.61E-02

∞

Solve time: 7.20E+02
L
error: 2.71E-02

∞

Solve time: 1.19E+03
L
error: 1.08E-02

S32

∞

Solve time: 2.26E+03
L
error: 2.02E-02

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) ARK8, Hybrid-IIa, CL

Figure 3.14: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid ARK8 methods using CL quadratures applied to
error are included for each solution.
the hohlraum test problem. The solve time (in seconds) and the L

∞

98

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 5.68E+02
L
error: 1.07E-01

∞

Solve time: 6.03E+02
L
error: 6.82E-02

∞

Solve time: 7.24E+02
L
error: 2.01E-02

∞

Solve time: 1.15E+03
L
error: 7.72E-03

S16

∞

Solve time: 2.31E+03
L
error: 8.44E-03

(a) ARK8, Hybrid-Ia, TN

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 5.77E+02
L
error: 1.03E-01

∞

Solve time: 5.96E+02
L
error: 4.71E-02

∞

Solve time: 7.24E+02
L
error: 2.20E-02

∞

Solve time: 1.15E+03
L
error: 7.83E-03

S16

∞

Solve time: 2.31E+03
L
error: 8.44E-03

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) ARK8, Hybrid-IIa, TN

Figure 3.15: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid ARK8 methods using TN quadratures applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solution.

∞

99

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 2.43E+02
L
error: 1.32E-01

∞

Solve time: 2.72E+02
L
error: 6.25E-02

∞

Solve time: 3.73E+02
L
error: 3.55E-02

∞

Solve time: 7.99E+02
L
error: 1.49E-02

S32

∞

Solve time: 1.99E+03
L
error: 1.99E-02

(a) STDG3, hybrid-a, CL

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 2.46E+02
error: 1.31E-01
L

∞

Solve time: 2.75E+02
error: 5.87E-02
L

∞

Solve time: 3.78E+02
error: 1.69E-02
L

∞

Solve time: 7.46E+02
error: 5.84E-03
L

S16

∞

Solve time: 2.10E+03
error: 8.26E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) STDG3, hybrid-a, TN

Figure 3.16: Comparison of scalar flux density (top rows) and deviation of scalar flux density from ref-
erence solution (bottom rows) of hybrid and non-hybrid STDG3 methods using CL and TN quadratures
applied to the hohlraum test problem. The solve time (in seconds) and the L
error are included for
each solution.

∞

100

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 2.81E+02
L
error: 1.36E-01

∞

Solve time: 3.17E+02
L
error: 6.53E-02

∞

Solve time: 4.60E+02
L
error: 3.36E-02

∞

Solve time: 9.97E+02
L
error: 2.02E-02

S32

∞

Solve time: 2.80E+03
L
error: 2.09E-02

(a) STDG5, hybrid-a, CL

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 2.81E+02
error: 1.34E-01
L

∞

Solve time: 3.18E+02
error: 7.78E-02
L

∞

Solve time: 4.61E+02
error: 2.32E-02
L

∞

Solve time: 9.92E+02
error: 8.26E-03
L

S16

∞

Solve time: 2.81E+03
error: 8.37E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) STDG5, hybrid-a, TN

Figure 3.17: Comparison of scalar flux density (top rows) and deviation of scalar flux density from ref-
erence solution (bottom rows) of hybrid and non-hybrid STDG5 methods using CL and TN quadratures
applied to the hohlraum test problem. The solve time (in seconds) and the L
error are included for
each solution.

∞

101

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 6.29E+02
L
error: 1.30E-01

∞

Solve time: 6.78E+02
L
error: 6.18E-02

∞

Solve time: 8.65E+02
L
error: 3.51E-02

∞

Solve time: 1.52E+03
L
error: 1.48E-02

S32

∞

Solve time: 3.26E+03
L
error: 1.99E-02

(a) IDC3, Hybrid-Ia, CL

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 6.37E+02
L
error: 1.02E-01

∞

Solve time: 6.85E+02
L
error: 4.79E-02

∞

Solve time: 8.78E+02
L
error: 2.72E-02

∞

Solve time: 1.56E+03
L
error: 1.13E-02

S32

∞

Solve time: 3.26E+03
L
error: 1.99E-02

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) IDC3, Hybrid-Ib, CL

Figure 3.18: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid IDC3 methods using CL quadratures applied to
error are included for each solution.
the hohlraum test problem. The solve time (in seconds) and the L

∞

102

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 6.35E+02
L
error: 1.29E-01

∞

Solve time: 6.83E+02
L
error: 5.83E-02

∞

Solve time: 8.63E+02
L
error: 1.69E-02

∞

Solve time: 1.53E+03
L
error: 5.82E-03

S16

∞

Solve time: 3.30E+03
L
error: 8.27E-03

(a) IDC3, Hybrid-Ia, TN

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 6.40E+02
L
error: 1.03E-01

∞

Solve time: 7.02E+02
L
error: 4.29E-02

∞

Solve time: 8.70E+02
L
error: 1.31E-02

∞

Solve time: 1.51E+03
L
error: 5.20E-03

S16

∞

Solve time: 3.30E+03
L
error: 8.27E-03

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) IDC3, Hybrid-Ib, TN

Figure 3.19: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid IDC3 methods using TN quadratures applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solution.

∞

103

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 7.80E+02
L
error: 1.38E-01

∞

Solve time: 8.33E+02
L
error: 6.56E-02

∞

Solve time: 1.06E+03
L
error: 3.38E-02

∞

Solve time: 2.03E+03
L
error: 2.04E-02

S32

∞

Solve time: 4.39E+03
L
error: 2.10E-02

(a) IDC5, Hybrid-Ia, CL

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 8.40E+02
L
error: 1.03E-01

∞

Solve time: 8.43E+02
L
error: 3.92E-02

∞

Solve time: 1.06E+03
L
error: 2.41E-02

∞

Solve time: 2.08E+03
L
error: 1.28E-02

S32

∞

Solve time: 4.39E+03
L
error: 2.10E-02

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) IDC5, Hybrid-Ib, CL

Figure 3.20: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid IDC5 methods using CL quadratures applied to
error are included for each solution.
the hohlraum test problem. The solve time (in seconds) and the L

∞

104

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 7.83E+02
L
error: 1.36E-01

∞

Solve time: 9.35E+02
L
error: 7.76E-02

∞

Solve time: 1.11E+03
L
error: 2.33E-02

∞

Solve time: 1.90E+03
L
error: 8.34E-03

S16

∞

Solve time: 4.39E+03
L
error: 8.38E-03

(a) IDC5, Hybrid-Ia, TN

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 7.94E+02
L
error: 1.03E-01

∞

Solve time: 8.44E+02
L
error: 3.94E-02

∞

Solve time: 1.06E+03
L
error: 1.32E-02

∞

Solve time: 1.91E+03
L
error: 4.90E-03

S16

∞

Solve time: 4.39E+03
L
error: 8.38E-03

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) IDC5, Hybrid-Ib, TN

Figure 3.21: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid IDC5 methods using TN quadratures applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solution.

∞

105

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 7.46E+02
L
error: 1.29E-01

∞

Solve time: 7.95E+02
L
error: 6.14E-02

∞

Solve time: 9.79E+02
L
error: 3.49E-02

∞

Solve time: 1.64E+03
L
error: 1.49E-02

S32

∞

Solve time: 3.28E+03
L
error: 1.99E-02

(a) LS-IDC3, Hybrid-Ia, CL

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 7.54E+02
error: 1.02E-01
L

∞

Solve time: 8.02E+02
error: 4.78E-02
L

∞

Solve time: 9.96E+02
error: 2.71E-02
L

∞

Solve time: 1.69E+03
error: 1.14E-02
L

S32

∞

Solve time: 3.28E+03
error: 1.99E-02
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) LS-IDC3, Hybrid-Ib, CL

Figure 3.22: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid LS-IDC3 methods using CL quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

106

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 7.27E+02
L
error: 1.29E-01

∞

Solve time: 7.70E+02
L
error: 5.86E-02

∞

Solve time: 9.62E+02
L
error: 1.70E-02

∞

Solve time: 1.60E+03
L
error: 5.82E-03

S16

∞

Solve time: 3.26E+03
L
error: 8.27E-03

(a) LS-IDC3, Hybrid-Ia, TN

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 7.31E+02
error: 1.03E-01
L

∞

Solve time: 7.81E+02
error: 4.30E-02
L

∞

Solve time: 9.60E+02
error: 1.31E-02
L

∞

Solve time: 1.63E+03
error: 5.20E-03
L

S16

∞

Solve time: 3.26E+03
error: 8.27E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) LS-IDC3, Hybrid-Ib, TN

Figure 3.23: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid LS-IDC3 methods using TN quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

107

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 7.88E+02
L
error: 1.38E-01

∞

Solve time: 8.39E+02
L
error: 6.59E-02

∞

Solve time: 1.06E+03
L
error: 3.37E-02

∞

Solve time: 1.95E+03
L
error: 2.04E-02

S32

∞

Solve time: 4.42E+03
L
error: 2.09E-02

(a) LS-IDC5, Hybrid-Ia, CL

S64,2

S64,4

S64,8

S64,16

∞

Solve time: 8.18E+02
error: 1.03E-01
L

∞

Solve time: 8.52E+02
error: 3.90E-02
L

∞

Solve time: 1.08E+03
error: 2.40E-02
L

∞

Solve time: 1.97E+03
error: 1.29E-02
L

S32

∞

Solve time: 4.42E+03
error: 2.09E-02
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) LS-IDC5, Hybrid-Ib, CL

Figure 3.24: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid LS-IDC5 methods using CL quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

108

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 1.01E+03
L
error: 1.33E-01

∞

Solve time: 1.06E+03
L
error: 7.74E-02

∞

Solve time: 1.28E+03
L
error: 2.31E-02

∞

Solve time: 2.16E+03
L
error: 8.21E-03

S16

∞

Solve time: 4.36E+03
L
error: 8.36E-03

(a) LS-IDC5, Hybrid-Ia, TN

S32,1

S32,2

S32,4

S32,8

∞

Solve time: 1.01E+03
error: 1.03E-01
L

∞

Solve time: 1.07E+03
error: 3.91E-02
L

∞

Solve time: 1.29E+03
error: 1.31E-02
L

∞

Solve time: 2.11E+03
error: 4.83E-03
L

S16

∞

Solve time: 4.36E+03
error: 8.36E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) LS-IDC5, Hybrid-Ib, TN

Figure 3.25: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid LS-IDC5 methods using TN quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

109

−1

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10

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c
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)
(cid:126)x
d
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c
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s
i
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)
(cid:126)x
d
(

2
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25

29

27
Runtime (seconds)
(a) SDIRK2, Hybrid-Ia, CL

211

213

−1

10

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10

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10

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)
(cid:126)x
d
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s
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)
(cid:126)x
d
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25

29

27
Runtime (seconds)
(b) SDIRK2, Hybrid-Ia, TN

211

25

27
Runtime (seconds)

29

211

213

(c) SDIRK2, Hybrid-IIa, CL

25

29

27
Runtime (seconds)
(d) SDIRK2, Hybrid-IIa, TN

211

213

213

Figure 3.26: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the SDIRK2 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

110

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

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−1

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−2

10

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c
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(cid:126)x
d
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)
(cid:126)x
d
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25

29

27
Runtime (seconds)
(a) SDIRK3, Hybrid-Ia, CL

211

213

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10

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)
(cid:126)x
d
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c
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s
i
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)
(cid:126)x
d
(

2
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25

29

27
Runtime (seconds)
(b) SDIRK3, Hybrid-Ia, TN

211

25

27
Runtime (seconds)

29

211

213

(c) SDIRK3, Hybrid-IIa, CL

25

29

27
Runtime (seconds)
(d) SDIRK3, Hybrid-IIa, TN

211

213

213

Figure 3.27: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the SDIRK3 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

111

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

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−1

10

−2

10

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c
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o
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)
(cid:126)x
d
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)
(cid:126)x
d
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25

27
Runtime (seconds)

29

211

213

(a) Kværnø7, Hybrid-Ia, CL

−1

10

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10

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)
(cid:126)x
d
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10

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10

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c
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s
i
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)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(b) Kværnø7, Hybrid-Ia, TN

211

213

25

27
Runtime (seconds)

29

211

213

(c) Kværnø7, Hybrid-IIa, CL

25

27
Runtime (seconds)

29

211

213

(d) Kværnø7, Hybrid-IIa, TN

Figure 3.28: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the Kværnø7 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

112

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

10

−1

10

−2

10

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c
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f
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o
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c
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s
i
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)
(cid:126)x
d
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e
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)
(cid:126)x
d
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2
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10

−2

10

−3

10

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c
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f
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s
i
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)
(cid:126)x
d
(

2
L

25

27

29

211

213

25

27

29

211

213

Runtime (seconds)
(a) ARK8, Hybrid-Ia, CL

Runtime (seconds)
(b) ARK8, Hybrid-Ia, TN

−1

10

−2

10

−3

10

e
c
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o
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s
i
d
)
(cid:126)x
d
(

2
L

25

27

29

211

213

25

27

29

211

213

Runtime (seconds)
(c) ARK8, Hybrid-IIa, CL

Runtime (seconds)
(d) ARK8, Hybrid-IIa, TN

Figure 3.29: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the ARK8 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

113

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

10

−1

10

−2

10

e
c
n
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f
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r
o
t
e
c
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a
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s
i
d
)
(cid:126)x
d
(

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e
c
n
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f
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c
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a
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s
i
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)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(a) STDG3, Hybrid-a, CL

211

213

−1

10

−2

10

−3

10

e
c
n
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r
e
f
e
r
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c
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s
i
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)
(cid:126)x
d
(

2
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−1

10

−2

10

−3

10

e
c
n
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r
e
f
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c
n
a
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s
i
d
)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(b) STDG3, Hybrid-a, TN

211

213

25

29

27
Runtime (seconds)
(c) STDG5, Hybrid-a, CL

211

213

25

29

27
Runtime (seconds)
(d) STDG5, Hybrid-a, TN

211

213

Figure 3.30: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the STDG3 and STDG5 integrators applied to the
holraum test problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are
shown in red and other hybrid methods are shown in blue. Note the differences in scales.

114

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8Ia methods for the reconstruction procedures considered here. However, the use of other reconstruction

procedures that are significantly more expensive than these may impact the relatively small efficiency

advantage observed here for the hybrid-Ib methods.

In order for the hybrid approach to be effective in realistic scenarios, it is important to determine

how the resolutions of the quadratures applied to the collided and uncollided components should be

chosen to balance reductions in solve time with increases in solution error. In an attempt to characterize

the computational efficiency of the hybrid methods, the second set of results presented here plots the

solve time required by each method against the error of the solution as measured against a reference

solution. Here errors are computed for the scalar flux in the L2 norm. If method A lies below and to

the left of method B, then method A requires less solve time and yields a smaller error than method B:

in this case we say that method A is more efficient than method B. For each integrator, hybrid method,

and angular quadrature type we survey a wide variety of angular quadrature resolutions, considering
only cases where Nc < Nu. Results for DIRK methods are shown in figures 3.26 to 3.29, results for STDG
methods are shown in figure 3.30, results for IDC methods are shown in figures 3.31 and 3.32, and results

for LS-IDC methods are shown in figures 3.33 and 3.34.

We make two primary observations: (i) hybrid methods using angular quadratures that differ in order

by a factor of 4 tend to consistently produce good computational efficiency relative to standard discrete

ordinates methods and other hybrid methods; and (ii) in the vast majority of cases hybrid methods us-

ing angular quadratures of order 2N and N /2 for the uncollided and collided components, respectively,

yield solutions with comparable or smaller errors in less time that a standard discrete ordinates method

using an angular quadrature of order N . In most cases, the aforementioned hybrid methods reduce

solve time by a factor of between two and three for the hohlraum test problem compared to standard

discrete ordinates methods. The efficiency gains offered by the hybrid methods depends strongly on the

specific form of the hybrid method, the time integrator and timestep size used, and the type of angular

quadrature. As noted before, the hybrid-IIa DIRK methods again yield consistently smaller errors than

the corresponding hybrid-Ia DIRK methods for all DIRK schemes considered. Similarly, the hybrid-Ib

IDC methods yield marginally smaller errors than the corresponding hybrid-Ia IDC methods.

Larger efficiency gains are observed for hybrid methods based on lower-order time integrators and

CL quadratures, and smaller efficiency gains are observed for hybrid methods based on higher-order

time integrators and TN quadratures, particularly when higher-resolution angular quadratures are used.

115

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(cid:126)x
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(cid:126)x
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25

29

27

211
Runtime (seconds)
(a) IDC3, Hybrid-Ia, CL

27

29

211
Runtime (seconds)
(c) IDC3, Hybrid-Ib, CL

−1

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10

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(cid:126)x
d
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25

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s
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)
(cid:126)x
d
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25

27

29

211
Runtime (seconds)
(b) IDC3, Hybrid-Ia, TN

27

29

211
Runtime (seconds)
(d) IDC3, Hybrid-Ib, TN

213

213

Figure 3.31: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the IDC3 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

116

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

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(cid:126)x
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(cid:126)x
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26

210

28
Runtime (seconds)
(a) IDC5, Hybrid-Ia, CL

212

210

28
Runtime (seconds)
(c) IDC5, Hybrid-Ib, CL

212

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10

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(cid:126)x
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26

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)
(cid:126)x
d
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26

210

28
Runtime (seconds)
(b) IDC5, Hybrid-Ia, TN

212

210

28
Runtime (seconds)
(d) IDC5, Hybrid-Ib, TN

212

214

214

Figure 3.32: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the IDC5 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

117

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

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(cid:126)x
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27

29

211
Runtime (seconds)

213

(a) LS-IDC3, Hybrid-Ia, CL

25

27

29

211
Runtime (seconds)

213

(c) LS-IDC3, Hybrid-Ib, CL

−1

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(cid:126)x
d
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25

29

27

211
Runtime (seconds)
(b) LS-IDC3, Hybrid-Ia, TN

27

29

211
Runtime (seconds)
(d) LS-IDC3, Hybrid-Ib, TN

213

213

Figure 3.33: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the LS-IDC3 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

118

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

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(a) LS-IDC5, Hybrid-Ia, CL

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(c) LS-IDC5, Hybrid-Ib, CL

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(b) LS-IDC5, Hybrid-Ia, TN

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(d) LS-IDC5, Hybrid-Ib, TN

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214

214

Figure 3.34: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the LS-IDC5 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

119

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8We believe that this disparity is caused mainly by a combination of two factors. First, while it is observed

that efficiency gains are inversely correlated with the order of the time integrator that is used, we believe

that the important factor here is the timestep size, not any property of the integrator such as its order of

accuracy. Recall that all of the tests conducted here are performed using a constant CFL for all methods,

so that higher-order methods use larger timestep sizes than lower-order methods. A larger timestep

permits a larger proportion of particles within the system to experience collision events during each

timestep, pushing more of the solution into the collided component.

The second contributing factor is the efficiency of the angular quadratures themselves. Figures 3.26

to 3.34 compare hybrid discrete ordinates methods to standard discrete ordinates methods using the

same type of angular quadrature. This comparison implicitly ignores the fact that the angular quadra-

tures considered here (CL and TN ) have radically different properties and hence behave differently when

applied to certain problems. Comparisons between standard discrete ordinates methods using CL and

TN quadratures and hybrid discrete ordinates methods using CL and TN quadratures that differ by a

factor of 4 are shown in figures 3.35 to 3.39. Since the distribution of quadrature nodes varies signifi-

cantly between the two quadrature types, TN quadratures require many fewer nodes to attain a certain

error tolerance than CL quadratures when applied to the hohlraum problem. This has two effects: First,

the tolerances at which the hybrid and nonhybrid methods are compared is different for TN and CL

quadratures. Second, the difference in accuracy between angular quadratures that differ by a factor of 4

is different for TN and CL quadratures. We expect that this second fact contributes to the disparity in the

efficiency gains observed fro different quadratures. While choosing angular quadratures that differ in

order by a factor of 4 seems to produce acceptable results here, it may be the case that a different choice

of angular quadrature resolutions, potentially dependent on the type of quadrature used and the level of

refinement, may yield better results than what is observed here.

One important observation to be made from figures 3.35 to 3.39 is that any efficiency advantage de-

rived from the hybrid approach is dependent on the angular quadrature that is used. Just as the choice

of angular quadrature can significantly impact the accuracy of standard discrete ordinates methods,

making a prudent choice of angular quadratures for the collided and uncollided components of the hy-

brid methods remains crucial to the overall efficiency of the methods. While hybrid methods using CL

quadratures can be more computationally efficient that standard discrete ordinates methods using CL

quadratures, standard discrete ordinates methods using TN quadratures can be more computationally

120

CL non-hybrid
CL hybrid-a

TN non-hybrid
TN hybrid-a

CL non-hybrid
CL hybrid-a

TN non-hybrid
TN hybrid-a

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(a) STDG3

(b) STDG5

Figure 3.35: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black and
gray) and hybrid (red and blue) discrete ordinates methods based on STDG integrators and various angu-
lar resolutions applied to the hohlraum test problem. CL (gray and blue) and TN (black and red) angular
quadratures are considered.

efficient than hybrid methods using CL quadratures.

Overall, the results presented here suggest that the hybrid approach can be an effective technique for

reducing the computation time of standard discrete ordinates methods. However this comes at the cost

of increased memory consumption. The recommendation that a standard discrete ordinates method

using an angular quadrature of order N be replaced by a hybrid discrete ordinates method using angu-

lar quadratures of order 2N and N /2 in the uncollided and collided components, respectively, increases

memory consumption in two ways. First, because the number of quadrature nodes scales proportion-

ally to the second power of the order of the angular quadrature, storing the uncollided components of

the stages requires four times the memory of the corresponding nonhybrid method. Second, storing

the collided components of the stages requires one-fourth the memory of the corresponding nonhybrid

method, but this is in addition to the storage of the uncollided component. The result is that, for each

time integrator, replacing a standard discrete ordinates method with a hybrid discrete ordinates method

increases memory consumption by a factor of approximately 4.25. This value may be marginally lower

for certain methods, such as the hybrid-IIa DIRK methods, where only one collided stage must be stored

at any given time. These methods will nevertheless increase memory consumption by a factor of at least

4 over the corresponding standard discrete ordinates method. In cases where traditional discrete ordi-

121

    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8CL non-hybrid
CL hybrid-Ia

TN non-hybrid
TN hybrid-Ia

CL non-hybrid
CL hybrid-Ia

TN non-hybrid
TN hybrid-Ia

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(a) SDIRK2, Hybrid-Ia

(b) SDIRK3, Hybrid-Ia

CL non-hybrid
CL hybrid-Ia

TN non-hybrid
TN hybrid-Ia

CL non-hybrid
CL hybrid-Ia

TN non-hybrid
TN hybrid-Ia

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(c) Kværnø7, Hybrid-Ia

25

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Runtime (seconds)

(d) ARK8, Hybrid-Ia

Figure 3.36: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black and
gray) and hybrid (red and blue) discrete ordinates methods based on DIRK integrators and various angu-
lar resolutions applied to the hohlraum test problem. CL (gray and blue) and TN (black and red) angular
quadratures are considered.

122

    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8CL non-hybrid
CL hybrid-IIa

TN non-hybrid
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CL non-hybrid
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(a) SDIRK2, Hybrid-IIa

(b) SDIRK3, Hybrid-IIa

CL non-hybrid
CL hybrid-IIa

TN non-hybrid
TN hybrid-IIa

CL non-hybrid
CL hybrid-IIa

TN non-hybrid
TN hybrid-IIa

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Runtime (seconds)

(c) Kværnø7, Hybrid-IIa

(d) ARK8, Hybrid-IIa

Figure 3.37: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black and
gray) and hybrid (red and blue) discrete ordinates methods based on DIRK integrators and various angu-
lar resolutions applied to the hohlraum test problem. CL (gray and blue) and TN (black and red) angular
quadratures are considered.

123

    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8−1

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CL non-hybrid
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(a) IDC3, Hybrid-Ia

(b) IDC3, Hybrid-Ib

CL non-hybrid
CL hybrid-Ia

TN non-hybrid
TN hybrid-Ia

CL non-hybrid
CL hybrid-Ib

TN non-hybrid
TN hybrid-Ib

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(c) IDC5, Hybrid-Ia

(d) IDC5, Hybrid-Ib

Figure 3.38: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black and
gray) and hybrid (red and blue) discrete ordinates methods based on IDC integrators and various angular
resolutions applied to the hohlraum test problem. CL (gray and blue) and TN (black and red) angular
quadratures are considered.

124

    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8−1

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(a) LS-IDC3, Hybrid-Ia

(b) LS-IDC3, Hybrid-Ib

CL non-hybrid
CL hybrid-Ia

TN non-hybrid
TN hybrid-Ia

CL non-hybrid
CL hybrid-Ib

TN non-hybrid
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(c) LS-IDC5, Hybrid-Ia

(d) LS-IDC5, Hybrid-Ib

Figure 3.39: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black and
gray) and hybrid (red and blue) discrete ordinates methods based on LS-IDC integrators and various
angular resolutions applied to the hohlraum test problem. CL (gray and blue) and TN (black and red)
angular quadratures are considered.

125

    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8    248163264124816328-216-432-864-164-18-216-432-8nates methods are already limited by available system memory, the savings in solve time that the hybrid

methods discussed here provide may be overshadowed by the burded of increased memory consump-

tion.

126

NYSTRÖM RECONSTRUCTION AND TWO-GRID DEFECT CORRECTION FOR HYBRID METHODS

CHAPTER 4

4.1 Introduction

At its most basic level, the discrete ordinates method is based on the principle of collocation. Many

methods for the numerical solution of integral equations have been developed based on this principle.

The purpose of this chapter is to examine several classes of hybrid methods derived from the collision-

based splitting (3.16) that incorporate various approaches for extending the ideas behind collocation

methods. We focus on methods related to two approaches in particular: Nyström interpolation and two-

grid iterative methods.

We provide brief descriptions of Nyström interpolation and two-grid iterative methods below. These

descriptions are based on those of [17, Section 4.1] and [17, Section 6.2], respectively, and we refer the

interested reader to the reference [17] for more detailed descriptions and analyses of these methods. Our

descriptions are given in the context of a canonical integral equation of the second kind:

λf (x)−(cid:90)

K (x, y)f (y)d y = g (x),

(x ∈ D).

D

(4.1)
Here f : D → (cid:82) is the unknown function whose values are sought, and the kernel K : D × D → (cid:82) of the
integral operator, the function g : D → (cid:82), and the scalar λ are assumed to be known. The set D is assumed
to be a manifold embedded within a finite dimensional Euclidean space (cid:82)M , e.g., the unit two-sphere

(cid:83)2.

4.1.1 Nyström interpolation

Collocation methods use numerical quadrature to provide an approximation to the solution of an equa-

tion of the form (4.1) on a discrete set of points; specifically, at the nodes of the quadrature that is ap-

plied. If solution values are desired at other points, then either these points must be incorporated into

the quadrature that is used or the solution values at the quadrature nodes must be extended to the de-

sired points in some manner after the values of the collocation solution have been computed. The first

approach can only be used if the points at which the solution is desired are known a priori. As we will

discuss, implementing the second approach is not necessarily straightforward.

127

Let(cid:169)xn,j , wn,j

(cid:170)Nn

j=1

⊂ D × (cid:82) be a sequence of quadratures (indexed here by n) such that

wn,j f(cid:161)xn,j

(cid:162) →(cid:90)

Nn(cid:88)

j=1

f (x)d x

D

(4.2)

as n → ∞ for any f belonging to a suitable space of functions (e.g., the space C (D) of continuous func-

tions). Approximations fn to the solution f of (4.1), each corresponding to one of the quadratures in

the sequence, can be defined by replacing the integral operator in (4.1) with an approximation using the

quadrature:

λfn(x)− Nn(cid:88)

j=1

wn,j K(cid:161)x, xn,j

(cid:162)fn

(cid:161)xn,j

(cid:162) = g (x).

(4.3)

There are two important observations to be made at this point. The first is that each function fn is

defined on the entire domain of integration D. The second is that each of the functions fn can be defined

uniquely through its values at the associated quadrature nodes. More specifically, for each n the function

fn is defined uniquely by the set of values

(cid:162)(cid:111)Nn

j=1

(cid:110)

fn

(cid:161)xn,j
(cid:161)xn,j
(cid:162)fn

(4.4)

(4.5)

(cid:161)i = 1,..., Nn

(cid:162).

satisfying

λfn

(cid:161)xn,i

(cid:162)− Nn(cid:88)

wn,j K(cid:161)xn,i , xn,j

(cid:162),
(cid:162) = g(cid:161)xn,i
lution. The question at this point is: once the discrete values(cid:169)fn

j=1

Since the nodal values (4.4) satisfy (4.5) exactly, these values correspond to those of the collocation so-
j=1 are known, how does one

(cid:161)xn,j

(cid:162)(cid:170)Nn

extend these values to obtain an approximate solution value for an arbitrary point in D?

Extending a discrete set of values to other points within some space is a problem that is commonly

solved using various forms of interpolation or extrapolation. A general interpolation problem may be

roughly stated as follows:

Given a set of values {y j }N
j=1
to some set of functions F such that f (x j ) = y j for all j = 1,..., N .

⊂ (cid:82) on a set of points {x j }N
j=1

⊂ X ⊂ (cid:82)n find f : X → (cid:82) belonging

The space X and the set of functions F to which f should belong will depend on the problem. However, it

is well-known that general forms of interpolation are not necessarily well-behaved in all situations (c.f.,

128

for example, [67, Chapter 7]). The situation is particularly bleak for interpolation problems on higher

dimensional manifolds such as (cid:83)2, where solutions may not exist [170].

This generic approach to interpolation was used to define the reconstruction procedures in chap-

ter 3. The nature of these reconstruction procedures is a result of the difficulties inherent to interpola-

tion problems on spaces such as (cid:83)2. In addition to the fact that the reconstruction will typically need

to be tailored to the properties of the specific quadrature that is being used, other sacrifices must of-

ten be made: Not all methods will preserve properties of the solution such as non-negativity, conserve

quantities of the solution such as the current density, or actually interpolate the discrete values.

The key observation to make is that there is more information to work with in the case of collocation

methods for integral equations than there is for a generic interpolation problem. More specifically, we

know that the set of points (4.4) that we would like to interpolate satisfies the system of equations (4.5)

and, in particular, that the approximate solution fn satisfying (4.3) is defined uniquely by the point values

(4.4) obtained from (4.5). Hence using the discrete values (4.4) in (4.3) yields an expression for any value

g (x)+ Nn(cid:88)

j=1

wn,j K(cid:161)x, xn,j

(cid:162)fn

(cid:161)xn,j

(cid:162).

of fn in D:

fn(x) = 1

λ

(4.6)

The formula (4.6), originally presented in [151], is called the Nyström interpolation formula.

4.1.2 Two-grid iteration

The motivation for discussing collocation methods in the context of a sequence of quadratures satisfying

the property (4.2) is that the approximate solutions fn satisfying (4.3) converge (in some sense) to the
true solution f of (4.1) as n → ∞. In theory, this implies that it is possible to obtain an approximate

solution to (4.1) with any level of accuracy by simply refining the quadrature that is used until the desired

level of accuracy has been attained. However a simple brute-force approach is not necessarily effective

in practice.

Computing each approximate solution fn requires solving a linear system of equations given by (4.5),

whose dimension goes to infinity as n does. This means that computing the approximate solutions be-

comes significantly more expensive as the number of quadrature nodes used increases. There are two

factors that can make this effect particularly problematic. The first is that the rate at which the number

of quadrature nodes increases is related to the dimension of the domain of integration D: when the di-

129

mension of D is large, the number of quadrature nodes will generally increase very rapidly. Additionally,

if the kernel K is very complicated then the linear system (4.5) may be dense and lack a clear sense of

structure that can be exploited to reduce the computational cost of each solve. Compounding the scal-

ing of the number of quadrature nodes with the scaling of the computational cost associated with dense

linear algebra solves can render many problems exceedingly difficult to solve to any meaningful degree

of accuracy.

Two-grid methods seek to reduce the significant computational cost associated with solving the lin-

ear system (4.5) when the number of quadrature nodes is large. This is done using a residual-correction

approach that is similar to that used by IDC and other defect correction methods. Whereas IDC meth-

ods use high- and low-order approximations on the same quadrature grid, two-grid methods vary the

quadrature nodes themselves. The effect is nevertheless the same in both cases: the higher-resolution

approximation is only applied when computing the residual, and cost savings are obtained by inverting

a simpler, lower-resolution approximation at each iteration.

The two-grid method that we will consider for (4.1) can be derived as follows. For each n, define the

numerical integration operator Kn by

(cid:161)Kn f(cid:162)(x) = Nn(cid:88)
wn,j K(cid:161)x, xn,j
(cid:162)fn = g .
(cid:161)λ−Kn

j=1

(cid:162)f(cid:161)xn,j

(cid:162).

(4.7)

(4.8)

Then (4.3) can be written as

Suppose that f (k)

n

is some approximation to the solution fn of (4.8). Here f (k)
n

is some initial guess or,

as the superscript would suggest, a previous iterate of the two-grid process. If the residual and error are

defined by

and

n = g −(cid:161)λ−Kn

r (k)

(cid:162)f (k)

n

respectively, then the error satisfies the equation

n = fn − f (k)
e(k)
n ,
(cid:162)e(k)
(cid:161)λ−Kn
n = r (k)
n .

(4.9)

(4.10)

(4.11)

If one solves (4.11) using a standard collocation approach, then the desired solution fn is obtained

exactly through (4.10). But this would provide no reduction in computational cost relative to solving

130

(4.8) directly because the dimension of the linear systems that must be solved in each case are the same.
Instead, Kn is replaced by Km with m < n to obtain
(cid:161)λ−Km
(cid:162)e(k)
m = r (k)
n ,

(4.12)

which is solved using a collocation approximation in terms of the lower-resolution quadrature. This
yields a lower-resolution approximation e(k)

m to e(k)

n satisfying

(cid:161)xm,i

(cid:162)− Nm(cid:88)

j=1

λe(k)
m

for i = 1,..., Nm. Once the point values(cid:169)e(k)

(cid:162)e(k)

(cid:161)xm,j

wm,j K(cid:161)xm,i , xm,j
(cid:162)(cid:170)Nm
j=1 have been extended to a representation for e(k)
m

(cid:162) = r (k)

(cid:161)xm,j

(cid:161)xm,i

(cid:162),

(4.13)

m

m

n

on all of D, the approximate solution f (k)

n can be updated using
n + e(k)
m .

f (k+1)

= f (k)

n

(4.14)

This process is then repeated until some convergence tolerance is satisfied.

The values of the function e(k)

j=1 using the
Nyström interpolation formula (4.6). However, our initial efforts at constructing defect correction meth-

m can be recovered from the point values(cid:169)e(k)

m

(cid:161)xm,j

(cid:162)(cid:170)Nm

ods for the hybrid splitting (3.16) used the reconstruction procedures discussed in chapter 3. Results for

these defect correction methods are included here for completeness. While we did not observe any be-

havior in the numerical tests that were conducted that we believe would suggest a lack of convergence,

we note that it is not clear which of the reconstruction methods discussed previously would be guaran-

teed to produce convergent iterations in all situations. For standard two-grid methods as outlined above

using Nyström interpolation, the following can be said:

Theorem 1. For a sequence of quadratures that are convergent in the sense of (4.2), the two-grid iteration
is convergent, i.e., f (k)

n → fn as k → ∞, for m sufficiently large.

Proof. See [17, Theorem 6.2.1].

■

While theorem 1 states that the two-grid method defined by (4.9), (4.12), and (4.14) converges for m

sufficiently large, this theorem says little about the nature of this convergence. An examination of the

proof given in [17, Theorem 6.2.1] reveals that this two-grid method does not converge monotonically:

Instead the method is expected to follow an even-odd pattern of convergence where every other iterate

131

converges monotonically but the overall iteration does not. We will see that the two-grid IDC methods

that are presented in this work display similar behavior in certain situations.

It should be noted that a different two-grid method can be devised whose convergence behavior is

superior to that of the method considered here. A description of this alternative method can be found

in [17, Section 6.2.2], along with a proof that the method converges monotonically with a geometric rate

of convergence. The discussion of two-grid methods in the present work is not intended to be an ex-

haustive comparison of all methods that can be derived using different combinations of IDC and hybrid

techniques. Instead, it is meant to serve as a proof-of-concept that motivates future investigations into

methods in which defect correction approaches targeted towards different discretization errors are com-

bined into a single iterative procedure. Therefore, while it is possible to construct two-grid IDC methods

based on this alternative technique, we reserve the consideration of such methods to future extensions

of this work.

4.2 Method descriptions

We now turn our attention to describing hybrid methods for the transport equation (1.2) using the

ideas behind Nyström interpolation and two-grid methods in combination with several classes of time-

integration schemes.

4.2.1 Nyström reconstruction

We first describe modifications of the hybrid methods introduced in chapter 3 that have a strong simi-

larity to the approach used in Nyström interpolation. In this context it is not necessarily clear that the

term “interpolation” is appropriate, so we use the term Nyström reconstruction when referring to these

methods. These methods are assigned the subtype c to denote the fact that a Nyström reconstruction

is used. Because Nyström methods rely on the structure of the equation that one is trying to solve, the

form of these reconstruction procedures depends on the time integrator used and the manner in which

the splitting is applied. However, the general idea behind each specific instance is the same.

132

First-order hybrid method with Nyström reconstruction

The collided and uncollided components of the first-order hybrid method with Nyström reconstruction
are computed using (3.20) as before. However, instead of defining the reconstructed flux ψn∗ explicitly

through (3.21), we return to the system (3.11) that is solved for the standard discrete ordinates method

with an implicit Euler discretization in time. For the sake of consistency of notation, we first re-write the

system (3.11) in the form

(cid:181)
L+ 1
∆t

(cid:182)

−SP

ψn = 1
∆t

ψn−1 + qn,

(4.15)

ignoring the spatial discretization. The Nyström reconstruction is performed by using the collided and

uncollided components obtained from (3.20) to approximate the scattering source in (4.15):

(cid:182)

(cid:181)
Lu + 1
∆t

(cid:161)Puψn

ψn∗ =Su

u +Pcψn

c

(cid:162)+ 1

∆t

ψn−1∗ + qn.

(4.16)

The reconstructed angular flux distribution is defined implicitly by (4.16), which can be solved using

a single transport sweep. We note that although the structure of (4.16) shares a significant degree of

similarity with that of the source iteration method, (4.16) is not used to define an iteration procedure.

Updating algorithm 3.1 to include the use of Nyström reconstruction yields algorithm 4.1.

The Nyström reconstruction procedure has a number of advantages. The first advantage is the flex-

ibility of the method: Because this procedure is defined entirely through the operators of which the

transport equation is composed, there is no explicit dependence on the problem geometry or choice of

angular quadratures. This also means that hybrid methods leveraging this reconstruction can be used

in existing discrete ordinates implementations without requiring significant modifications. Second, the

accuracy of this procedure tends to be superior to that of the relabeling procedures described in sec-

tion 3.3.2 – see section 4.4 for comparisons. Finally, when the Nyström reconstruction is used, the hybrid

method naturally reduces to the non-hybrid, standard discrete ordinates method when the quadratures

applied to the collided and uncollided components are the same. We emphasize two facts here: (i) this

result is independent of the specific quadrature that is applied, and (ii) not all reconstruction procedures

satisfy the above property, including the hyperinterpolation method used for CL quadratures.

Proposition 5. Suppose that the same quadrature is applied to the collided and uncollided components
of the hybrid method (3.20) and that ψn∗ is computed through (4.16). Then ψn∗ = ψn
c and ψn∗ = ψn
where ψn is the solution of the standard discrete ordinates system (4.15).

u + ψn

133

Algorithm 4.1 Hybrid Euler timestep with Nyström reconstruction.

input: Initial condition ψn−1∗
1: Compute ψn

u by solving (3.20a):

.

(cid:182)

(cid:181)
Lu + 1
∆t

ψn−1∗

+ qn.

u = 1
ψn
∆t
(cid:182)

2: Compute ψn

c by solving (3.20b):

(cid:181)
Lc + 1
∆t

−ScPc

c =ScPuψn
ψn
u .

case Use a reconstruction operator R:

3: switch reconstruction method do
4:
5:
6:
7:

ψn∗ ← ψn
u

end case
case Use the Nyström reconstruction (4.16) by solving:

+Rψn
c .

(cid:182)

(cid:181)
Lu + 1
∆t

(cid:161)Puψn

ψn∗ =Su

u +Pcψn

c

(cid:162)+ 1

∆t

ψn−1∗

+ qn.

end case
8:
9: end switch

return: ψn∗

Proof. Since the angular quadrature is assumed to be the same for both the collided and uncollided com-

ponents, we drop the subscripts used on the operators in the hybrid equations with the understanding
that Lu =Lc =L, and similarly for S and P. For simplicity, we do not include the terms due to the in-
flow of radiation at the boundary, in which case the operator L is linear. Using the equations in (3.20) to

replace the terms on the right side of (4.16) implies that

(cid:181)
L+ 1
∆t

(cid:182)

ψn∗ =(cid:181)

L+ 1
∆t

(cid:182)

c +(cid:181)

ψn

L+ 1
∆t

(cid:182)

u =(cid:181)

ψn

L+ 1
∆t

(cid:182)(cid:161)ψn
u + ψn

c

(cid:162),

(4.17)

from which the result ψn∗ = ψn
c follows through inversion of the streaming operator. The fact that
ψn∗ = ψn follows by simply combining the terms of the scattering source in (4.16) and observing that the
■
resulting equation for ψn∗ is the same as that of equation (4.15) for ψn.

u + ψn

Results similar to proposition 5 can be stated for the higher-order methods discussed in the remain-

der of this section. These results are essentially a side-effect of the fact that the splitting (3.16) is exact on

the continuum level, and hence the hybrid approach using identical quadratures and Nyström recon-

struction yields a matrix inversion identity for each method.

134

Hybrid-Ic DIRK method

The collided and uncollided components of the hybrid-Ic DIRK method are computed using the same

equations (3.25) as the hybrid-Ia DIRK method. The Nyström reconstruction procedure is performed at

each stage using the following modified form of (3.24b):

(cid:104)Luψ(i )∗ −Su
(cid:179)Puψ(i )
ψ(i )∗ = ψn−1∗ − ∆t ai i
u +Pcψ(i )
(cid:162)ψ(j )∗ − q(j )(cid:105)
(cid:104)(cid:161)Lu −SuPu
i−1(cid:88)

− ∆t

c

,

ai j

j=1

(cid:180)− q(i )(cid:105)

(4.18)

formed for each stage in sequence. Once the reconstructed stage values(cid:169)ψ(i )∗ (cid:170)s

for i = 1,..., s. Note that, in contrast to the hybrid-Ia DIRK method, the reconstruction must be per-
i=1 are known, the so-
lution for the timestep is computed using (3.28) or is obtained directly from ψ(s)∗ if the DIRK scheme

is stiffly accurate. A result similar to that of proposition 5 for the hybrid-Ic DIRK method is stated in

proposition 6. Updating algorithm 3.3 to include the use of Nyström reconstruction yields algorithm 4.2.

Proposition 6. Suppose that the same quadrature is applied to the collided and uncollided components
of the hybrid-I DIRK method (3.25) and that the reconstructed stage values ψ(i )∗ for i = 1,..., s are com-
puted through (4.18). Then the reconstructed stage values satisfy ψ(i )∗ = ψ(i )
c and ψ(i )∗ = ψ(i ) for
u + ψ(i )
i = 1,..., s, where ψ(i ) are the stages for the standard discrete ordinates method (3.24b).

Proof. As in the proof of proposition 5, we drop the subscripts used on the operators of the hybrid equa-

tions and omit the terms due to the inflow of radiation at the boundary. We proceed by induction on the
stage index i : Suppose that i ≥ 1 and that proposition 6 is true for each stage with index less than i . From

(3.25c) and (3.25d) it follows that

1

ψn−1∗ + q(i ) =(cid:181)
(cid:180) =(cid:181)
SP(cid:179)

∆t ai i

c

u + ψ(i )
ψ(i )
(cid:182)

ψ(i )∗ =(cid:181)
=(cid:181)

(cid:181)
L+ 1
∆t ai i

(cid:182)
(cid:182)

ψ(i )

ψ(i )

j=1

j=1

ai j
ai i
ai j
ai i

u + i−1(cid:88)
c + i−1(cid:88)
(cid:180)+ i−1(cid:88)
(cid:180)

j=1

,

c

u + ψ(i )
ψ(i )
u + ψ(i )
ψ(i )

c

L+ 1
∆t ai i
L+ 1
∆t ai i
(cid:182)(cid:179)
(cid:182)(cid:179)

,

(cid:104)Lψ(j )
u − q(j )(cid:105)
(cid:104)Lψ(j )
c −SP(cid:179)
(cid:161)L−SP(cid:162)(cid:179)

ai j
ai i

L+ 1
∆t ai i
L+ 1
∆t ai i

Replacing the terms on the right side of (4.18) using (4.19) yields

(cid:180)(cid:105)

.

c + ψ(j )
ψ(j )

u

u + ψ(j )
ψ(j )

c − ψ(j )∗

(4.19a)

(4.19b)

(4.20a)

(4.20b)

(cid:180)

135

Algorithm 4.2 Hybrid-I DIRK timestep with Nyström reconstruction.

input: Initial condition ψn−1∗
1: for i = 1,..., s do
Compute ψ(i )
2:

.

u by solving (3.25c):

(cid:182)

(cid:181)
Lu + 1
∆t ai i

ψ(i )
u

= 1

∆t ai i

ψn−1∗

− i−1(cid:88)
j=1

ai j
ai i

(cid:183)
Luψ(j )
u

(cid:184)+ q(i ).

− q(j )

3:

Compute ψ(i )

c by solving (3.25d):

(cid:181)
Lc + 1
∆t ai i

(cid:182)

−ScPc

ψ(i )
c

= − i−1(cid:88)
j=1

ai j
ai i

(cid:183)(cid:161)Lc −ScPc

(cid:162)ψ(j )

c

(cid:184)+ScPuψ(i )

u .

−ScPuψ(j )
u

4: end for

5: switch hybrid method do
6:
7:

case hybrid-Ia:

if the DIRK scheme is stiffly accurate then

← ψ(s)
u .
← ψ(s)
c .

ψn
u
ψn
c

else

(cid:104)Luψ(i )
− q(i )(cid:105)
s(cid:88)
(cid:104)(cid:161)Lc −ScPc
(cid:162)ψ(i )
i=1

bi

u

c

.

−ScPuψ(i )
u

(cid:105)

.

← ψn−1∗
− ∆t
s(cid:88)
← −∆t
i=1

bi

8:

9:
10:

11:

12:

13:

14:
15:

16:
17:
18:

19:

20:

21:
22:

23:

ψn

u is defined by (3.25a): ψn
u

ψn

c is defined by (3.25b): ψn
c

end if
ψn∗ ← ψn
u

end case

+Rψn
c .

case hybrid-Ic:

for i = 1,..., s do

Compute ψ(i )∗ by solving (4.18):

(cid:182)

(cid:181)
Lu + 1
∆t ai i

end for

if the DIRK scheme is stiffly accurate then

ψn∗ ← ψ(s)∗ .
ψn∗ is defined by (3.28): ψn∗ ← ψn−1∗

else

end if
24:
end case
25:
26: end switch

return: ψn∗

ψ(i )∗ = 1
∆t ai i

ψn−1∗

+Su

(cid:179)Puψ(i )

u

+Pcψ(i )
c

(cid:180)+ q(i ) − i−1(cid:88)

j=1

ai j
ai i

(cid:183)(cid:161)Lu −SuPu

(cid:162)ψ(j )∗ − q(j )

(cid:184)

.

s(cid:88)
i=1

bi

− ∆t

(cid:104)(cid:161)Lu −SuPu

(cid:162)ψ(i )∗ − q(i )(cid:105)

.

136

Algorithm 4.3 Hybrid-II DIRK timestep with Nyström reconstruction.

(cid:183)(cid:161)Lu −SuPu

(cid:162)ψ(j )∗ − q(j )

(cid:184)+ q(i ).

ai j
ai i

− i−1(cid:88)
j=1
(cid:182)

−ScPc

ψ(i )
c

=ScPuψ(i )
u .

.

input: Initial condition ψn−1∗
1: for i = 1,..., s do
Compute ψ(i )
(cid:181)
u by solving (3.26a):
2:
Lu + 1
∆t ai i

ψ(i )
u

(cid:182)

ψn−1∗

= 1

∆t ai i

Compute ψ(i )

c by solving (3.26b):(cid:181)

Lc + 1
∆t ai i

switch hybrid method do

case hybrid-IIa:
ψ(i )∗ ← ψ(i )
u

end case

+Rψ(i )
c .

3:

4:
5:

6:
7:

8:

9:

case hybrid-IIc:

Compute ψ(i )∗ by solving (4.18):

(cid:182)

(cid:181)
Lu + 1
∆t ai i

ψ(i )∗ = 1
∆t ai i

ψn−1∗

+Su

(cid:179)Puψ(i )

u

+Pcψ(i )
c

(cid:180)+ q(i ) − i−1(cid:88)

j=1

ai j
ai i

(cid:183)(cid:161)Lu −SuPu

(cid:162)ψ(j )∗ − q(j )

(cid:184)

.

end case
end switch

10:
11:
12: end for

13: if the DIRK scheme is stiffly accurate then
14:
15: else

ψn∗ ← ψ(s)∗ .
ψn∗ is defined by (3.28): ψn∗ ← ψn−1∗

16:

− ∆t

17: end if

return: ψn∗

(cid:104)(cid:161)Lu −SuPu

(cid:162)ψ(i )∗ − q(i )(cid:105)

.

s(cid:88)
i=1

bi

where the second equality follows from the inductive hypothesis, or, for the case when i = 1, from the fact
that the sum contains zero terms. Hence the result ψ(i )∗ = ψ(i )
c follows from the inversion of the
streaming operator. The fact that ψ(i )∗ = ψ(i ) follows by simply combining the collided and uncollided
terms of the scattering source in (4.18) and observing that the resulting equation for ψ(i )∗ is the same as
■
that of the equation (3.24b) for ψ(i ).

u + ψ(i )

137

Hybrid-IIc DIRK method

The collided and uncollided components of the hybrid-IIc DIRK method are computed using the same
equations (3.26) as the hybrid-IIa DIRK method. The reconstructed stage values ψ(i )∗ are defined im-

plicitly using the same equation (4.18) as the reconstructed stage values of the hybrid-Ic DIRK method.

Equation (3.28) may also be used if the DIRK scheme is not stiffly-accurate. A result similar to that of

proposition 5 for the hybrid-IIc DIRK method is stated in proposition 7. Updating algorithm 3.4 to in-

clude the use of Nyström reconstruction yields algorithm 4.3.

Proposition 7. Suppose that the same quadrature is applied to the collided and uncollided components
of the hybrid-II DIRK method (3.26) and that the reconstructed stage values ψ(i )∗ for i = 1,..., s are com-
puted through (4.18). Then the reconstructed stage values satisfy ψ(i )∗ = ψ(i )
c and ψ(i )∗ = ψ(i ) for
u + ψ(i )
i = 1,..., s, where ψ(i ) are the stages for the standard discrete ordinates method (3.24b).

Proof. The proof proceeds in the same manner as proposition 6 with the relations (4.19) replaced by

1

ψn−1∗ + q(i ) =(cid:181)
(cid:180) =(cid:181)
SP(cid:179)

u + ψ(i )
ψ(i )

c

∆t ai i

(cid:182)
(cid:182)

L+ 1
∆t ai i
L+ 1
∆t ai i

u + i−1(cid:88)

j=1

ψ(i )

ψ(i )
c .

(cid:104)(cid:161)L−SP(cid:162)ψ(i )∗ − q(j )(cid:105)

ai j
ai i

,

(4.21a)

(4.21b)

■

Hybrid-c STDG method

The collided and uncollided components of the hybrid-c STDG method are computed using the same

equations (3.46) as the hybrid-a STDG method. The coefficients of the reconstructed approximation
ψh,n∗ are defined implicitly through

u Ψh,n∗ = Sh,n
Lh,n

u

PuΨh,n

u + PcΨh,n

c

(4.22)

(cid:179)

(cid:180)+ Qh,n.

The approximate solution at the time tn+1 is obtained by evaluating Ψh,n∗ at tn+1. A result analogous to
that of proposition 5 for the hybrid-c STDG method is stated in proposition 8. Updating algorithm 3.5 to

include the use of Nyström reconstruction yields algorithm 4.4.

138

Algorithm 4.4 Hybrid STDG timestep with Nyström reconstruction.

input: Initial condition ψn∗.

1: Compute Ψh,n

u

by solving (3.46a):

2: Compute Ψh,n

c

by solving (3.46b):

(cid:179)

= Qh,n.

u Ψh,n
Lh,n
u
(cid:180)

Lh,n

c

− Sh,n

c Ph,n

c

Ψh,n

c

= Sh,n

c Ph,n

c Ψh,n
u .

3: switch hybrid method do
4:

case hybrid-a:

Ψh,n∗ ← Ψh,n
u

+ RΨh,n

.

c

end case
case hybrid-c:

5:
6:
7:

8:

Compute Ψh,n∗ by solving (4.22):

u Ψh,n∗ = Sh,n
Lh,n

u

end case
9:
10: end switch

return: ψh,n∗

(cid:175)(cid:175)(cid:175)t=tn+1

(cid:179)PuΨh,n

u

+ PcΨh,n
c

(cid:180)+ Qh,n.

Proposition 8. Suppose that the same quadrature is applied to the collided and uncollided components of
the hybrid STDG method (3.46) and that the coefficients of the reconstructed approximation Ψh,n∗ are com-
puted through (4.22). Then the coefficients of the reconstructed approximation satisfy Ψh,n∗ = Ψh,n
u + Ψh,n
and Ψh,n∗ = Ψh,n where Ψh,n are the coefficients for the standard discrete ordinates method (3.35).

c

Proof. The proof follows the same form as that of proposition 5.

■

Hybrid-Ic IDC method

The construction of a Nyström reconstruction procedure for hybrid-I IDC methods is less obvious. Nys-

tröm reconstruction procedures were constructed for implicit Euler, DIRK, and STDG methods by us-

ing the approximations of the collided and uncollided stage values obtained by solving the split system

to approximate the scattering sources in the equations for the stages of the corresponding (un-split or

non-hybrid) discrete ordinates system. The approximations of the collided and uncollided stage values

for the hybrid-I IDC methods are computed by solving (3.53) and the corresponding expression for the

139

stage values of the non-hybrid IDC method is (3.51). At first glance, one might therefore suggest that a

Nyström reconstruction procedure for hybrid-I IDC methods should be constructed based on (3.51).

Upon closer inspection, it becomes clear that the collided and uncollided fluxes obtained from (3.53)

are incompatible with the form of (3.51) and that another approach must be taken. The reason that the

level; i.e.,(cid:169)ψ(cid:96),[p−1](cid:170)N

collided and uncollided components obtained from solving (3.53) are inconsistent with the form of (3.51)
is due to the fact that the operator L is applied to the provisional solutions from the previous correction
(cid:96)=1. This inconsistency can be avoided by defining a Nyström reconstruction pro-
cedure for hybrid-I IDC methods using the non-hybrid collocation scheme. The justification for this is

the observation that the provisional solutions computed by the IDC method converge to the stage values

of the associated collocation scheme.

Applying the collocation approximation (2.8) to the discrete ordinates system (3.9) yields

(cid:98)ψ = (cid:98)ψ0 − ∆t (Θ⊗ (L−SP))(cid:98)ψ+ ∆t (Θ⊗I)(cid:98)q,
q1(cid:180)T
ψ1(cid:180)T

ψN(cid:180)T(cid:184)

(cid:98)qT =(cid:183)(cid:179)

,...,

and

(cid:179)

,...,

(cid:98)ψT =(cid:183)(cid:179)

(cid:179)

q N(cid:180)T(cid:184)

where

are the vectors of stage values and sources, respectively, with qn = q (tn), (cid:98)ψ0 = 1N ⊗ ψ0, and 1N is the
N -dimensional column vector of ones. Let (cid:98)ψu and (cid:98)ψc denote the stacked vectors of provisional solu-

(4.23)

(4.24)

(4.25)

(4.26)

tions obtained from (3.53) after p correction iterations; i.e.,

u

u

ψ1,[p]

(cid:179)(cid:98)ψ[p]
(cid:179)(cid:98)ψ[p]∗

(cid:180)T =(cid:183)(cid:179)
(cid:180)T
and similarly for (cid:98)ψc. The reconstructed stages
(cid:180)T =(cid:183)(cid:179)(cid:98)ψ1,[p]∗
(cid:162)(cid:105)(cid:98)ψ[p]∗ = (cid:98)ψ0∗ + ∆t(cid:161)Θ⊗Su

(cid:162)+ ∆t(cid:161)Θ⊗Lu

(cid:180)T

,...,

(cid:179)

(cid:180)T(cid:184)

,

ψN ,[p]

u

,...,

∗

(cid:180)T(cid:184)

(cid:179)(cid:98)ψN ,[p]
(cid:162)(cid:98)ψ[p]
u +(cid:161)I ⊗Pc

for the hybrid-Ic IDC method are defined implicitly through the following modified form of (4.23):

(cid:104)(cid:161)I ⊗Iu
(cid:162)(cid:98)q, (4.27)
where (cid:98)ψ0∗ = 1N ⊗ ψ0∗. As before, the operator on the left of (4.27) can be inverted using a single trans-

(cid:105)+ ∆t(cid:161)Θ⊗Iu

(cid:162)(cid:104)(cid:161)I ⊗Pu

(cid:162)(cid:98)ψ[p]

c

port sweep. Updating algorithm 3.7 to include the use of the Nyström reconstruction (4.27) yields algo-

rithm 4.5.

140

Algorithm 4.5 Hybrid-Ia and hybrid-Ic IDC timestep.

input: Initial condition ψ0∗, number of correction iterations P.

u

1: for p = 0,...,P do
← ψ0∗.
2:
← 0.

ψ0,[p]
ψ0,[p]

3:
c
4: end for
5: for n = 1,..., N do
Compute ψn,[0]
6:
u

7:

Compute ψn,[0]

c

(cid:46) Set initial condition.

(cid:46) Prediction steps.

by solving (3.52a):

(cid:181)
Lu + 1
hn ∆t

(cid:182)

ψn,[0]

u

= 1
hn ∆t

ψn−1,[0]

u

+ qn.

by solving (3.52b):

(cid:181)
Lc + 1
hn ∆t

−ScPc

(cid:182)

ψn,[0]

c

= 1
hn ∆t

ψn−1,[0]

u

+ScPuψn,[0]

u

.

8: end for
9: for p = 1,...,P do
10:

Compute ψn,[p]

for n = 1,..., N do
(cid:181)
u
Lu + 1
hn ∆t

(cid:182)

(cid:46) Correction iterations.

by solving (3.53a):

ψn,[p]

u

= 1
hn ∆t

ψn−1,[p]

u

+ N(cid:88)
(cid:96)=1

(cid:181)
δn,(cid:96) − γn,(cid:96)
hn

(cid:182)

Luψ

(cid:96),[p−1]
u

+ N(cid:88)
(cid:96)=1

γn,(cid:96)
hn

(cid:96)

.

q

by solving (3.53b):

(cid:181)
Compute ψn,[p]
Lc + 1
hn ∆t

c
−ScPc

(cid:182)

ψn,[p]

c

= 1
hn ∆t

+ N(cid:88)
(cid:96)=1

c

ψn−1,[p]
(cid:181)
δn,(cid:96) − γn,(cid:96)
hn

+ScPuψn,[p]
(cid:182)(cid:183)

u

(Lc −ScPc) ψ

(cid:96),[p−1]
c

−ScPuψ

(cid:96),[p−1]
u

(cid:184)

.

11:

12:

17:
18:
19:

20:

end for

13:
14: end for

15: switch hybrid method do
16:
← ψN ,[P]

case hybrid-Ia:

ψN ,[P]

.

c

u

∗

end case
case hybrid-Ic:

+RψN ,[P]
Compute (cid:98)ψ[P]∗ by solving (4.27):
(cid:163)(cid:161)I ⊗Iu

(cid:162)+ ∆t(cid:161)Θ⊗Lu

(cid:162)(cid:164)(cid:98)ψ[P]∗ = (cid:98)ψ0∗ + ∆t(cid:161)Θ⊗Su

(cid:46) Reconstruction for final solution.

(cid:162)(cid:104)(cid:161)I ⊗Pu

(cid:162)(cid:98)ψ[P]

u

+(cid:161)I ⊗Pc

(cid:162)(cid:98)ψ[P]

c

(cid:105)+ ∆t(cid:161)Θ⊗Iu

(cid:162)(cid:98)q.

end case
21:
22: end switch

return: ψN ,[P]

∗

141

The hybrid-Ic IDC method has not been implemented at this time, and we include the description

of this method here only for the sake of completeness. We anticipate that the accuracy of the hybrid-Ic

IDC method would be similar to that of other integrators such as DIRK and STDG methods that leverage

a Nyström reconstruction. The extension of proposition 5 to the hybrid-Ic IDC method requires a some-

what weaker statement: One would only be able to say that the non-hybrid and hybrid-Ic IDC methods

produce identical results when the same quadrature is used if both IDC methods have been iterated to

convergence with respect to the corresponding collocation solution1. The disadvantage of the hybrid-Ic

IDC method is that it effectively requires the application of two sweep algorithms: one for computing the

collided and uncollided stages that operates on each of the temporal degrees of freedom independently,

and one for computing the reconstructed stages which are fully coupled.

The coupling that is present in the time dimension in the reconstruction step is similar to that of

STDG methods, and similar approaches can be applied to solve the reconstruction system (4.27). Our

current implementation of the methods described in this work leverages separate solvers for Runge-

Kutta methods and STDG methods. While we acknowledge that it is possible that this speaks to a design

deficiency of our implementation, we have generally found that the performance of the RK and STDG

solvers is maximized when different data layouts are used for each implementation that are tailored

towards their particular memory access patterns. It is not clear how using a sub-optimal data layout

for some parts of the hybrid-Ic IDC method would affect the overall performance of the method, or if

it may be better to transpose the data layout between parts of the method. Because of this complexity,

we have avoided the implementation of the hybrid-Ic IDC method in favor of pursuing work on the two-

grid or hybrid-II IDC methods discussed in section 4.2.2, for which the implementation of a Nyström

reconstruction procedure is much simpler.

4.2.2 Two-grid IDC methods

The hybrid-II IDC methods seek to leverage the defect correction properties of IDC methods to decrease

the error in the hybrid approach due to the use of a lower-resolution angular discretization in the collided

component. These methods have a strong similarity with standard two-grid methods for integral equa-

tions of the second kind (c.f. section 4.1.2). The construction of these methods is based on three ideas:

1To be clear, the term collocation solution in this case refers to the collocation approximation (4.23)

with respect to time, and not the discrete ordinates approximation in angle.

142

(i) apply the collision-based splitting (3.16) to the exact error equation (2.14c) for the discrete ordinates

system (3.9); (ii) compute a set of reconstructed provisional solutions at each correction iteration using

some reconstruction procedure; and (iii) apply the operators corresponding to the high-resolution dis-

crete ordinates approximation to the reconstructed provisional solutions to compute the residual terms

for each correction iteration. This last step is the fundamental mechanic of the defect correction ap-

proach: the residual should always be computed with respect to the operators corresponding to the

higher-resolution or higher-order approximation that is ultimately desired. In this case, the residual is

computed with respect to the operators obtained by applying a collocation approximation in time and a

high-resolution discrete ordinates approximation in angle.

The derivation here alternates between elements of the derivation of the IDC method given in sec-

tion 2.2.2 and the derivation of the hybrid discrete ordinates methods given in section 3.1.3, while at-

tempting to follow both for some measure of consistency. We begin at the continuum level: Let ψ(t) de-
(t) denote a provisional solution at correction level p − 1.

note the exact solution of (3.2) and let ψ(p−1)

∗

Let

and

e(p−1)(t) = ψ(t)− ψ(p−1)

∗

(t)

r (p−1)(t) = ∂t ψ(p−1)

(t)− q(t)
denote the error and residual, respectively, of the provisional solution ψ(p−1)
(2.14c) for ψ(p−1)

(t)+ (L−SP) ψ(p−1)

takes the form

∗

∗

∗

∗

∂t e(p−1)(t) = −(cid:161)(cid:101)L−SP(cid:162)e(p−1)(t)− ∂t

(cid:90) t

t0

(4.28)

(4.29)

. The exact error equation

r (p−1)(τ)d τ.

(4.30)

Note that the inflow of radiation at the boundary is accounted for through the residual term: the stream-
ing operator that is applied to the error e(p−1)(t) is that corresponding to the case of vacuum boundary
conditions.

Applying the collision-based splitting (3.16) to (4.30) with e(p−1) = e

(p−1)
u

+ e

(p−1)
c

term treated as a source term yields

(p−1)
u
(p−1)
c

∂t e

∂t e

(cid:90) t

(p−1)
u

= −(cid:101)Le
− ∂t
= −(cid:161)(cid:101)L−SP(cid:162)e
t0
(p−1)
c

r (p−1)(τ)d τ,
(p−1)
+SPe
u

.

143

and the residual

(4.31a)

(4.31b)

High- and low-resolution discrete ordinates approximations are now applied to the uncollided and col-

lided components of (4.31), respectively, to obtain

(cid:90) t
= −(cid:101)Lue(p−1)
− ∂t
(cid:162)e(p−1)
= −(cid:161)(cid:101)Lc −ScPc

t0

u

c

u

∂t e(p−1)
∂t e(p−1)

c

r (p−1)(τ)d τ,
+ScPue(p−1)

u

.

(4.32a)

(4.32b)

Note that the high-resolution discrete ordinates approximation used for the uncollided flux is applied to

the residual term.

Two cases were considered for the hybrid-I IDC methods discussed in section 3.2.3: relabeling only

the final step of the final correction iteration (hybrid-Ia) and relabeling after each substep of each correc-

tion iteration (hybrid-Ib). While relabeling only the final step can be advantageous when the relabeling

procedure that is used is very computationally expensive to apply, the two-grid methods described here

require a reconstruction to be performed for each stage of each correction iteration in order to compute

the residuals of the provisional solutions. It is possible to defer the reconstruction of all substeps for a

given correction level to the end of that correction iteration, but there is no performance advantage to

be gained from doing so. However in this case, there are advantages to applying the reconstruction pro-

cedure between substeps. Namely, computing the reconstructed solution after each substep results in a

simpler implementation and a reduction in the number of data vectors that must be stored, particularly

for the LS-IDC methods discussed in section 4.2.3. For these reasons, we consider only two-grid methods

wherein a reconstruction is applied after each substep.

When the reconstruction is applied between each substep, the initial set of provisional solutions are

given by (3.57). Applying implicit Euler approximations to the time derivatives in (4.32) with the assump-

tion that the initial conditions for the nth substep are given by en−1,[p−1]

∗

and zero for the uncollided

and collided components of the error, respectively, and expanding out the residual term yields

u

+(cid:179)
= −hn∆t (cid:101)Luen,[p−1]
N(cid:88)
− ∆t
(cid:104)(cid:161)(cid:101)Lc −ScPc
(cid:96)=1
= −hn∆t

(cid:180)
ψn−1,[p]
− ψn,[p−1]
(cid:104)(cid:161)Lu −SuPu
− q (cid:96)(cid:105)
∗
∗
(cid:162)ψ
(cid:96),[p−1]
∗
(cid:162)en,[p−1]
−ScPuen,[p−1]

γn,(cid:96)

,

u

c

en,[p−1]

u

en,[p−1]

c

(cid:105)

.

(4.33a)

(4.33b)

Correction iterations for the two-grid defect correction methods, which we also refer to as the hybrid-II

IDC methods, are based on the error equations (4.33).

144

Algorithm 4.6 Hybrid-II IDC and LS-IDC prediction.

input: Initial condition ψ0∗, number of correction iterations P.

1: for p = 0,...,P do
ψ0,[p]∗ ← ψ0∗.
2:
3: end for
4: for n = 1,..., N do
Compute ψn,[0]
5:
u

6:

Compute ψn,[0]

c

(cid:46) Set initial condition.

(cid:46) Prediction steps.

(cid:182)

ψn,[0]

u

= 1
hn ∆t

ψn−1,[0]
∗

+ qn.

(cid:182)

−ScPc

ψn,[0]

c

=ScPuψn,[0]

u

.

by solving (3.57a):

by solving (3.57b):

(cid:181)
Lu + 1
hn ∆t
(cid:181)
Lc + 1
hn ∆t

7:
8:

9:
10:
11:

12:

switch hybrid method do
ψn,[0]∗ ← ψn,[0]

case hybrid-IIb:

u

+Rψn,[0]

.

c

end case
case hybrid-IIc:

(cid:181)
Compute ψn,[0]∗
Lu + 1
hn ∆t

(cid:182)

by solving (4.16):
=Su

ψn,[0]∗

(cid:179)Puψn,[0]

u

+Pcψn,[0]

c

(cid:180)+ 1

hn ∆t

ψn−1,[0]
∗

+ qn.

end case
end switch

13:
14:
15: end for

Two subtypes of hybrid-II IDC methods are considered: b and c. The hybrid-IIb IDC method em-
ploys a reconstruction operator R that maps directly from the collided quadrature set to the uncollided

quadrature set. The reconstruction operator is applied to the collided and uncollided components of the

error approximation; that is, the reconstructed error approximation for each substep is given by

en,[p−1]
∗

= en,[p−1]

u

+Ren,[p−1]

c

.

This approximation of the error is used to define the updated provisional solution

ψn,[p]

∗

= ψn,[p−1]

∗

+ en,[p−1]

∗

.

(4.34)

(4.35)

The hybrid-IIc IDC method leverages a Nyström reconstruction procedure for the initial set of provi-

sional solutions that is analogous to that of (4.16). The correction iterations of the hybrid-IIc IDC method

use a Nyström reconstruction procedure that is constructed using the error equation for the non-hybrid

145

IDC method discussed in section 3.2.3. The non-hybrid IDC method described in section 3.2.3 is writ-

ten in a form that avoids explicitly constructing the error approximation; i.e., one solves (3.51) for the

corrected solution directly. One can nevertheless write out the equation for the error approximation

explicitly:

en,[p−1] = −hn∆t(cid:161)(cid:101)L−SP(cid:162)en,[p−1] +(cid:179)

ψn−1,[p] − ψn,[p−1](cid:180)

N(cid:88)

(cid:96)=1

− ∆t

γn,(cid:96)

(cid:104)
(L−SP) ψ(cid:96),[p−1] − q (cid:96)(cid:105)

.

(4.36)

The reconstruction equation for the hybrid-IIc IDC method is obtained by using the collided and uncol-

lided components of the error from (4.33) to approximate the scattering source in (4.36):

en,[p−1]
∗

= −hn∆t (cid:101)Luen,[p−1]
+(cid:179)

∗
ψn−1,[p]
∗

∗

+ hn∆tSu

(cid:180)− ∆t

− ψn,[p−1]

(cid:179)Puen,[p−1]
N(cid:88)

u

γn,(cid:96)

(cid:96)=1

(cid:180)

+Pcen,[p−1]
(cid:104)(cid:161)Lu −SuPu
(cid:162)ψ
(cid:96),[p−1]
∗

c

− q (cid:96)(cid:105)

.

(4.37)

Using (4.35) in (4.37) yields the following equation from which the updated solution ψn,[p]

∗

can be ob-

tained directly:

ψn,[p]

∗

∗

=− hn∆tLuψn,[p]
+ hn∆tLuψn,[p−1]

+ hn∆tSu
N(cid:88)

− ∆t

∗

γn,(cid:96)

(cid:96)=1

(cid:179)Puen,[p−1]

u

(cid:104)(cid:161)Lu −SuPu

+Pcen,[p−1]
(cid:96),[p−1]
∗

(cid:162)ψ

c

(cid:180)+ ψn−1,[p]
− q (cid:96)(cid:105)

∗

.

(4.38)

A result analogous to that of proposition 5 for the hybrid-IIc IDC method is stated in proposition 9. The

algorithm for the prediction stage of the hybrid-IIb and hybrid-IIc IDC timesteps is outlined in algo-

rithm 4.6, and the algorithm for the correction stages is outlined in algorithm 4.7.

Proposition 9. Suppose that the same quadrature is used to compute the initial provisional solutions
ψn,[0]

for the hybrid-II IDC method and that the reconstructed provisional solutions ψn,[0]∗

and ψn,[0]

are

u

c

computed using the Nyström reconstruction formula (4.16). Suppose further that this quadrature is used to

compute the collided and uncollided components of the error approximations of the hybrid-II IDC method

(4.33) and that the reconstructed error approximations en,[p−1]
are computed through (4.37). Then the re-
constructed error approximations satisfy en,[p−1]
= en,[p−1]
= en,[p−1] for each
substep n = 1,..., N, where en,[p−1] is the error approximation for the standard IDC discrete ordinates
method (4.36).

and en,[p−1]

+ en,[p−1]

∗

∗

∗

u

c

146

Proof. As before, we drop the subscripts used on the operators of the hybrid equations and omit the

terms due to the inflow of radiation at the boundary. Since the initial provisional solutions are com-

puted using the implicit Euler method, proposition 5 implies that the result holds for the initial set of

provisional solutions.

The remainder of the proof proceeds by induction on the substep index n and the correction index
p. Suppose that n ≥ 1, p > 1, and that proposition 9 is true for each provisional solution with substep

index less than n or correction index less than p. From (4.33) it follows that

(cid:104)(cid:161)L−SP(cid:162)ψ

− q (cid:96)(cid:105)

,

(cid:96),[p−1]
∗

(cid:161)I + hn∆tL(cid:162)en,[p−1]
(cid:161)I + hn∆tL(cid:162)en,[p−1]

u

c

∗

∗

(cid:180)

− ψn,[p−1]

= ψn−1,[p]

N(cid:88)
− ∆t
γn,(cid:96)
(cid:96)=1
+Pen,[p−1]
=(cid:161)I + hn∆tL(cid:162)(cid:179)
+ en,[p−1]

= hn∆tS(cid:179)Pen,[p−1]
(cid:161)I + hn∆tL(cid:162)en,[p−1]
= en,[p−1]

∗

u

u

c

.

Replacing the terms on the right side of (4.37) using (4.39) yields

(cid:180)

,

+ en,[p−1]

c

en,[p−1]

(4.39a)

(4.39b)

(4.40)

∗

from which the result en,[p−1]
fact that en,[p−1]
tering source in (4.37) and observing that the resulting equation for en,[p−1]
equation (4.36) for en,[p−1].

follows by inversion of the streaming operator. The
= en,[p−1] follows by simply combining the collided and uncollided terms of the scat-
is the same as that of the
■

∗

∗

u

c

4.2.3 Two-grid LS-IDC methods

As was the case for the non-hybrid and hybrid-I methods discussed in section 3.2, the hybrid-II LS-IDC

methods are obtained through simple modifications of the residual terms in the correction equations for

the hybrid-II IDC methods. Modifying the residual terms in (4.33) yields the following equations from

which the components of the error approximation for the hybrid-IIb and hybrid-IIc LS-IDC methods are

computed:

en,[p−1]

u

en,[p−1]

c

u

= −hn∆t (cid:101)Luen,[p−1]
+(cid:179)
= −hn∆t(cid:161)(cid:101)Lc −ScPc

(cid:104)(cid:161)Lu −SuPu
n−1(cid:88)
(cid:162)ψ
− ∆t
(cid:180)− ∆t
(cid:104)(cid:161)Lu −SuPu
N(cid:88)
− ψn,[p−1]
(cid:162)en,[p−1]
(cid:96)=n
+ hn∆tScPuen,[p−1]

ψn−1,[p]
∗

(cid:96)=1

γn,(cid:96)

γn,(cid:96)

∗

.

(cid:96),[p]∗

u

c

− q (cid:96)(cid:105)
(cid:162)ψ

(cid:96),[p−1]
∗

− q (cid:96)(cid:105)

,

(4.41a)

(4.41b)

147

Algorithm 4.7 Hybrid-II IDC correction.
1: for p = 1,...,P do
2:

Compute e

for n = 1,..., N do
n,[p−1]
(cid:182)
u

(cid:181)(cid:101)Lu + 1

e

n,[p−1]
u

hn ∆t

by solving (4.33a):

(cid:179)
ψn−1,[p]
∗

= 1
hn ∆t

(cid:46) Correction iterations.

− ψn,[p−1]

∗

(cid:180)− N(cid:88)

(cid:96)=1

γn,(cid:96)
hn

(cid:183)(cid:161)Lu −SuPu

(cid:162)ψ

(cid:96),[p−1]
∗

− q

(cid:96)(cid:184)

.

3:

4:

5:
6:

7:
8:
9:

10:

Compute e

n,[p]
c

by solving (4.33a):

(cid:181)(cid:101)Lc + 1

hn ∆t

(cid:182)

−ScPc

n,[p−1]
c

e

=ScPue

n,[p−1]
u

.

switch hybrid method do

+ e

n,[p−1]
u

+Re

n,[p−1]
c

.

case hybrid-IIb:
∗

∗ ← ψn,[p−1]
ψn,[p]
end case
case hybrid-IIc:
∗

(cid:181)
Compute ψn,[p]
Lu + 1
hn ∆t

ψn,[p]

(cid:182)

∗

(cid:179)Pue
by solving (4.38):
=Su
− N(cid:88)
(cid:96)=1

n,[p−1]
u
γn,(cid:96)
hn

(cid:180)+ 1

+Pce

(cid:183)(cid:161)Lu −SuPu

n,[p−1]
(cid:162)ψ
c

hn ∆t
− q

(cid:96),[p−1]
∗

+Luψn,[p−1]

∗

ψn−1,[p]
∗
(cid:96)(cid:184)

.

end case
end switch

11:
12:
13:
14: end for

end for

return: ψN ,[P]

∗

The reconstructed error approximations for the hybrid-IIb LS-IDC method are defined by (4.34). For

the hybrid-IIc LS-IDC method, reconstructed error approximations for each correction are computed by

solving a modified form of (4.37):

en,[p−1]
∗

= −hn∆t (cid:101)Luen,[p−1]
− ∆t

n−1(cid:88)

+ hn∆t

(cid:104)(cid:161)Lu −SuPu

γn,(cid:96)

∗

(cid:96)=1

(cid:179)Puen,[p−1]
(cid:162)ψ

u
(cid:96),[p]∗

+Pcen,[p−1]
− q (cid:96)(cid:105)− ∆t
N(cid:88)

c

ψn−1,[p]
∗

(cid:180)+(cid:179)
(cid:104)(cid:161)Lu −SuPu

− ψn,[p−1]
∗
(cid:162)ψ
(cid:96),[p−1]
∗

(cid:180)

γn,(cid:96)

(cid:96)=n

− q (cid:96)(cid:105)

(4.42)

Alternatively, one may compute the corrected solution directly using

(cid:179)

ψn,[p]

∗

(cid:180)+ hn∆tSu
− ψn−1,[p]
− q (cid:96)(cid:105)− ∆t
(cid:162)ψ

(cid:104)(cid:161)Lu −SuPu

(cid:179)Puen,[p−1]
N(cid:88)

(cid:96),[p]∗

∗

u

γn,(cid:96)

(cid:180)+ ψn−1,[p]
− q (cid:96)(cid:105)
(cid:162)ψ

(cid:96),[p−1]
∗

∗

(4.43)

,

c

+Pcen,[p−1]
(cid:104)(cid:161)Lu −SuPu

ψn,[p]

∗

=− hn∆tLu
n−1(cid:88)
− ∆t

γn,(cid:96)

(cid:96)=1

(cid:96)=n

148

Algorithm 4.8 Hybrid-II LS-IDC correction.
1: for p = 1,...,P do
2:

Compute e

for n = 1,..., N do
n,[p−1]
(cid:182)
u

(cid:181)(cid:101)Lu + 1

e

hn ∆t

n,[p−1]
u

Compute e

n,[p]
c

by solving (4.33a):

∗

− ψn,[p−1]
(cid:162)ψ
(cid:161)Lu −SuPu

(cid:180)+ N(cid:88)
(cid:96)=1
− N(cid:88)
(cid:96)=n

(cid:96),[p]∗

γn,(cid:96)
hn

(cid:96)

q

γn,(cid:96)
hn

(cid:182)

n,[p−1]
c

e

=ScPue

n,[p−1]
u

.

by solving (4.41a):

= 1
hn ∆t

(cid:179)
ψn−1,[p]
∗
− n−1(cid:88)
(cid:96)=1
(cid:181)(cid:101)Lc + 1

γn,(cid:96)
hn

−ScPc

hn ∆t

+ e

n,[p−1]
u

+Re

n,[p−1]
c

.

3:

4:

5:
6:

7:
8:
9:

10:

(cid:46) Correction iterations.

(cid:161)Lu −SuPu

(cid:162)ψ

(cid:96),[p−1]
∗

.

switch hybrid method do

case hybrid-IIb:
∗

∗ ← ψn,[p−1]
ψn,[p]
end case
case hybrid-IIc:
∗

Compute ψn,[p]

(cid:182)

(cid:181)
Lu + 1
hn ∆t

ψn,[p]

∗

by solving (4.43):

(cid:179)Pue
=Su
− n−1(cid:88)
(cid:96)=1

γn,(cid:96)
hn

n,[p−1]
u

+Pce
(cid:161)Lu −SuPu

n,[p−1]
c
(cid:162)ψ

(cid:96),[p]∗

(cid:180)+ 1
− N(cid:88)
(cid:96)=n

hn ∆t

+ N(cid:88)
+Luψn,[p−1]
(cid:96)=1
(cid:96),[p−1]
∗

ψn−1,[p]
∗
(cid:161)Lu −SuPu

(cid:162)ψ

∗

.

γn,(cid:96)
hn

γn,(cid:96)
hn

(cid:96)

q

end case
end switch

11:
12:
13:
14: end for

end for

return: ψN ,[P]

∗

which is analogous to (4.38). A result analogous to that of proposition 5 for the hybrid-IIc LS-IDC method

is stated in proposition 10. The algorithm for the prediction stage of the LS-IDC hybrid-II methods is the

same as that of algorithm 4.6, and the algorithm for the correction stage of the LS-IDC hybrid-II methods

is outlined in algorithm 4.8.

Proposition 10. Suppose that the same quadrature is used to compute the initial provisional solutions
for the hybrid-II LS-IDC method and that the reconstructed provisional solutions ψn,[0]∗
ψn,[0]

and ψn,[0]

u

c

are computed using the Nyström reconstruction formula (4.16). Suppose further that this quadrature is

used to compute the collided and uncollided components of the error approximations of the hybrid-II

149

LS-IDC method (4.41) and that the reconstructed error approximations en,[p−1]
(4.42). Then the reconstructed error approximations en,[p−1]
en,[p−1]
∗

and
= en,[p−1] for each substep n = 1,..., N, where en,[p−1] is the error approximation for the stan-

= en,[p−1]

satisfy en,[p−1]

∗

are computed through

∗

∗

+ en,[p−1]

c

u

dard IDC discrete ordinates method (4.36).

Proof. The proof is essentially the same as that of proposition 9.

■

4.3 Numerical tests for Nyström methods

The same set of tests used in section 3.4 are repeated for the hybrid methods using Nyström recon-

struction procedures. Results for two-grid methods (i.e., hybrid-II IDC and LS-IDC methods) are given

in section 4.4. Descriptions of the problems and test procedures are not repeated here – see section 3.4

for detailed descriptions of these tests.

4.3.1 One spatial dimension

We first examine the convergence behavior of hybrid methods using Nyström reconstruction procedures.

The results of the convergence tests for DIRK integrators are shown in figures 4.2 to 4.5 and for STDG

integrators in figure 4.1.

The hybrid methods introduced in chapter 3 displayed two phenomena in these convergence tests:

convergence saturation and order reduction. The saturation behavior of the hybrid methods remains

unchanged when Nyström reconstruction procedures are used, but significant differences are observed

with respect to the order reduction behavior of the methods. Whereas the S32,2 hybrid methods using

reconstruction procedures based on polynomial interpolation were observed to converge with second-

order accuracy after the saturation point, we observe that S32,2 hybrid methods using Nyström recon-

struction procedures converge with fifth-order accuracy after the saturation point. This suggests a con-
vergence rate of 2Nc + 1 for hybrid methods using Nyström reconstruction procedures and an angular
quadrature of order Nc in the collided component. Preliminary investigations into the order-reduction
phenomenon observed in the hybrid methods support this observation2: Specifically, investigations into

2Since the results of these investigations were not fully available when the present work was pub-
lished, this claim is, unfortunately, stated without justification. We hope to include a thorough justifica-
tion for the observed order reduction phenomenon in future publications.

150

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

−2

10

−4

10

−6

10

−8

10

−10

10

−12

2
L

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

−14

10

−2

10

−4

10

−6

10

−8

10

−10

10

−14

10

−12

2
L

10

S8
S4
S2
S32

S32,8 a
S32,4 a
S32,2 a

S32,8 c
S32,4 c
S32,2 c

Order 2
Order 3

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

(a) STDG3

S8
S4
S2
S32

S32,8 a
S32,4 a
S32,2 a

S32,8 c
S32,4 c
S32,2 c

Order 2
Order 4
Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) STDG5

Figure 4.1: L2 convergence studies of hybrid STDG methods for a sequence of smooth 1D problems un-
der a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various
values of ε approaching 0. The error in the scalar flux of each solution is approximated by comparing to
a reference solution using 218 spatial cells and an S32 discrete ordinates approximation in angle. Point
marks for hybrid solutions whose convergence is unhindered lie within the black boxes.

151

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

−2

10

−4

10

−6

10

−8

10

−10

10

−12

2
L

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 Ic
S32,4 Ic
S32,2 Ic

Order 2

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(a) SDIRK2, Hybrid-I

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

S32,8 IIa
S32,4 IIa
S32,2 IIa

S32,8 IIc
S32,4 IIc
S32,2 IIc

Order 2

−14

10

−2

10

−4

10

−6

10

−8

10

−10

10

−14

10

−12

2
L

10

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) SDIRK2, Hybrid-II

Figure 4.2: L2 convergence studies of SDIRK2 hybrid methods for a sequence of smooth 1D problems un-
der a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various
values of ε approaching 0. The error in the scalar flux of each solution is approximated by comparing to
a reference solution using 218 spatial cells and an S32 discrete ordinates approximation in angle. Point
marks for hybrid solutions whose convergence is unhindered lie within the black boxes.

152

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

−2

10

−4

10

−6

10

−8

10

−10

10

−12

2
L

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 Ic
S32,4 Ic
S32,2 Ic

Order 2
Order 3

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(a) SDIRK3, Hybrid-I

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

S32,8 IIa
S32,4 IIa
S32,2 IIa

S32,8 IIc
S32,4 IIc
S32,2 IIc

Order 2
Order 3

−14

10

−2

10

−4

10

−6

10

−8

10

−10

10

−14

10

−12

2
L

10

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) SDIRK3, Hybrid-II

Figure 4.3: L2 convergence studies of SDIRK3 hybrid methods for a sequence of smooth 1D problems un-
der a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various
values of ε approaching 0. The error in the scalar flux of each solution is approximated by comparing to
a reference solution using 218 spatial cells and an S32 discrete ordinates approximation in angle. Point
marks for hybrid solutions whose convergence is unhindered lie within the black boxes.

153

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

−2

10

−4

10

−6

10

−8

10

−10

10

−12

2
L

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 Ic
S32,4 Ic
S32,2 Ic

Order 2
Order 4
Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

(a) Kværnø7, Hybrid-I
ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

S32,8 IIa
S32,4 IIa
S32,2 IIa

S32,8 IIc
S32,4 IIc
S32,2 IIc

Order 2
Order 4
Order 5

−14

10

−2

10

−4

10

−6

10

−8

10

−10

10

−14

10

−12

2
L

10

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) Kværnø7, Hybrid-II

Figure 4.4: L2 convergence studies of Kværnø7 hybrid methods for a sequence of smooth 1D problems
under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for vari-
ous values of ε approaching 0. The error in the scalar flux of each solution is approximated by comparing
to a reference solution using 218 spatial cells and an S32 discrete ordinates approximation in angle. Point
marks for hybrid solutions whose convergence is unhindered lie within the black boxes.

154

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

−2

10

−4

10

−6

10

−8

10

−10

10

−12

2
L

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

S8
S4
S2
S32

S32,8 Ia
S32,4 Ia
S32,2 Ia

S32,8 Ic
S32,4 Ic
S32,2 Ic

Order 2
Order 4
Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(a) ARK8, Hybrid-I

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

S32,8 IIa
S32,4 IIa
S32,2 IIa

S32,8 IIc
S32,4 IIc
S32,2 IIc

Order 2
Order 4
Order 5

−14

10

−2

10

−4

10

−6

10

−8

10

−10

10

−14

10

−12

2
L

10

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) ARK8, Hybrid-II

Figure 4.5: L2 convergence studies of ARK8 hybrid methods for a sequence of smooth 1D problems under
a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time 0.5, and a CFL of 8 for various
values of ε approaching 0. The error in the scalar flux of each solution is approximated by comparing to
a reference solution using 218 spatial cells and an S32 discrete ordinates approximation in angle. Point
marks for hybrid solutions whose convergence is unhindered lie within the black boxes.

155

the order-reduction phenomenon indicate the existence of superconvergence behavior in the angular

moments of the hybrid solution with respect to the length of time that elapses between applications of

the reconstruction operation. The results suggest that the truncation error in the zeroth angular moment
corresponds to a global convergence rate of 2Nc + 1. Since the values obtained through reconstruction
procedures based on polynomial interpolation contain contributions from all of the angular moments

of the collided component, the superconvergence behavior of the zeroth moment is obscured by the

lower convergence rates of the higher-degree angular moments. In contrast, the Nyström reconstruction

procedures use only the zeroth angular moment of the collided component, and hence the superconver-

gence behavior in the zeroth moment is preserved during reconstruction. The fact that the truncation

error of the hybrid methods using Nyström reconstruction procedures is observed to be significantly

higher than that of previous methods alleviates concerns over order reduction for most practical pur-

poses. However, the issue of convergence saturation remains.

Overall, the use of Nyström reconstruction procedures improves the accuracy of the hybrid methods

by a significant margin. Whereas the S32,4 hybrid methods using reconstruction procedures based on

polynomial interpolation displayed increases in error over the reference S32 approximations in a small
number of tests (specifically, for fifth-order methods and ε = 1E−1), the S32,4 hybrid methods using Nys-
tröm reconstruction procedures do not display any discernible increase in error over the reference S32

approximation. While saturation is still a concern for S32,2 hybrid methods, accuracy in regimes with

low to moderate collision rates is significantly improved due to the fact that the Nyström reconstruction

procedures yield truncation errors with a much higher order of accuracy than reconstruction procedures

based on polynomial interpolation.

4.3.2 Two spatial dimensions

Hybrid methods using Nyström reconstruction procedures are now applied to the hohlraum test prob-

lem in order to evaluate their computational efficiency. The problem setup remains the same as before;

i.e., the final time, CFL, and spatial meshes used are the same as those outlined in section 3.4.2. Results

are presented in three forms as before, with minor modifications to each presentation.

First, we show plots comparing the scalar flux density and the error distribution of standard dis-

crete ordinates methods and hybrid discrete ordinates methods using Nyström reconstruction proce-

156

dures and a selection of angular quadrature resolutions. For CL quadratures, standard discrete ordinates

methods using an angular quadrature of order 64 are compared with hybrid discrete ordinates methods

using an angular quadrature of order 64 in the uncollided component and angular quadratures of var-

ious resolutions in the collided component. For TN quadratures, standard discrete ordinates methods

using an angular quadrature of order 32 are compared with hybrid discrete ordinates methods using an

angular quadrature of order 32 in the uncollided component and angular quadratures of various resolu-

tions in the collided component. Results for DIRK methods are shown in figures 4.6 to 4.13 and results

for STDG methods are shown in figures 4.14 and 4.15. The solve time (in seconds) and an approximation

∞

of the L

error are included for each solution.

As before, we observe that hybrid methods require significantly less solve time than the standard

discrete ordinates methods that they are compared to, and the reduced solve times are accompanied by

increases in solution error when very coarse angular approximations are applied to the collided com-

ponent of the hybrid methods. However, the structure of the additional error introduced by the hybrid

splitting is substantially different when Nyström reconstruction procedures are used than was observed

in the hybrid methods in chapter 3. Whereas the use of lower-resolution angular quadratures in the

collided component of the hybrid methods in chapter 3 introduced prominent ray effects in the scalar

flux distributions, it is notable that no visibly apparent ray effects due to the lower-resolution angular

quadrature can be observed in the scalar flux distributions obtained from hybrid methods using Nys-

tröm reconstruction procedures. Indeed, the plots of the error distributions clearly show that the Nys-

tröm reconstruction procedures yield errors that are more smoothly distributed over the spatial domain.

The disparities that are observed in the error distributions of the various hybrid methods due to

the choice of reconstruction procedure stem from inherent differences between the reconstruction ap-

proaches. The reconstruction procedures used in chapter 3 are applied locally at each spatial point and

map the values of the collided component to the higher-resolution angular quadrature without applying

any scaling to the original or resulting values. Thus any ray effects present in the collided component

are transferred into the reconstructed solution. The Nyström reconstruction procedures are composed

of two essential components that deviate significantly from this paradigm. First, an approximation of

the scattering source is computed using the collided and uncollided components: In doing so, the values

of the collided component are scaled by the material cross section in each region. For regions with very

low collision probabilities (where the low-resolution angular quadrature would be expected to exhibit

157

S64,2

S64,4

S64,8

S64,16

Runtime: 6.48E+02
∞
L
error: 3.68E-02

Runtime: 6.66E+02
∞
L
error: 7.39E-03

Runtime: 7.58E+02
∞
L
error: 6.13E-03

Runtime: 1.11E+03
∞
L
error: 5.96E-03

S64

Runtime: 7.40E+03
∞
L
error: 5.93E-03

(a) SDIRK2, Hybrid-Ic, CL

S64,2

S64,4

S64,8

S64,16

Runtime: 6.48E+02
∞
L
error: 3.89E-02

Runtime: 6.72E+02
∞
L
error: 1.22E-02

Runtime: 7.72E+02
∞
L
error: 5.96E-03

Runtime: 1.11E+03
∞
L
error: 5.95E-03

S64

Runtime: 7.40E+03
∞
L
error: 5.93E-03

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) SDIRK2, Hybrid-IIc, CL

Figure 4.6: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid SDIRK2 methods using CL quadratures applied to
error are included for each solution.
the hohlraum test problem. The solve time (in seconds) and the L

∞

158

S32,1

S32,2

S32,4

S32,8

Runtime: 6.52E+02
∞
L
error: 3.69E-02

Runtime: 6.70E+02
∞
L
error: 3.59E-03

Runtime: 7.75E+02
∞
L
error: 1.62E-03

Runtime: 1.11E+03
∞
L
error: 1.52E-03

S32

Runtime: 7.46E+03
∞
L
error: 1.50E-03

(a) SDIRK2, Hybrid-Ic, TN

S32,1

S32,2

S32,4

S32,8

Runtime: 6.51E+02
∞
error: 3.89E-02
L

Runtime: 6.71E+02
∞
error: 8.97E-03
L

Runtime: 7.65E+02
∞
error: 2.22E-03
L

Runtime: 1.13E+03
∞
error: 1.51E-03
L

S32

Runtime: 7.46E+03
∞
error: 1.50E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) SDIRK2, Hybrid-IIc, TN

Figure 4.7: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid SDIRK2 methods using TN quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

159

S64,2

S64,4

S64,8

S64,16

Runtime: 6.74E+02
∞
L
error: 4.04E-02

Runtime: 6.94E+02
∞
L
error: 1.18E-02

Runtime: 7.99E+02
∞
L
error: 9.71E-03

Runtime: 1.13E+03
∞
L
error: 7.79E-03

S64

Runtime: 7.53E+03
∞
L
error: 7.57E-03

(a) SDIRK3, Hybrid-Ic, CL

S64,2

S64,4

S64,8

S64,16

Runtime: 6.86E+02
∞
L
error: 4.00E-02

Runtime: 7.09E+02
∞
L
error: 1.19E-02

Runtime: 8.04E+02
∞
L
error: 7.80E-03

Runtime: 1.16E+03
∞
L
error: 7.68E-03

S64

Runtime: 7.53E+03
∞
L
error: 7.57E-03

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) SDIRK3, Hybrid-IIc, CL

Figure 4.8: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid SDIRK3 methods using CL quadratures applied to
error are included for each solution.
the hohlraum test problem. The solve time (in seconds) and the L

∞

160

S32,1

S32,2

S32,4

S32,8

Runtime: 6.85E+02
∞
L
error: 4.04E-02

Runtime: 7.22E+02
∞
L
error: 5.35E-03

Runtime: 8.06E+02
∞
L
error: 2.49E-03

Runtime: 1.15E+03
∞
L
error: 2.21E-03

S32

Runtime: 7.63E+03
∞
L
error: 2.27E-03

(a) SDIRK3, Hybrid-Ic, TN

S32,1

S32,2

S32,4

S32,8

Runtime: 7.42E+02
∞
error: 4.00E-02
L

Runtime: 7.35E+02
∞
error: 7.84E-03
L

Runtime: 8.40E+02
∞
error: 2.73E-03
L

Runtime: 1.21E+03
∞
error: 2.40E-03
L

S32

Runtime: 7.63E+03
∞
error: 2.27E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) SDIRK3, Hybrid-IIc, TN

Figure 4.9: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid SDIRK3 methods using TN quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

161

S64,2

S64,4

S64,8

S64,16

Runtime: 9.50E+02
∞
L
error: 2.52E-02

Runtime: 9.74E+02
∞
L
error: 1.49E-02

Runtime: 1.06E+03
∞
L
error: 1.34E-02

Runtime: 1.46E+03
∞
L
error: 8.12E-03

S64

Runtime: 8.95E+03
∞
L
error: 8.05E-03

(a) Kværnø7, Hybrid-Ic, CL

S64,2

S64,4

S64,8

S64,16

Runtime: 9.64E+02
∞
error: 1.28E-02
L

Runtime: 9.87E+02
∞
error: 8.32E-03
L

Runtime: 1.09E+03
∞
error: 8.09E-03
L

Runtime: 1.46E+03
∞
error: 8.04E-03
L

S64

Runtime: 8.95E+03
∞
error: 8.05E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) Kværnø7, Hybrid-IIc, CL

Figure 4.10: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid Kværnø7 methods using CL quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

162

S32,1

S32,2

S32,4

S32,8

Runtime: 9.47E+02
∞
L
error: 2.52E-02

Runtime: 9.99E+02
∞
L
error: 1.10E-02

Runtime: 1.09E+03
∞
L
error: 3.74E-03

Runtime: 1.50E+03
∞
L
error: 1.98E-03

S32

Runtime: 8.72E+03
∞
L
error: 2.22E-03

(a) Kværnø7, Hybrid-Ic, TN

S32,1

S32,2

S32,4

S32,8

Runtime: 9.86E+02
∞
error: 1.28E-02
L

Runtime: 1.03E+03
∞
error: 2.25E-03
L

Runtime: 1.12E+03
∞
error: 2.29E-03
L

Runtime: 1.51E+03
∞
error: 2.25E-03
L

S32

Runtime: 8.72E+03
∞
error: 2.22E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) Kværnø7, Hybrid-IIc, TN

Figure 4.11: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid Kværnø7 methods using TN quadratures applied
to the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solu-
tion.

∞

163

S64,2

S64,4

S64,8

S64,16

Runtime: 1.11E+03
∞
L
error: 2.41E-02

Runtime: 1.13E+03
∞
L
error: 1.21E-02

Runtime: 1.23E+03
∞
L
error: 9.82E-03

Runtime: 1.68E+03
∞
L
error: 7.75E-03

S64

Runtime: 9.31E+03
∞
L
error: 7.49E-03

(a) ARK8, Hybrid-Ic, CL

S64,2

S64,4

S64,8

S64,16

Runtime: 1.17E+03
∞
L
error: 7.11E-02

Runtime: 1.19E+03
∞
L
error: 2.34E-02

Runtime: 1.32E+03
∞
L
error: 7.56E-03

Runtime: 1.74E+03
∞
L
error: 7.50E-03

S64

Runtime: 9.31E+03
∞
L
error: 7.49E-03

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) ARK8, Hybrid-IIc, CL

Figure 4.12: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid ARK8 methods using CL quadratures applied to
error are included for each solution.
the hohlraum test problem. The solve time (in seconds) and the L

∞

164

S32,1

S32,2

S32,4

S32,8

Runtime: 1.12E+03
∞
L
error: 2.41E-02

Runtime: 1.15E+03
∞
L
error: 9.55E-03

Runtime: 1.24E+03
∞
L
error: 3.01E-03

Runtime: 1.72E+03
∞
L
error: 2.31E-03

S32

Runtime: 9.68E+03
∞
L
error: 2.37E-03

(a) ARK8, Hybrid-Ic, TN

S32,1

S32,2

S32,4

S32,8

Runtime: 1.20E+03
∞
L
error: 7.11E-02

Runtime: 1.21E+03
∞
L
error: 1.22E-02

Runtime: 1.36E+03
∞
L
error: 3.74E-03

Runtime: 1.78E+03
∞
L
error: 2.61E-03

S32

Runtime: 9.68E+03
∞
L
error: 2.37E-03

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) ARK8, Hybrid-IIc, TN

Figure 4.13: Comparison of scalar flux density (top rows) and deviation of scalar flux density from refer-
ence solution (bottom rows) of hybrid and non-hybrid ARK8 methods using TN quadratures applied to
the hohlraum test problem. The solve time (in seconds) and the L
error are included for each solution.

∞

165

S64,2

S64,4

S64,8

S64,16

Runtime: 5.04E+02
∞
L
error: 6.44E-02

Runtime: 4.96E+02
∞
L
error: 1.50E-02

Runtime: 5.95E+02
∞
L
error: 9.68E-03

Runtime: 9.61E+02
∞
L
error: 7.80E-03

S64

Runtime: 8.50E+03
∞
L
error: 7.50E-03

(a) STDG3, hybrid-c, CL

S32,1

S32,2

S32,4

S32,8

Runtime: 4.64E+02
∞
error: 6.45E-02
L

Runtime: 4.96E+02
∞
error: 8.53E-03
L

Runtime: 5.89E+02
∞
error: 2.85E-03
L

Runtime: 9.74E+02
∞
error: 2.27E-03
L

S32

Runtime: 8.51E+03
∞
error: 2.27E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) STDG3, hybrid-c, TN

Figure 4.14: Comparison of scalar flux density (top rows) and deviation of scalar flux density from ref-
erence solution (bottom rows) of hybrid and non-hybrid STDG3 methods using CL and TN quadratures
applied to the hohlraum test problem. The solve time (in seconds) and the L
error are included for
each solution.

∞

166

S64,2

S64,4

S64,8

S64,16

Runtime: 5.45E+02
∞
L
error: 6.10E-02

Runtime: 5.77E+02
∞
L
error: 2.01E-02

Runtime: 7.21E+02
∞
L
error: 1.69E-02

Runtime: 1.25E+03
∞
L
error: 8.06E-03

S64

Runtime: 1.13E+04
∞
L
error: 7.85E-03

(a) STDG5, hybrid-c, CL

S32,1

S32,2

S32,4

S32,8

Runtime: 5.47E+02
∞
error: 6.11E-02
L

Runtime: 5.80E+02
∞
error: 1.37E-02
L

Runtime: 7.23E+02
∞
error: 4.61E-03
L

Runtime: 1.25E+03
∞
error: 2.07E-03
L

S32

Runtime: 1.22E+04
∞
error: 2.29E-03
L

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

0
−1
−2
−3
−4
−1
−2
−3
−4
−5

(b) STDG5, hybrid-c, TN

Figure 4.15: Comparison of scalar flux density (top rows) and deviation of scalar flux density from ref-
erence solution (bottom rows) of hybrid and non-hybrid STDG5 methods using CL and TN quadratures
applied to the hohlraum test problem. The solve time (in seconds) and the L
error are included for
each solution.

∞

167

−1

10

−2

10

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(a) SDIRK2, Hybrid-Ic, CL

211

213

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(b) SDIRK2, Hybrid-Ic, TN

211

25

27
Runtime (seconds)

29

211

213

(c) SDIRK2, Hybrid-IIc, CL

25

29

27
Runtime (seconds)
(d) SDIRK2, Hybrid-IIc, TN

211

213

213

Figure 4.16: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the SDIRK2 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

168

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

10

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(a) SDIRK3, Hybrid-Ic, CL

211

213

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(b) SDIRK2, Hybrid-Ic, TN

211

25

27
Runtime (seconds)

29

211

213

(c) SDIRK3, Hybrid-IIc, CL

25

29

27
Runtime (seconds)
(d) SDIRK3, Hybrid-IIc, TN

211

213

213

Figure 4.17: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the SDIRK3 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

169

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

10

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

27
Runtime (seconds)

29

211

213

(a) Kværnø7, Hybrid-Ic, CL

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(b) Kværnø7, Hybrid-Ic, TN

211

213

25

27
Runtime (seconds)

29

211

213

(c) Kværnø7, Hybrid-IIc, CL

25

27
Runtime (seconds)

29

211

213

(d) Kværnø7, Hybrid-IIc, TN

Figure 4.18: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the Kværnø7 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

170

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

10

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

27

29

211

213

25

27

29

211

213

Runtime (seconds)
(a) ARK8, Hybrid-Ic, CL

Runtime (seconds)
(b) ARK8, Hybrid-Ic, TN

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

27

29

211

213

25

27

29

211

213

Runtime (seconds)
(c) ARK8, Hybrid-IIc, CL

Runtime (seconds)
(d) ARK8, Hybrid-IIc, TN

Figure 4.19: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on the ARK8 integrator applied to the holraum test
problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are shown in red
and other hybrid methods are shown in blue. Note the differences in scales.

171

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

10

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(a) STDG3, Hybrid-c, CL

211

213

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

29

27
Runtime (seconds)
(b) STDG3, Hybrid-c, TN

211

213

25

29

27
Runtime (seconds)
(c) STDG5, Hybrid-c, CL

211

213

25

29

27
Runtime (seconds)
(d) STDG5, Hybrid-c, TN

211

213

Figure 4.20: Solve times in seconds versus L2 (d(cid:126)x) distance to reference for standard (black) and hybrid
(red and blue) discrete ordinates methods based on STDG3 and STDG5 integrators applied to the hol-
raum test problem. Hybrid methods using angular quadratures that differ in order by a factor of 4 are
shown in red and other hybrid methods are shown in blue. Note the differences in scales.

172

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-82481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8ray effects) this scaling has a damping effect. The scattering source is therefore concentrated in regions

with higher collision probabilities where the low-resolution angular quadrature is likely to be sufficient

for acceptable accuracy. Hence, the amount of ray effects present in the scattering source is expected

to be limited. The computed approximation of the scattering source can nevertheless be inaccurate

due to ray effects in neighboring regions even though ray effects may not be present in the scattering

source itself. The second component of the Nyström reconstruction procedures is the inversion of the

high-resolution streaming operator via a transport sweep. This operation is non-local and distributes

particles from the approximate scattering source – and any errors contained therein – throughout the

spatial domain. Because this operation is computed using the higher-resolution angular quadrature, the

amount of ray effects introduced in this step is limited, resulting in the smoother error distributions that

are observed.

In addition to the differences observed in the overall structure of the error distributions, the Nyström

reconstruction procedures also yield solutions that are overall more accurate than the reconstruction

procedures considered in chapter 3. For example, the S32,2 hybrid-Ic DIRK methods using TN quadra-

∞

tures yield solutions with L

errors that are smaller than those of the corresponding hybrid-Ia DIRK

methods by factors of between 7 and 13. On the other hand, the Nyström reconstruction procedures are

generally more expensive than the procedures used in chapter 3. When Nyström reconstruction pro-

cedures are used, two transport sweeps must be performed for each stage using the higher-resolution

angular quadrature: one to compute the uncollided component and another to computed the recon-

structed solution. This means that hybrid methods using Nyström reconstruction procedures require at

most twice the solve time of the hybrid methods described in chapter 3.

Comparisons of the computational efficiency of hybrid methods using Nyström reconstruction pro-

cedures for a broader range of angular resolutions are shown for DIRK methods in figures 4.16 to 4.19

and for STDG methods in figure 4.20. As before, we observe that hybrid methods using angular quadra-

tures that differ in order by a factor of 4 tend to consistently lead to good computational efficiency rel-

ative to standard discrete ordinates methods and other hybrid methods. However, since using a Nys-

tröm reconstruction procedure significantly increases the overall accuracy of the hybrid solutions, the

efficiency comparisons that can be made against standard discrete ordinates methods are similarly im-

proved. In order to attain a level of accuracy comparable to or better than that of a standard discrete

ordinates method using an angular quadrature of order N , the hybrid methods in chapter 3 required

173

that an angular quadrature of order 2N be used in the uncollided component. This is no longer the case

when Nyström reconstruction procedures are used: We observe that hybrid methods using Nyström re-

construction procedures and angular quadratures of order N and N /4 in the uncollided and collided

components, respectively, yield solutions with comparable errors in less time than a standard discrete

ordinates method using an angular quadrature of order N . These hybrid methods reduce solve times by

factors of between 4 and 8 for the hohlraum test problem, with larger reductions observed when higher-

resolution angular quadratures are used.

We note that using angular quadratures that differ in order by a factor of 8 may also be an accept-

able choice in certain cases. However, the variability observed in the efficacy of these methods prevents

us from issuing a general recommendation for their use. When the angular quadratures applied to the

components of the hybrid methods differ in order by a factor of 8 or more, the vast majority of the overall

solve time of the methods is spent on the uncollided component. Hence further coarsening the reso-

lution of the angular quadrature that is applied to the collided component yields diminishing returns

in computational cost savings. Weighing this fact against the potential uncertainty in the accuracy of

these methods, we find that using angular quadratures that differ in order by a factor of 4 is likely a better

choice in most cases.

In chapter 3 it was observed that hybrid-IIa DIRK methods produced significantly more accurate

solutions than hybrid-Ia DIRK methods. The difference in accuracy between hybrid-IIc and hybrid-Ic

DIRK methods does not appear to be quite as significant. When angular quadratures that differ in order

by a factor of 4 or less are used, hybrid-IIc and hybrid-Ic DIRK methods yield solutions with comparable

levels of accuracy. When angular quadratures that differ in order by a factor of more than 8 are used,

hybrid-IIc DIRK methods yield marginally smaller errors than hybrid-Ic DIRK methods. However, the

difference in accuracy between hybrid-IIc and hybrid-Ic DIRK methods is much less than the difference

that was observed between hybrid-IIa and hybrid-Ia DIRK methods.

Comparisons between standard discrete ordinates methods using CL and TN quadratures and hy-

brid discrete ordinates methods using CL and TN quadratures that differ by a factor of 4 are shown in

figures 4.21 to 4.23. Hybrid methods using Nyström reconstruction procedures and the reconstruction

procedures described in chapter 3 are included. In each case, using a Nyström reconstruction approach

improves the accuracy of the hybrid methods. While the difference is comparatively small for hybrid-II

DIRK methods, the improvements observed for STDG hybrid methods and hybrid-I DIRK methods are

174

CL non-hybrid
CL hybrid-Ia
CL hybrid-Ic

TN non-hybrid
TN hybrid-Ia
TN hybrid-Ic

CL non-hybrid
CL hybrid-Ia
CL hybrid-Ic

TN non-hybrid
TN hybrid-Ia
TN hybrid-Ic

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

27
Runtime (seconds)

29

211

213

25

27
Runtime (seconds)

29

211

213

(a) STDG3

(b) STDG5

Figure 4.21: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black and
gray), hybrid-a (red and blue), and hybrid-c (pink and teal) discrete ordinates methods based on STDG
integrators and using various angular quadrature resolutions applied to the hohlraum test problem. CL
(gray, blue, and teal) and TN (black, red, and pink) angular quadratures are considered.

quite significant. It is clear that the computational cost of applying the Nyström reconstruction proce-

dures is appreciably larger than that of the reconstruction procedures used in chapter 3. Hybrid methods

using Nyström reconstruction procedures require between 15 to 55 percent more solve time than pre-

vious methods when angular quadratures that differ in order by a factor of four are used. However, the

increase in computational cost is more than outweighed by the increase in accuracy observed in most

cases.

Nyström reconstruction procedures improve the accuracy and computational efficiency of the hy-

brid methods by a clear margin. As a result, the effective memory requirements of the hybrid methods

are significantly reduced. The recommendation given in chapter 3 that a standard discrete ordinates

method using an angular quadrature of order N be replaced with a hybrid discrete ordinates method

using angular quadratures of order 2N and N /2 for the uncollided and collided components, respec-

tively, yields an increase in memory consumption by a factor of approximately 4.25. The revised rec-

ommendations given for hybrid methods using Nyström reconstructions – that angular quadratures of

order N and N /4 be used for the uncollided and collided components, respectively – requires only a mi-

nor increase in memory footprint over standard discrete ordinates methods. Since the resolution of the

175

 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16CL non-hybrid
CL hybrid-Ia
CL hybrid-Ic

TN non-hybrid
TN hybrid-Ia
TN hybrid-Ic

CL non-hybrid
CL hybrid-Ia
CL hybrid-Ic

TN non-hybrid
TN hybrid-Ia
TN hybrid-Ic

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

27
Runtime (seconds)

29

211

213

25

27
Runtime (seconds)

29

211

213

(a) SDIRK2, Hybrid-I

(b) SDIRK3, Hybrid-I

CL non-hybrid
CL hybrid-Ia
CL hybrid-Ic

TN non-hybrid
TN hybrid-Ia
TN hybrid-Ic

CL non-hybrid
CL hybrid-Ia
CL hybrid-Ic

TN non-hybrid
TN hybrid-Ia
TN hybrid-Ic

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

27
Runtime (seconds)

29

211

213

25

27

29

211

213

Runtime (seconds)

(c) Kværnø7, Hybrid-I

(d) ARK8, Hybrid-I

Figure 4.22: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black and
gray), hybrid-Ia (red and blue), and hybrid-Ic (pink and teal) discrete ordinates methods based on DIRK
integrators and using various angular quadrature resolutions applied to the hohlraum test problem. CL
(gray, blue, and teal) and TN (black, red, and pink) angular quadratures are considered.

176

 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16CL non-hybrid
CL hybrid-IIa
CL hybrid-IIc

TN non-hybrid
TN hybrid-IIa
TN hybrid-IIc

CL non-hybrid
CL hybrid-IIa
CL hybrid-IIc

TN non-hybrid
TN hybrid-IIa
TN hybrid-IIc

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

27
Runtime (seconds)

29

211

213

25

27
Runtime (seconds)

29

211

213

(a) SDIRK2, Hybrid-II

(b) SDIRK3, Hybrid-II

CL non-hybrid
CL hybrid-IIa
CL hybrid-IIc

TN non-hybrid
TN hybrid-IIa
TN hybrid-IIc

CL non-hybrid
CL hybrid-IIa
CL hybrid-IIc

TN non-hybrid
TN hybrid-IIa
TN hybrid-IIc

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

25

27
Runtime (seconds)

29

211

213

25

27

29

211

213

Runtime (seconds)

(c) Kværnø7, Hybrid-II

(d) ARK8, Hybrid-II

Figure 4.23: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black and
gray), hybrid-IIa (red and blue), and hybrid-IIc (pink and teal) discrete ordinates methods based on DIRK
integrators and using various angular quadrature resolutions applied to the hohlraum test problem. CL
(gray, blue, and teal) and TN (black, red, and pink) angular quadratures are considered.

177

 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16 248163264   12481632  4-18-216-432-84-18-216-432-88-216-432-864-168-216-432-864-16angular quadrature used for the uncollided component is the same as that of the corresponding stan-

dard discrete ordinates method, storing the uncollided components of the stages requires no increase in

memory footprint. The only increase comes from storing the collided components of the stages, which

requires one-sixteenth the memory of the corresponding non-hybrid method. The result is that, for each

time integrator, replacing a standard discrete ordinates method with a hybrid discrete ordinates method

using Nyström reconstruction increases memory consumption by approximately 6 percent.

4.4 Numerical tests for two-grid methods

The same set of tests used in sections 3.4 and 4.3 are repeated for two-grid methods based on IDC

and LS-IDC integrators. Descriptions of the problems and test procedures are not repeated here – see

section 3.4 for detailed descriptions of these tests.

4.4.1 One spatial dimension

We first examine the convergence behavior of the two-grid methods. Convergence tests for two-grid

methods are presented in a format that is different than that of previous convergence tests. In previous

convergence tests, results for one or more hybrid methods based on a single time-integration scheme

using angular quadratures of order 2, 4, and 8 in the collided component were presented in each figure.

Here each figure shows results for a single time integrator, hybrid method, and fixed angular resolution

with varying numbers of correction iterations. While all of the hybrid methods shown in these figures are

computed using IDC5 or LS-IDC5 integrators, the standard discrete ordinates methods that are included

are computed using the collocation integrator associated with each defect correction method, which

corresponds to the STDG5 scheme in both cases.

As additional correction iterations are performed, the solutions obtained from the IDC methods

would be expected to converge to the solutions obtained from the associated collocation scheme. How-

ever, the order of accuracy of the IDC methods is expected to be less than that of the collocation scheme

until sufficiently many correction iterations have been performed. In this case, one prediction and four

correction iterations are required in order to observe fifth-order accuracy from the IDC methods. As dis-

cussed in section 2.3.3, the linear stability of IDC methods is generally somewhat less robust than that

of the associated collocation scheme, and additional correction iterations can improve the stability of

the methods and diminish the extent of any order reduction that may occur. This is particularly true for

178

LS-IDC methods, which tend to converge more slowly towards the collocation solution than traditional

IDC methods.

The colors and styles of point marks used for hybrid methods in these convergence plots are selected

based on the above observations. Hybrid methods using four and five correction iterations are the focus

of these convergence plots, and hence a large starburst symbol has been used to emphasize these meth-

ods: Point marks for methods using other numbers of correction iterations are denoted using circles of

different sizes. The color that is used for each point mark emphasizes which solutions would ideally have

accuracy comparable to that of the associated collocation solution. Since the IDC5 and LS-IDC5 meth-

ods attain their maximum theoretical order of accuracy after four correction iterations, methods using

more than four correction iterations should have accuracy similar to that of the collocation scheme un-

less their accuracy has been inhibited in some manner. Therefore, a red color is used for point marks of

hybrid methods using more than four correction iterations, and a blue color is used for point marks of

hybrid methods using four or fewer correction iterations. The result is that, unless the accuracy of the

methods has been inhibited in some way, point marks for hybrid methods should satisfy the following

conditions: (i) red-colored marks should lie mostly or entirely within the black squares, and (ii) starburst

marks of any color should lie close to the black squares, though it would not necessarily be expected that

they lie within the black squares.

We also note that a different norm is used to compute solution errors than was used in previous con-

vergence tests. Errors in previous convergence tests were measured using the scalar flux of the solution

with respect to the norm (2.52) at the final simulation time. Here errors are measured using the angular

flux of the solution with respect to the semi-discrete L2(d(cid:126)xd(cid:126)Ω) norm

k=1 is the high-resolution angular quadrature used for the reference solution and the
uncollided components of the hybrid solutions (S32 in this case). The motivation for this is based on the

fact that the metric (4.44) tends to be somewhat more sensitive than (2.52). Hence behavior that may

otherwise be obscured when considering only the scalar flux should be more readily apparent in these

convergence tests.

The results of the convergence tests for methods based on the IDC5 integrator are shown in fig-

ures 4.24 and 4.26 and results for methods based on the LS-IDC5 integrator are shown in figures 4.25

179

(cid:195) Ku(cid:88)

k=1

where(cid:169)(cid:126)Ωu,k , ωu,k

e(t) =
(cid:170)Ku

(cid:90)

(cid:104)

(cid:179)

(cid:180)− ψref(cid:179)

ωu,k

X

ψ

(cid:126)x,(cid:126)Ωu,k , t

(cid:126)x,(cid:126)Ωu,k , t

(cid:180)(cid:105)2

(cid:33)1/2

d(cid:126)x

,

(4.44)

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

0 Corr.
1 Corr.
2 Corr.
3 Corr.
4 Corr.

5 Corr.
6 Corr.
7 Corr.
8 Corr.
9 Corr.

Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

(a) IDC5, hybrid-IIb, S32,2
ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

0 Corr.
1 Corr.
2 Corr.
3 Corr.
4 Corr.

5 Corr.
6 Corr.
7 Corr.
8 Corr.
9 Corr.

Order 5

−2

10

−4

10

−6

10

−8

10

−10

10

−12

10

−14

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)

(cid:126)Ω
d
(cid:126)x
d
(

2
L

−2

10

−4

10

−6

10

−8

10

−10

10

−12

10

−14

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)

(cid:126)Ω
d
(cid:126)x
d
(

2
L

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) IDC5, hybrid-IIc, S32,2

Figure 4.24: L2 convergence studies of IDC5 hybrid-IIb and hybrid-IIc S32,2 methods for a sequence
of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time
0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar flux of each solution is
approximated by comparing to a reference solution using 218 spatial cells and an S32 discrete ordinates
approximation in angle. Point marks for hybrid solutions whose convergence is unhindered lie within
the black boxes.

180

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

0 Corr.
1 Corr.
2 Corr.
3 Corr.
4 Corr.

5 Corr.
6 Corr.
7 Corr.
8 Corr.
9 Corr.

Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

(a) LS-IDC5, hybrid-IIb, S32,2
ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

0 Corr.
1 Corr.
2 Corr.
3 Corr.
4 Corr.

5 Corr.
6 Corr.
7 Corr.
8 Corr.
9 Corr.

Order 5

−2

10

−4

10

−6

10

−8

10

−10

10

−12

10

−14

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)

(cid:126)Ω
d
(cid:126)x
d
(

2
L

−2

10

−4

10

−6

10

−8

10

−10

10

−12

10

−14

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)

(cid:126)Ω
d
(cid:126)x
d
(

2
L

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) LS-IDC5, hybrid-IIc, S32,2

Figure 4.25: L2 convergence studies of LS-IDC5 hybrid-IIb and hybrid-IIc S32,2 methods for a sequence
of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time
0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar flux of each solution is
approximated by comparing to a reference solution using 218 spatial cells and an S32 discrete ordinates
approximation in angle. Point marks for hybrid solutions whose convergence is unhindered lie within
the black boxes.

181

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

0 Corr.
1 Corr.
2 Corr.
3 Corr.
4 Corr.

5 Corr.
6 Corr.
7 Corr.
8 Corr.
9 Corr.

Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

(a) IDC5, hybrid-IIb, S32,4
ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

0 Corr.
1 Corr.
2 Corr.
3 Corr.
4 Corr.

5 Corr.
6 Corr.
7 Corr.
8 Corr.
9 Corr.

Order 5

−2

10

−4

10

−6

10

−8

10

−10

10

−12

10

−14

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)

(cid:126)Ω
d
(cid:126)x
d
(

2
L

−2

10

−4

10

−6

10

−8

10

−10

10

−12

10

−14

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)

(cid:126)Ω
d
(cid:126)x
d
(

2
L

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) IDC5, hybrid-IIc, S32,4

Figure 4.26: L2 convergence studies of IDC5 hybrid-IIb and hybrid-IIc S32,4 methods for a sequence
of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time
0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar flux of each solution is
approximated by comparing to a reference solution using 218 spatial cells and an S32 discrete ordinates
approximation in angle. Point marks for hybrid solutions whose convergence is unhindered lie within
the black boxes.

182

ε = 1E+0

ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

0 Corr.
1 Corr.
2 Corr.
3 Corr.
4 Corr.

5 Corr.
6 Corr.
7 Corr.
8 Corr.
9 Corr.

Order 5

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

ε = 1E+0

(a) LS-IDC5, hybrid-IIb, S32,4
ε = 1E−1

ε = 1E−2

ε = 1E−3

S8
S4
S2
S32

0 Corr.
1 Corr.
2 Corr.
3 Corr.
4 Corr.

5 Corr.
6 Corr.
7 Corr.
8 Corr.
9 Corr.

Order 5

−2

10

−4

10

−6

10

−8

10

−10

10

−12

10

−14

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)

(cid:126)Ω
d
(cid:126)x
d
(

2
L

−2

10

−4

10

−6

10

−8

10

−10

10

−12

10

−14

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)

(cid:126)Ω
d
(cid:126)x
d
(

2
L

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

27

29

211 213 215 217

Number of spatial cells

Number of spatial cells

Number of spatial cells

Number of spatial cells

(b) LS-IDC5, hybrid-IIc, S32,4

Figure 4.27: L2 convergence studies of LS-IDC5 hybrid-IIb and hybrid-IIc S32,4 methods for a sequence
of smooth 1D problems under a diffusive scaling with σt = σs = 1, spatial domain z ∈ [−4,4], final time
0.5, and a CFL of 8 for various values of ε approaching 0. The error in the scalar flux of each solution is
approximated by comparing to a reference solution using 218 spatial cells and an S32 discrete ordinates
approximation in angle. Point marks for hybrid solutions whose convergence is unhindered lie within
the black boxes.

183

and 4.27. Overall, it is clear that the two-grid defect correction procedures significantly increase the

accuracy of the hybrid methods. The saturation and order reduction that was observed with previous

hybrid methods are both significantly reduced by the two-grid approach.

The degree of improvement that is observed strongly depends on the method. For the hybrid-IIb

methods, the order reduction phenomenon that was observed in previous hybrid methods is effectively

eliminated. Whereas reduction to second-order accuracy was observed with previous S32,2 hybrid meth-

ods using reconstruction procedures based on polynomial interpolation, the hybrid-IIb IDC and LS-IDC

methods are able to recover the expected fifth-order convergence rate. Convergence saturation is still ob-

served in the hybrid-IIb methods, though the error level at which this phenomenon occurs in the hybrid-

IIb IDC and LS-IDC methods is significantly smaller than was observed for previous hybrid methods.

Reductions in the saturation error of the S32,2 hybrid-IIb methods as additional correction iterations

are performed can be clearly observed in figures 4.24a and 4.25a. However, this saturation persists very

strongly in many tests even when more than four correction iterations are used, and the amount by which

the error in the saturation regime is reduced for each additional correction iteration is rather small. It is

interesting to note that the convergence behavior of the hybrid-IIb methods in the saturation regime is

distinctly different for IDC and LS-IDC integrators. In particular, the convergence of the hybrid-IIb IDC5

method is non-monotonic and somewhat erratic, while the behavior of the hybrid-IIb LS-IDC5 method

is much more consistent. At this time it is not clear what properties of the IDC and LS-IDC methods

may affect this behavior, though we note that the erratic behavior of the IDC5 method does not seem

unreasonable given the fact that it is known that the two-grid method on which the hybrid-II methods

are based does not converge monotonically (c.f. section 4.1.2 and [17]).

We observe that the hybrid-IIc methods are overall much more accurate than the hybrid-IIb meth-

ods. A significant degree of convergence saturation is observed for S32,2 hybrid-IIb methods, and a

moderate amount of saturation is observed for S32,4 hybrid-IIb methods below very small tolerances.

In contrast, the S32,2 hybrid-IIc methods show errors that could indicate a very small amount of sat-
uration behavior for only two data points (for ε = 1E−1 and mesh sizes of 211 and 212), and the S32,4
hybrid-IIc methods do not display any discernible measure of saturation behavior in any of these tests.

It is interesting to note that the advantage of using a Nyström reconstruction procedure for hybrid meth-

ods based on DIRK and STDG integrators was rather different than what is observed here. For hybrid

methods based on DIRK and STDG integrators, the advantage of using a Nyström reconstruction pro-

184

cedure was isolated to the order-reduction behavior of the methods – the saturation behavior remained

nearly identical. For hybrid-II IDC and LS-IDC methods, the two-grid approach eliminates the impact

of the order reduction phenomenon rather effectively irrespective of the reconstruction procedure that

is used, and the Nyström reconstruction procedure seems to primarily serve to reduce the impact of the

saturation phenomenon.

Comparing the IDC and LS-IDC methods, we observe differences in overall accuracy similar to what

was observed in section 2.4.1. Specifically, the LS-IDC methods generally produce larger errors for a

given number of correction iterations than the equivalent IDC method. We further observe that: (i) ig-

noring the impact of the saturation phenomenon, the difference in accuracy between the IDC and LS-

IDC methods is typically rather small, and (ii) the difference in accuracy seems to vary depending on the

value of ε, with larger differences observed for larger values of ε. In every case, an LS-IDC method using
p + 1 correction iterations yields an error comparable to or better than that of the corresponding IDC

method using p correction iterations. The increased computational cost of performing an additional

correction iteration is likely to be a worthwhile penalty in certain applications in exchange for the sig-

nificantly decreased memory requirements of the LS- IDC methods. This penalty is also expected to be

much less pronounced in cases where other considerations, such as the use of a combined space-time

defect correction approach, may motivate the use of additional correction iterations to reduce the error

contributions from components other than the time discretization.

4.4.2 Two spatial dimensions

Two-grid methods are now applied to the hohlraum test problem in order to evaluate their computa-

tional efficiency. The problem setup remains the same as before; i.e., the final time, CFL, and spatial

meshes used are the same as those outlined in section 3.4.2. As before, results are presented in three

forms, but modifications are made to each collection of figures to better illuminate the characteristics of

the two-grid methods.

First, we show plots comparing the error distribution of standard discrete ordinates methods and

two-grid discrete ordinates methods using a selection of angular quadrature resolutions. For CL quadra-

tures, standard discrete ordinates methods using an angular quadrature of order 64 are compared with

two-grid discrete ordinates methods using an angular quadrature of order 64 in the uncollided compo-

185

nent and angular quadratures of various resolutions in the collided component. For TN quadratures,

standard discrete ordinates methods using an angular quadrature of order 32 are compared with two-

grid discrete ordinates methods using an angular quadrature of order 32 in the uncollided component

and angular quadratures of various resolutions in the collided component. As in the convergence tests

in the preceding section, we present results for IDC and LS-IDC methods using a range of correction

iterations.

Results are shown for hybrid methods based on the IDC3 scheme in figures 4.28 to 4.31, for hybrid

methods based on the LS-IDC3 scheme in figures 4.32 to 4.35, for hybrid methods based on the IDC5

scheme in figures 4.36 to 4.39, and for hybrid methods based on the LS-IDC5 scheme in figures 4.40

to 4.43. In these figures, each of the four leftmost columns shows the error distribution for one of the

two-grid methods and the rightmost column shows the error distribution for the corresponding stan-

dard discrete ordinates method. The solutions in each row are computed using the specified number of

correction iterations.

Many of the same characteristics observed for the hybrid methods examined previously can be ob-

served here as well. Specifically, we observe that the hybrid methods require significantly less solve time

than the standard discrete ordinates methods using the same number of correction iterations, and the

reduced solve times are accompanied by increases in solution error when very coarse angular approxi-

mations are applied to the collided component of the hybrid methods and only a small number of cor-

rection iterations are used. We also observe that using a Nyström reconstruction procedure yields an

error profile that is more smoothly distributed over the spatial domain and does not contain the ray ef-

fects from the low-resolution angular quadrature that are found in the solutions of methods using other

reconstruction approaches.

In each case, the error of the hybrid methods is visibly reduced as additional correction iterations are

performed, and many methods are able to recover most of the accuracy of the standard discrete ordinates

approximation that they are compared to within several correction iterations. However, this reduction in

error is not accompanied by a change in the overall character of the error distribution. For example, ray

effects are still present in the hybrid-IIb solutions due to the lower-resolution angular quadrature that is

applied to the collided component, though they are reduced in magnitude after each iteration. Similarly,

the additional error in the hybrid-IIc solutions remains rather smoothly distributed over the spatial do-

main. We also observe that the amount by which each correction iteration reduces each solution’s error

186

is different for IDC and LS-IDC schemes. In particular, the error for IDC schemes is often marginally

smaller than for LS-IDC schemes. However, as in the convergence tests in section 4.4.1, the convergence

∞

of the IDC schemes is not consistent: For example, in figure 4.38 the L

error increases between one

and two correction iterations for both the S32,2 and S32,4 methods. The LS-IDC schemes to not appear

to exhibit this phenomenon.

Comparisons of the computational efficiency of hybrid-II methods for a broader range of angular

quadrature resolutions are shown for hybrid methods based on the IDC3 scheme in figures 4.44 to 4.47,

for hybrid methods based on the IDC5 scheme in figures 4.48 to 4.51, for hybrid methods based on the

LS-IDC3 scheme in figures 4.52 to 4.55, and for hybrid methods based on the LS-IDC5 scheme in fig-

ures 4.56 to 4.59.

Because the prediction steps are computed using the implicit Euler method, the efficiency results

obtained for the prediction steps are similar to what was observed with previous methods. However,

the accuracy of the hybrid methods improves significantly as additional correction iterations are per-

formed. These results suggest that using even very low resolution angular quadratures in the collided

component of the two-grid methods is reasonable provided that sufficiently many correction iterations

are performed. In theory, these methods could be iterated until some prescribed convergence tolerance

has been satisfied: this is how traditional two-grid methods for integral equations are used. However,

we are interested in asking whether it is possible to avoid applying an arbitrary number of correction

iterations. More specifically, we ask in what cases is it reasonable to apply only the minimum number of

correction iterations required to attain the maximum order of accuracy possible from the IDC scheme,

and no more, without sacrificing accuracy because of the hybrid splitting? We remind the reader that the

minimum number of correction iterations required for the IDC3 and LS-IDC3 schemes is two, and the

minimum number required for the IDC5 and LS-IDC5 schemes is four.

In previous sections of this work it was observed that hybrid methods using angular quadratures that

differ in order by a factor of four consistently produced good computational efficiency relative to stan-

dard discrete ordinates methods and other hybrid methods. We observe that this is also the case for the

two-grid methods discussed here. In particular, we observe that two-grid methods using the minimum

number of correction iterations prescribed by the IDC scheme and angular quadratures of order N and

N /4 in the uncollided and collided components respectively, yield solutions with comparable errors in

less time than a standard discrete ordinates method using an angular quadrature of order N . This obser-

187

S64,2

S64,4

S64,8

S64,16

Runtime: 1.69E+02
∞
L
error: 1.02E-01

Runtime: 1.86E+02
∞
L
error: 4.42E-02

Runtime: 2.57E+02
∞
L
error: 2.50E-02

Runtime: 5.16E+02
∞
L
error: 1.08E-02

S64

Runtime: 5.23E+03
∞
L
error: 7.11E-03

Runtime: 5.19E+02
∞
L
error: 2.24E-02

Runtime: 5.27E+02
∞
L
error: 1.09E-02

Runtime: 6.73E+02
∞
L
error: 7.91E-03

Runtime: 1.23E+03
∞
L
error: 8.06E-03

Runtime: 1.01E+04
∞
L
error: 7.43E-03

Runtime: 7.96E+02
∞
error: 8.00E-03
L

Runtime: 8.50E+02
∞
error: 7.93E-03
L

Runtime: 1.09E+03
∞
error: 7.47E-03
L

Runtime: 1.90E+03
∞
error: 7.60E-03
L

Runtime: 1.47E+04
∞
error: 7.48E-03
L

Runtime: 1.12E+03
∞
error: 7.47E-03
L

Runtime: 1.19E+03
∞
error: 7.63E-03
L

Runtime: 1.46E+03
∞
error: 7.45E-03
L

Runtime: 2.58E+03
∞
error: 7.48E-03
L

Runtime: 1.90E+04
∞
error: 7.50E-03
L

Runtime: 1.43E+03
∞
L
error: 7.51E-03

Runtime: 1.50E+03
∞
L
error: 7.54E-03

Runtime: 1.96E+03
∞
L
error: 7.49E-03

Runtime: 3.23E+03
∞
L
error: 7.49E-03

Runtime: 2.30E+04
∞
L
error: 7.50E-03

s
n
o
i
t
c
e
r
r
o
C
0

s
n
o
i
t
c
e
r
r
o
C
1

s
n
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i
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c
e
r
r
o
C
2

s
n
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i
t
c
e
r
r
o
C
3

s
n
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c
e
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r
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C
4

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.28: Deviation of scalar flux density from reference solution for hybrid-IIb IDC3 methods using
CL quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

188

S64,2

S64,4

S64,8

S64,16

Runtime: 3.21E+02
∞
L
error: 5.94E-02

Runtime: 3.47E+02
∞
L
error: 1.36E-02

Runtime: 4.15E+02
∞
L
error: 7.54E-03

Runtime: 6.34E+02
∞
L
error: 7.22E-03

S64

Runtime: 5.23E+03
∞
L
error: 7.11E-03

Runtime: 8.32E+02
∞
L
error: 8.37E-03

Runtime: 8.32E+02
∞
L
error: 7.85E-03

Runtime: 9.43E+02
∞
L
error: 7.41E-03

Runtime: 1.35E+03
∞
L
error: 7.41E-03

Runtime: 1.01E+04
∞
L
error: 7.43E-03

Runtime: 1.26E+03
∞
error: 7.51E-03
L

Runtime: 1.38E+03
∞
error: 7.55E-03
L

Runtime: 1.48E+03
∞
error: 7.47E-03
L

Runtime: 2.06E+03
∞
error: 7.47E-03
L

Runtime: 1.47E+04
∞
error: 7.48E-03
L

Runtime: 1.80E+03
∞
error: 7.49E-03
L

Runtime: 1.81E+03
∞
error: 7.51E-03
L

Runtime: 2.01E+03
∞
error: 7.49E-03
L

Runtime: 2.76E+03
∞
error: 7.49E-03
L

Runtime: 1.90E+04
∞
error: 7.50E-03
L

Runtime: 2.21E+03
∞
L
error: 7.50E-03

Runtime: 2.27E+03
∞
L
error: 7.50E-03

Runtime: 2.54E+03
∞
L
error: 7.50E-03

Runtime: 3.44E+03
∞
L
error: 7.50E-03

Runtime: 2.30E+04
∞
L
error: 7.50E-03

s
n
o
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C
0

s
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1

s
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2

s
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3

s
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r
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C
4

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.29: Deviation of scalar flux density from reference solution for hybrid-IIc IDC3 methods using
CL quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

189

S32,1

S32,2

S32,4

S32,8

Runtime: 1.71E+02
∞
L
error: 1.02E-01

Runtime: 1.88E+02
∞
L
error: 4.00E-02

Runtime: 2.63E+02
∞
L
error: 1.27E-02

Runtime: 4.75E+02
∞
L
error: 5.05E-03

S32

Runtime: 5.23E+03
∞
L
error: 2.27E-03

Runtime: 4.97E+02
∞
L
error: 2.30E-02

Runtime: 5.58E+02
∞
L
error: 7.63E-03

Runtime: 6.46E+02
∞
L
error: 2.17E-03

Runtime: 1.06E+03
∞
L
error: 2.21E-03

Runtime: 1.01E+04
∞
L
error: 2.30E-03

Runtime: 8.48E+02
∞
error: 7.09E-03
L

Runtime: 8.46E+02
∞
error: 1.95E-03
L

Runtime: 1.08E+03
∞
error: 2.15E-03
L

Runtime: 1.74E+03
∞
error: 2.24E-03
L

Runtime: 1.41E+04
∞
error: 2.28E-03
L

Runtime: 1.14E+03
∞
error: 2.39E-03
L

Runtime: 1.18E+03
∞
error: 2.20E-03
L

Runtime: 1.43E+03
∞
error: 2.24E-03
L

Runtime: 2.21E+03
∞
error: 2.26E-03
L

Runtime: 1.84E+04
∞
error: 2.27E-03
L

Runtime: 1.45E+03
∞
L
error: 2.28E-03

Runtime: 1.55E+03
∞
L
error: 2.26E-03

Runtime: 1.81E+03
∞
L
error: 2.26E-03

Runtime: 2.75E+03
∞
L
error: 2.26E-03

Runtime: 2.27E+04
∞
L
error: 2.27E-03

s
n
o
i
t
c
e
r
r
o
C
0

s
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o
i
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c
e
r
r
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C
1

s
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i
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c
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r
r
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C
2

s
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t
c
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r
r
o
C
3

s
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c
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r
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C
4

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.30: Deviation of scalar flux density from reference solution for hybrid-IIb IDC3 methods using
TN quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

190

S32,1

S32,2

S32,4

S32,8

Runtime: 3.25E+02
∞
L
error: 5.94E-02

Runtime: 3.43E+02
∞
L
error: 7.65E-03

Runtime: 4.03E+02
∞
L
error: 2.55E-03

Runtime: 6.24E+02
∞
L
error: 2.23E-03

S32

Runtime: 5.23E+03
∞
L
error: 2.27E-03

Runtime: 7.99E+02
∞
L
error: 8.32E-03

Runtime: 8.30E+02
∞
L
error: 2.65E-03

Runtime: 9.46E+02
∞
L
error: 2.31E-03

Runtime: 1.36E+03
∞
L
error: 2.26E-03

Runtime: 1.01E+04
∞
L
error: 2.30E-03

Runtime: 1.33E+03
∞
error: 2.88E-03
L

Runtime: 1.37E+03
∞
error: 2.31E-03
L

Runtime: 1.55E+03
∞
error: 2.24E-03
L

Runtime: 2.06E+03
∞
error: 2.27E-03
L

Runtime: 1.41E+04
∞
error: 2.28E-03
L

Runtime: 1.74E+03
∞
error: 2.40E-03
L

Runtime: 1.82E+03
∞
error: 2.24E-03
L

Runtime: 2.01E+03
∞
error: 2.25E-03
L

Runtime: 2.74E+03
∞
error: 2.26E-03
L

Runtime: 1.84E+04
∞
error: 2.27E-03
L

Runtime: 2.34E+03
∞
L
error: 2.27E-03

Runtime: 2.29E+03
∞
L
error: 2.26E-03

Runtime: 2.54E+03
∞
L
error: 2.26E-03

Runtime: 3.41E+03
∞
L
error: 2.26E-03

Runtime: 2.27E+04
∞
L
error: 2.27E-03

s
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s
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1

s
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c
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r
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2

s
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i
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c
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r
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C
3

s
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i
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c
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r
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C
4

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.31: Deviation of scalar flux density from reference solution for hybrid-IIc IDC3 methods using
TN quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

191

S64,2

S64,4

S64,8

S64,16

Runtime: 1.70E+02
∞
L
error: 1.02E-01

Runtime: 1.95E+02
∞
L
error: 4.42E-02

Runtime: 2.57E+02
∞
L
error: 2.50E-02

Runtime: 5.13E+02
∞
L
error: 1.08E-02

S64

Runtime: 5.23E+03
∞
L
error: 7.11E-03

Runtime: 4.93E+02
∞
L
error: 4.90E-02

Runtime: 5.18E+02
∞
L
error: 2.12E-02

Runtime: 6.65E+02
∞
L
error: 1.03E-02

Runtime: 1.21E+03
∞
L
error: 8.15E-03

Runtime: 1.01E+04
∞
L
error: 7.40E-03

Runtime: 7.93E+02
∞
error: 2.37E-02
L

Runtime: 8.44E+02
∞
error: 9.69E-03
L

Runtime: 1.06E+03
∞
error: 7.45E-03
L

Runtime: 1.90E+03
∞
error: 7.66E-03
L

Runtime: 1.47E+04
∞
error: 7.47E-03
L

Runtime: 1.10E+03
∞
error: 1.16E-02
L

Runtime: 1.17E+03
∞
error: 7.80E-03
L

Runtime: 1.45E+03
∞
error: 7.47E-03
L

Runtime: 2.57E+03
∞
error: 7.50E-03
L

Runtime: 1.91E+04
∞
error: 7.49E-03
L

Runtime: 1.42E+03
∞
L
error: 7.63E-03

Runtime: 1.49E+03
∞
L
error: 7.62E-03

Runtime: 1.85E+03
∞
L
error: 7.46E-03

Runtime: 3.22E+03
∞
L
error: 7.49E-03

Runtime: 2.38E+04
∞
L
error: 7.49E-03

s
n
o
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r
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0

s
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i
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s
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s
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s
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−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.32: Deviation of scalar flux density from reference solution for hybrid-IIb LS-IDC3 methods us-
ing CL quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

192

S64,2

S64,4

S64,8

S64,16

Runtime: 3.28E+02
∞
L
error: 5.94E-02

Runtime: 3.36E+02
∞
L
error: 1.36E-02

Runtime: 3.98E+02
∞
L
error: 7.54E-03

Runtime: 6.18E+02
∞
L
error: 7.22E-03

S64

Runtime: 5.23E+03
∞
L
error: 7.11E-03

Runtime: 8.97E+02
∞
L
error: 2.74E-02

Runtime: 9.30E+02
∞
L
error: 8.68E-03

Runtime: 1.05E+03
∞
L
error: 7.36E-03

Runtime: 1.46E+03
∞
L
error: 7.40E-03

Runtime: 1.01E+04
∞
L
error: 7.40E-03

Runtime: 1.47E+03
∞
error: 1.20E-02
L

Runtime: 1.52E+03
∞
error: 7.58E-03
L

Runtime: 1.70E+03
∞
error: 7.43E-03
L

Runtime: 2.26E+03
∞
error: 7.44E-03
L

Runtime: 1.47E+04
∞
error: 7.47E-03
L

Runtime: 2.05E+03
∞
error: 7.49E-03
L

Runtime: 2.11E+03
∞
error: 7.53E-03
L

Runtime: 2.34E+03
∞
error: 7.47E-03
L

Runtime: 3.07E+03
∞
error: 7.47E-03
L

Runtime: 1.91E+04
∞
error: 7.49E-03
L

Runtime: 2.63E+03
∞
L
error: 7.49E-03

Runtime: 2.72E+03
∞
L
error: 7.51E-03

Runtime: 2.96E+03
∞
L
error: 7.49E-03

Runtime: 3.87E+03
∞
L
error: 7.49E-03

Runtime: 2.38E+04
∞
L
error: 7.49E-03

s
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s
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i
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c
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1

s
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2

s
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3

s
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−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.33: Deviation of scalar flux density from reference solution for hybrid-IIc LS-IDC3 methods us-
ing CL quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

193

S32,1

S32,2

S32,4

S32,8

Runtime: 1.70E+02
∞
L
error: 1.02E-01

Runtime: 1.87E+02
∞
L
error: 4.00E-02

Runtime: 2.52E+02
∞
L
error: 1.27E-02

Runtime: 4.71E+02
∞
L
error: 5.05E-03

S32

Runtime: 5.23E+03
∞
L
error: 2.27E-03

Runtime: 4.88E+02
∞
L
error: 4.91E-02

Runtime: 5.15E+02
∞
L
error: 1.73E-02

Runtime: 6.40E+02
∞
L
error: 4.95E-03

Runtime: 1.05E+03
∞
L
error: 2.06E-03

Runtime: 1.01E+04
∞
L
error: 2.30E-03

Runtime: 8.25E+02
∞
error: 2.37E-02
L

Runtime: 8.69E+02
∞
error: 6.73E-03
L

Runtime: 9.93E+02
∞
error: 1.93E-03
L

Runtime: 1.68E+03
∞
error: 2.15E-03
L

Runtime: 1.44E+04
∞
error: 2.28E-03
L

Runtime: 1.09E+03
∞
error: 1.16E-02
L

Runtime: 1.21E+03
∞
error: 2.83E-03
L

Runtime: 1.37E+03
∞
error: 2.09E-03
L

Runtime: 2.16E+03
∞
error: 2.20E-03
L

Runtime: 1.84E+04
∞
error: 2.27E-03
L

Runtime: 1.40E+03
∞
L
error: 5.71E-03

Runtime: 1.47E+03
∞
L
error: 2.11E-03

Runtime: 1.74E+03
∞
L
error: 2.19E-03

Runtime: 2.69E+03
∞
L
error: 2.24E-03

Runtime: 2.28E+04
∞
L
error: 2.26E-03

s
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0

s
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i
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c
e
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1

s
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c
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o
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2

s
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o
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3

s
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c
e
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C
4

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.34: Deviation of scalar flux density from reference solution for hybrid-IIb LS-IDC3 methods us-
ing TN quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

194

S32,1

S32,2

S32,4

S32,8

Runtime: 3.18E+02
∞
L
error: 5.94E-02

Runtime: 3.38E+02
∞
L
error: 7.65E-03

Runtime: 3.99E+02
∞
L
error: 2.55E-03

Runtime: 6.16E+02
∞
L
error: 2.23E-03

S32

Runtime: 5.23E+03
∞
L
error: 2.27E-03

Runtime: 9.02E+02
∞
L
error: 2.74E-02

Runtime: 9.31E+02
∞
L
error: 3.58E-03

Runtime: 1.05E+03
∞
L
error: 2.34E-03

Runtime: 1.45E+03
∞
L
error: 2.24E-03

Runtime: 1.01E+04
∞
L
error: 2.30E-03

Runtime: 1.48E+03
∞
error: 1.21E-02
L

Runtime: 1.52E+03
∞
error: 2.42E-03
L

Runtime: 1.73E+03
∞
error: 2.27E-03
L

Runtime: 2.26E+03
∞
error: 2.25E-03
L

Runtime: 1.44E+04
∞
error: 2.28E-03
L

Runtime: 2.10E+03
∞
error: 5.49E-03
L

Runtime: 2.12E+03
∞
error: 2.29E-03
L

Runtime: 2.32E+03
∞
error: 2.24E-03
L

Runtime: 3.06E+03
∞
error: 2.25E-03
L

Runtime: 1.84E+04
∞
error: 2.27E-03
L

Runtime: 2.64E+03
∞
L
error: 2.50E-03

Runtime: 2.74E+03
∞
L
error: 2.24E-03

Runtime: 2.99E+03
∞
L
error: 2.24E-03

Runtime: 3.91E+03
∞
L
error: 2.26E-03

Runtime: 2.28E+04
∞
L
error: 2.26E-03

s
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s
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s
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2

s
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3

s
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c
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C
4

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.35: Deviation of scalar flux density from reference solution for hybrid-IIc LS-IDC3 methods us-
ing TN quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

195

S64,2

S64,4

S64,8

S64,16

Runtime: 1.26E+02
∞
L
error: 1.04E-01

Runtime: 1.36E+02
∞
L
error: 4.46E-02

Runtime: 1.88E+02
∞
L
error: 2.70E-02

Runtime: 3.95E+02
∞
L
error: 1.29E-02

S64

Runtime: 4.01E+03
∞
L
error: 7.38E-03

Runtime: 3.35E+02
∞
L
error: 1.65E-02

Runtime: 3.56E+02
∞
L
error: 9.77E-03

Runtime: 4.56E+02
∞
L
error: 8.01E-03

Runtime: 8.68E+02
∞
L
error: 8.20E-03

Runtime: 7.74E+03
∞
L
error: 7.89E-03

Runtime: 5.44E+02
∞
error: 1.17E-02
L

Runtime: 5.74E+02
∞
error: 9.31E-03
L

Runtime: 7.70E+02
∞
error: 7.88E-03
L

Runtime: 1.33E+03
∞
error: 7.84E-03
L

Runtime: 1.14E+04
∞
error: 7.92E-03
L

Runtime: 7.54E+02
∞
error: 8.71E-03
L

Runtime: 7.95E+02
∞
error: 8.52E-03
L

Runtime: 9.98E+02
∞
error: 7.90E-03
L

Runtime: 1.91E+03
∞
error: 7.80E-03
L

Runtime: 1.55E+04
∞
error: 7.90E-03
L

Runtime: 9.57E+02
∞
L
error: 8.12E-03

Runtime: 1.00E+03
∞
L
error: 8.05E-03

Runtime: 1.33E+03
∞
L
error: 7.85E-03

Runtime: 2.26E+03
∞
L
error: 7.82E-03

Runtime: 1.84E+04
∞
L
error: 7.88E-03

Runtime: 1.16E+03
∞
L
error: 7.87E-03

Runtime: 1.29E+03
∞
L
error: 7.90E-03

Runtime: 1.49E+03
∞
L
error: 7.86E-03

Runtime: 2.72E+03
∞
L
error: 7.84E-03

Runtime: 2.17E+04
∞
L
error: 7.87E-03

s
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c
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C
0

s
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i
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e
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s
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2

s
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3

s
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e
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C
4

s
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c
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C
5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.36: Deviation of scalar flux density from reference solution for hybrid-IIb IDC5 methods using
CL quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

196

S64,2

S64,4

S64,8

S64,16

Runtime: 2.35E+02
∞
L
error: 5.90E-02

Runtime: 2.53E+02
∞
L
error: 1.43E-02

Runtime: 2.95E+02
∞
L
error: 8.63E-03

Runtime: 4.73E+02
∞
L
error: 7.50E-03

S64

Runtime: 4.01E+03
∞
L
error: 7.38E-03

Runtime: 5.68E+02
∞
L
error: 9.14E-03

Runtime: 5.86E+02
∞
L
error: 8.27E-03

Runtime: 7.13E+02
∞
L
error: 7.94E-03

Runtime: 1.01E+03
∞
L
error: 7.91E-03

Runtime: 7.74E+03
∞
L
error: 7.89E-03

Runtime: 8.87E+02
∞
error: 8.13E-03
L

Runtime: 9.16E+02
∞
error: 7.92E-03
L

Runtime: 1.05E+03
∞
error: 7.92E-03
L

Runtime: 1.54E+03
∞
error: 7.92E-03
L

Runtime: 1.14E+04
∞
error: 7.92E-03
L

Runtime: 1.21E+03
∞
error: 7.96E-03
L

Runtime: 1.27E+03
∞
error: 7.86E-03
L

Runtime: 1.42E+03
∞
error: 7.90E-03
L

Runtime: 2.07E+03
∞
error: 7.90E-03
L

Runtime: 1.55E+04
∞
error: 7.90E-03
L

Runtime: 1.63E+03
∞
L
error: 7.92E-03

Runtime: 1.58E+03
∞
L
error: 7.86E-03

Runtime: 1.79E+03
∞
L
error: 7.88E-03

Runtime: 2.57E+03
∞
L
error: 7.88E-03

Runtime: 1.84E+04
∞
L
error: 7.88E-03

Runtime: 1.86E+03
∞
L
error: 7.89E-03

Runtime: 1.92E+03
∞
L
error: 7.86E-03

Runtime: 2.16E+03
∞
L
error: 7.87E-03

Runtime: 3.25E+03
∞
L
error: 7.87E-03

Runtime: 2.17E+04
∞
L
error: 7.87E-03

s
n
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i
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c
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C
0

s
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o
i
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c
e
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C
1

s
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2

s
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i
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3

s
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C
4

s
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C
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−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.37: Deviation of scalar flux density from reference solution for hybrid-IIc IDC5 methods using
CL quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

197

S32,1

S32,2

S32,4

S32,8

Runtime: 1.26E+02
∞
L
error: 1.04E-01

Runtime: 1.38E+02
∞
L
error: 4.06E-02

Runtime: 1.83E+02
∞
L
error: 1.29E-02

Runtime: 3.78E+02
∞
L
error: 4.90E-03

S32

Runtime: 4.00E+03
∞
L
error: 2.30E-03

Runtime: 3.36E+02
∞
L
error: 1.66E-02

Runtime: 3.56E+02
∞
L
error: 4.52E-03

Runtime: 4.50E+02
∞
L
error: 2.31E-03

Runtime: 7.92E+02
∞
L
error: 2.53E-03

Runtime: 7.73E+03
∞
L
error: 2.40E-03

Runtime: 5.45E+02
∞
error: 1.07E-02
L

Runtime: 5.75E+02
∞
error: 6.47E-03
L

Runtime: 7.06E+02
∞
error: 2.89E-03
L

Runtime: 1.21E+03
∞
error: 2.58E-03
L

Runtime: 1.17E+04
∞
error: 2.35E-03
L

Runtime: 7.57E+02
∞
error: 6.54E-03
L

Runtime: 8.01E+02
∞
error: 3.58E-03
L

Runtime: 9.72E+02
∞
error: 2.59E-03
L

Runtime: 1.63E+03
∞
error: 2.45E-03
L

Runtime: 1.50E+04
∞
error: 2.31E-03
L

Runtime: 9.52E+02
∞
L
error: 3.03E-03

Runtime: 9.96E+02
∞
L
error: 2.46E-03

Runtime: 1.21E+03
∞
L
error: 2.35E-03

Runtime: 2.03E+03
∞
L
error: 2.34E-03

Runtime: 1.89E+04
∞
L
error: 2.30E-03

Runtime: 1.21E+03
∞
L
error: 2.28E-03

Runtime: 1.22E+03
∞
L
error: 2.20E-03

Runtime: 1.47E+03
∞
L
error: 2.28E-03

Runtime: 2.42E+03
∞
L
error: 2.30E-03

Runtime: 2.20E+04
∞
L
error: 2.29E-03

s
n
o
i
t
c
e
r
r
o
C
0

s
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o
i
t
c
e
r
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C
1

s
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i
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c
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r
o
C
2

s
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o
i
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c
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s
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C
4

s
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c
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C
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−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.38: Deviation of scalar flux density from reference solution for hybrid-IIb IDC5 methods using
TN quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

198

S32,1

S32,2

S32,4

S32,8

Runtime: 2.42E+02
∞
L
error: 5.91E-02

Runtime: 2.54E+02
∞
L
error: 6.50E-03

Runtime: 2.95E+02
∞
L
error: 2.52E-03

Runtime: 4.73E+02
∞
L
error: 2.16E-03

S32

Runtime: 4.00E+03
∞
L
error: 2.30E-03

Runtime: 5.83E+02
∞
L
error: 5.25E-03

Runtime: 6.22E+02
∞
L
error: 2.44E-03

Runtime: 6.74E+02
∞
L
error: 2.51E-03

Runtime: 1.01E+03
∞
L
error: 2.41E-03

Runtime: 7.73E+03
∞
L
error: 2.40E-03

Runtime: 8.97E+02
∞
error: 3.83E-03
L

Runtime: 9.17E+02
∞
error: 2.84E-03
L

Runtime: 1.05E+03
∞
error: 2.58E-03
L

Runtime: 1.54E+03
∞
error: 2.38E-03
L

Runtime: 1.17E+04
∞
error: 2.35E-03
L

Runtime: 1.22E+03
∞
error: 3.43E-03
L

Runtime: 1.25E+03
∞
error: 2.58E-03
L

Runtime: 1.42E+03
∞
error: 2.44E-03
L

Runtime: 2.07E+03
∞
error: 2.33E-03
L

Runtime: 1.50E+04
∞
error: 2.31E-03
L

Runtime: 1.54E+03
∞
L
error: 2.53E-03

Runtime: 1.61E+03
∞
L
error: 2.35E-03

Runtime: 1.96E+03
∞
L
error: 2.33E-03

Runtime: 2.72E+03
∞
L
error: 2.30E-03

Runtime: 1.89E+04
∞
L
error: 2.30E-03

Runtime: 1.86E+03
∞
L
error: 2.29E-03

Runtime: 1.92E+03
∞
L
error: 2.27E-03

Runtime: 2.16E+03
∞
L
error: 2.29E-03

Runtime: 3.07E+03
∞
L
error: 2.29E-03

Runtime: 2.20E+04
∞
L
error: 2.29E-03

s
n
o
i
t
c
e
r
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C
0

s
n
o
i
t
c
e
r
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1

s
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o
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2

s
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i
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c
e
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3

s
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i
t
c
e
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r
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4

s
n
o
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e
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r
o
C
5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.39: Deviation of scalar flux density from reference solution for hybrid-IIc IDC5 methods using
TN quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

199

S64,2

S64,4

S64,8

S64,16

Runtime: 3.05E+02
∞
L
error: 1.04E-01

Runtime: 1.36E+02
∞
L
error: 4.46E-02

Runtime: 1.85E+02
∞
L
error: 2.70E-02

Runtime: 3.81E+02
∞
L
error: 1.29E-02

S64

Runtime: 4.01E+03
∞
L
error: 7.38E-03

Runtime: 3.61E+02
∞
L
error: 4.38E-02

Runtime: 3.82E+02
∞
L
error: 1.62E-02

Runtime: 4.82E+02
∞
L
error: 9.84E-03

Runtime: 8.95E+02
∞
L
error: 8.91E-03

Runtime: 8.33E+03
∞
L
error: 7.73E-03

Runtime: 5.96E+02
∞
error: 1.84E-02
L

Runtime: 6.52E+02
∞
error: 9.23E-03
L

Runtime: 7.76E+02
∞
error: 8.00E-03
L

Runtime: 1.43E+03
∞
error: 8.13E-03
L

Runtime: 1.23E+04
∞
error: 7.81E-03
L

Runtime: 8.31E+02
∞
error: 8.09E-03
L

Runtime: 8.72E+02
∞
error: 8.31E-03
L

Runtime: 1.07E+03
∞
error: 7.90E-03
L

Runtime: 1.99E+03
∞
error: 7.92E-03
L

Runtime: 1.54E+04
∞
error: 7.83E-03
L

Runtime: 1.02E+03
∞
L
error: 8.19E-03

Runtime: 1.01E+03
∞
L
error: 8.12E-03

Runtime: 1.26E+03
∞
L
error: 7.95E-03

Runtime: 2.32E+03
∞
L
error: 7.92E-03

Runtime: 1.82E+04
∞
L
error: 7.84E-03

Runtime: 1.17E+03
∞
L
error: 7.93E-03

Runtime: 1.23E+03
∞
L
error: 7.90E-03

Runtime: 1.51E+03
∞
L
error: 7.87E-03

Runtime: 2.73E+03
∞
L
error: 7.83E-03

Runtime: 2.28E+04
∞
L
error: 7.85E-03

s
n
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c
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r
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0

s
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s
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2

s
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3

s
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4

s
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e
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−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.40: Deviation of scalar flux density from reference solution for hybrid-IIb LS-IDC5 methods us-
ing CL quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

200

S64,2

S64,4

S64,8

S64,16

Runtime: 2.37E+02
∞
L
error: 5.90E-02

Runtime: 2.48E+02
∞
L
error: 1.43E-02

Runtime: 2.95E+02
∞
L
error: 8.63E-03

Runtime: 4.76E+02
∞
L
error: 7.50E-03

S64

Runtime: 4.01E+03
∞
L
error: 7.38E-03

Runtime: 7.04E+02
∞
L
error: 2.27E-02

Runtime: 7.23E+02
∞
L
error: 8.74E-03

Runtime: 8.45E+02
∞
L
error: 7.83E-03

Runtime: 1.14E+03
∞
L
error: 7.76E-03

Runtime: 8.33E+03
∞
L
error: 7.73E-03

Runtime: 1.13E+03
∞
error: 8.92E-03
L

Runtime: 1.17E+03
∞
error: 8.06E-03
L

Runtime: 1.29E+03
∞
error: 7.84E-03
L

Runtime: 1.88E+03
∞
error: 7.82E-03
L

Runtime: 1.23E+04
∞
error: 7.81E-03
L

Runtime: 1.58E+03
∞
error: 8.04E-03
L

Runtime: 1.62E+03
∞
error: 7.92E-03
L

Runtime: 1.78E+03
∞
error: 7.84E-03
L

Runtime: 2.41E+03
∞
error: 7.84E-03
L

Runtime: 1.54E+04
∞
error: 7.83E-03
L

Runtime: 2.06E+03
∞
L
error: 7.92E-03

Runtime: 2.07E+03
∞
L
error: 7.87E-03

Runtime: 2.27E+03
∞
L
error: 7.85E-03

Runtime: 3.03E+03
∞
L
error: 7.84E-03

Runtime: 1.82E+04
∞
L
error: 7.84E-03

Runtime: 2.47E+03
∞
L
error: 7.87E-03

Runtime: 2.52E+03
∞
L
error: 7.86E-03

Runtime: 2.76E+03
∞
L
error: 7.85E-03

Runtime: 3.65E+03
∞
L
error: 7.85E-03

Runtime: 2.28E+04
∞
L
error: 7.85E-03

s
n
o
i
t
c
e
r
r
o
C
0

s
n
o
i
t
c
e
r
r
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C
1

s
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e
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2

s
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i
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c
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3

s
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i
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4

s
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C
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−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.41: Deviation of scalar flux density from reference solution for hybrid-IIc LS-IDC5 methods us-
ing CL quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

201

S32,1

S32,2

S32,4

S32,8

Runtime: 1.25E+02
∞
L
error: 1.04E-01

Runtime: 1.38E+02
∞
L
error: 4.06E-02

Runtime: 1.83E+02
∞
L
error: 1.29E-02

Runtime: 3.62E+02
∞
L
error: 4.90E-03

S32

Runtime: 4.00E+03
∞
L
error: 2.30E-03

Runtime: 3.60E+02
∞
L
error: 4.39E-02

Runtime: 3.80E+02
∞
L
error: 1.46E-02

Runtime: 5.01E+02
∞
L
error: 4.70E-03

Runtime: 8.14E+02
∞
L
error: 2.33E-03

Runtime: 7.83E+03
∞
L
error: 2.35E-03

Runtime: 6.18E+02
∞
error: 1.84E-02
L

Runtime: 6.66E+02
∞
error: 5.49E-03
L

Runtime: 7.65E+02
∞
error: 2.01E-03
L

Runtime: 1.34E+03
∞
error: 2.23E-03
L

Runtime: 1.16E+04
∞
error: 2.33E-03
L

Runtime: 8.34E+02
∞
error: 7.64E-03
L

Runtime: 8.68E+02
∞
error: 2.00E-03
L

Runtime: 1.04E+03
∞
error: 2.16E-03
L

Runtime: 1.69E+03
∞
error: 2.27E-03
L

Runtime: 1.51E+04
∞
error: 2.31E-03
L

Runtime: 1.07E+03
∞
L
error: 3.14E-03

Runtime: 1.11E+03
∞
L
error: 2.19E-03

Runtime: 1.33E+03
∞
L
error: 2.24E-03

Runtime: 2.11E+03
∞
L
error: 2.28E-03

Runtime: 1.91E+04
∞
L
error: 2.30E-03

Runtime: 1.29E+03
∞
L
error: 2.26E-03

Runtime: 1.37E+03
∞
L
error: 2.26E-03

Runtime: 1.60E+03
∞
L
error: 2.27E-03

Runtime: 2.56E+03
∞
L
error: 2.29E-03

Runtime: 2.20E+04
∞
L
error: 2.29E-03

s
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i
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c
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0

s
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o
i
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c
e
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s
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2

s
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c
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s
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4

s
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C
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−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.42: Deviation of scalar flux density from reference solution for hybrid-IIb LS-IDC5 methods us-
ing TN quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

202

S32,1

S32,2

S32,4

S32,8

Runtime: 2.36E+02
∞
L
error: 5.91E-02

Runtime: 2.47E+02
∞
L
error: 6.50E-03

Runtime: 2.95E+02
∞
L
error: 2.52E-03

Runtime: 4.76E+02
∞
L
error: 2.16E-03

S32

Runtime: 4.00E+03
∞
L
error: 2.30E-03

Runtime: 7.32E+02
∞
L
error: 2.27E-02

Runtime: 7.05E+02
∞
L
error: 2.94E-03

Runtime: 8.46E+02
∞
L
error: 2.30E-03

Runtime: 1.20E+03
∞
L
error: 2.29E-03

Runtime: 7.83E+03
∞
L
error: 2.35E-03

Runtime: 1.21E+03
∞
error: 8.94E-03
L

Runtime: 1.19E+03
∞
error: 2.40E-03
L

Runtime: 1.33E+03
∞
error: 2.24E-03
L

Runtime: 1.78E+03
∞
error: 2.31E-03
L

Runtime: 1.16E+04
∞
error: 2.33E-03
L

Runtime: 1.58E+03
∞
error: 3.46E-03
L

Runtime: 1.62E+03
∞
error: 2.22E-03
L

Runtime: 1.78E+03
∞
error: 2.28E-03
L

Runtime: 2.41E+03
∞
error: 2.30E-03
L

Runtime: 1.51E+04
∞
error: 2.31E-03
L

Runtime: 2.02E+03
∞
L
error: 2.34E-03

Runtime: 2.07E+03
∞
L
error: 2.27E-03

Runtime: 2.28E+03
∞
L
error: 2.29E-03

Runtime: 3.03E+03
∞
L
error: 2.29E-03

Runtime: 1.91E+04
∞
L
error: 2.30E-03

Runtime: 2.46E+03
∞
L
error: 2.22E-03

Runtime: 2.53E+03
∞
L
error: 2.28E-03

Runtime: 2.75E+03
∞
L
error: 2.29E-03

Runtime: 3.66E+03
∞
L
error: 2.29E-03

Runtime: 2.20E+04
∞
L
error: 2.29E-03

s
n
o
i
t
c
e
r
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0

s
n
o
i
t
c
e
r
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C
1

s
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i
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c
e
r
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o
C
2

s
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o
i
t
c
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C
3

s
n
o
i
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c
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C
4

s
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c
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C
5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

−1
−2
−3
−4
−5

Figure 4.43: Deviation of scalar flux density from reference solution for hybrid-IIc LS-IDC5 methods us-
ing TN quadratures and different numbers of correction iterations applied to the hohlraum test problem.
The solve time (in seconds) and the L

error are included for each solution.

∞

203

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

24

Prediction

1 Correction

2 Corrections

28

26
Runtime (seconds)

210

212

26

28

210

212

214

26

Runtime (seconds)

210

28
212
Runtime (seconds)

214

3 Corrections

4 Corrections

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

28
212
Runtime (seconds)

214

26

28

210

212

214

Runtime (seconds)

Figure 4.44: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIb (red and blue) discrete ordinates methods based on the IDC3 integrator using CL quadratures
applied to the hohlraum test problem with various numbers of correction iterations. Hybrid methods
using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

204

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

24

Prediction

1 Correction

2 Corrections

28

26
Runtime (seconds)

210

212

26

28

210

212

214

26

Runtime (seconds)

210

28
212
Runtime (seconds)

214

3 Corrections

4 Corrections

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

28
212
Runtime (seconds)

214

26

28

210

212

214

Runtime (seconds)

Figure 4.45: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIc (red and blue) discrete ordinates methods based on the IDC3 integrator using CL quadratures
applied to the hohlraum test problem with various numbers of correction iterations. Hybrid methods
using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

205

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

24

Prediction

1 Correction

2 Corrections

28

26
Runtime (seconds)

210

212

26

28

210

212

214

26

Runtime (seconds)

210

28
212
Runtime (seconds)

214

3 Corrections

4 Corrections

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

28
212
Runtime (seconds)

214

26

28

210

212

214

Runtime (seconds)

Figure 4.46: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIb (red and blue) discrete ordinates methods based on the IDC3 integrator using TN quadratures
applied to the hohlraum test problem with various numbers of correction iterations. Hybrid methods
using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

206

124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

24

Prediction

1 Correction

2 Corrections

28

26
Runtime (seconds)

210

212

26

28

210

212

214

26

Runtime (seconds)

210

28
212
Runtime (seconds)

214

3 Corrections

4 Corrections

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

28
212
Runtime (seconds)

214

26

28

210

212

214

Runtime (seconds)

Figure 4.47: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIc (red and blue) discrete ordinates methods based on the IDC3 integrator using TN quadratures
applied to the hohlraum test problem with various numbers of correction iterations. Hybrid methods
using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

207

124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8Prediction

1 Correction

2 Corrections

24

28

26
210
Runtime (seconds)

212 24

26

28

210

212

26

28

210

212

214

Runtime (seconds)

Runtime (seconds)

3 Corrections

4 Corrections

5 Corrections

−1

10

−2

10

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

212
28
Runtime (seconds)

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

214

Figure 4.48: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIb (red and blue) discrete ordinates methods based on the IDC5 integrator using CL quadratures
applied to the hohlraum test problem with various numbers of correction iterations. Hybrid methods
using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

208

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16Prediction

1 Correction

2 Corrections

24

28

26
210
Runtime (seconds)

212 24

26

28

210

212

26

28

210

212

214

Runtime (seconds)

Runtime (seconds)

3 Corrections

4 Corrections

5 Corrections

−1

10

−2

10

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

212
28
Runtime (seconds)

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

214

Figure 4.49: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIc (red and blue) discrete ordinates methods based on the IDC5 integrator using CL quadratures
applied to the hohlraum test problem with various numbers of correction iterations. Hybrid methods
using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

209

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16Prediction

1 Correction

2 Corrections

24

28

26
210
Runtime (seconds)

212 24

26

28

210

212

26

28

210

212

214

Runtime (seconds)

Runtime (seconds)

3 Corrections

4 Corrections

5 Corrections

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

212
28
Runtime (seconds)

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

214

Figure 4.50: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIb (red and blue) discrete ordinates methods based on the IDC5 integrator using TN quadratures
applied to the hohlraum test problem with various numbers of correction iterations. Hybrid methods
using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

210

124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8Prediction

1 Correction

2 Corrections

24

28

26
210
Runtime (seconds)

212 24

26

28

210

212

26

28

210

212

214

Runtime (seconds)

Runtime (seconds)

3 Corrections

4 Corrections

5 Corrections

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

212
28
Runtime (seconds)

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

214

Figure 4.51: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIc (red and blue) discrete ordinates methods based on the IDC5 integrator using TN quadratures
applied to the hohlraum test problem with various numbers of correction iterations. Hybrid methods
using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid methods
are shown in blue.

211

124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

24

Prediction

1 Correction

2 Corrections

28

26
Runtime (seconds)

210

212

26

28

210

212

214

26

Runtime (seconds)

210

28
212
Runtime (seconds)

214

3 Corrections

4 Corrections

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

28
212
Runtime (seconds)

214

26

28

210

212

214

Runtime (seconds)

Figure 4.52: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIb (red and blue) discrete ordinates methods based on the LS-IDC3 integrator using CL quadra-
tures applied to the hohlraum test problem with various numbers of correction iterations. Hybrid meth-
ods using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid
methods are shown in blue.

212

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

24

Prediction

1 Correction

2 Corrections

28

26
Runtime (seconds)

210

212

26

28

210

212

214

26

Runtime (seconds)

210

28
212
Runtime (seconds)

214

3 Corrections

4 Corrections

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

28
212
Runtime (seconds)

214

26

28

210

212

214

Runtime (seconds)

Figure 4.53: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIc (red and blue) discrete ordinates methods based on the LS-IDC3 integrator using CL quadra-
tures applied to the hohlraum test problem with various numbers of correction iterations. Hybrid meth-
ods using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid
methods are shown in blue.

213

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

24

Prediction

1 Correction

2 Corrections

28

26
Runtime (seconds)

210

212

26

28

210

212

214

26

Runtime (seconds)

210

28
212
Runtime (seconds)

214

3 Corrections

4 Corrections

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

28
212
Runtime (seconds)

214

26

28

210

212

214

Runtime (seconds)

Figure 4.54: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid-IIb (red and blue) discrete ordinates methods based on the LS-IDC3 integrator using TN
quadratures applied to the hohlraum test problem with various numbers of correction iterations. Hybrid
methods using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid
methods are shown in blue.

214

124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

24

Prediction

1 Correction

2 Corrections

28

26
Runtime (seconds)

210

212

26

28

210

212

214

26

Runtime (seconds)

210

28
212
Runtime (seconds)

214

3 Corrections

4 Corrections

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

28
212
Runtime (seconds)

214

26

28

210

212

214

Runtime (seconds)

Figure 4.55: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid-IIc (red and blue) discrete ordinates methods based on the LS-IDC3 integrator using TN
quadratures applied to the hohlraum test problem with various numbers of correction iterations. Hybrid
methods using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid
methods are shown in blue.

215

124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8Prediction

1 Correction

2 Corrections

24

28

26
210
Runtime (seconds)

212 24

26

28

210

212

26

28

210

212

214

Runtime (seconds)

Runtime (seconds)

3 Corrections

4 Corrections

5 Corrections

−1

10

−2

10

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

212
28
Runtime (seconds)

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

214

Figure 4.56: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIb (red and blue) discrete ordinates methods based on the LS-IDC5 integrator using CL quadra-
tures applied to the hohlraum test problem with various numbers of correction iterations. Hybrid meth-
ods using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid
methods are shown in blue.

216

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16Prediction

1 Correction

2 Corrections

24

28

26
210
Runtime (seconds)

212 24

26

28

210

212

26

28

210

212

214

Runtime (seconds)

Runtime (seconds)

3 Corrections

4 Corrections

5 Corrections

−1

10

−2

10

−1

10

−2

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

212
28
Runtime (seconds)

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

214

Figure 4.57: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid-IIc (red and blue) discrete ordinates methods based on the LS-IDC5 integrator using CL quadra-
tures applied to the hohlraum test problem with various numbers of correction iterations. Hybrid meth-
ods using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid
methods are shown in blue.

217

2481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-162481632644-28-416-216-832-232-432-1664-264-464-864-328-216-432-864-16Prediction

1 Correction

2 Corrections

24

28

26
210
Runtime (seconds)

212 24

26

28

210

212

26

28

210

212

214

Runtime (seconds)

Runtime (seconds)

3 Corrections

4 Corrections

5 Corrections

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

212
28
Runtime (seconds)

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

214

Figure 4.58: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid-IIb (red and blue) discrete ordinates methods based on the LS-IDC5 integrator using TN
quadratures applied to the hohlraum test problem with various numbers of correction iterations. Hybrid
methods using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid
methods are shown in blue.

218

124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8Prediction

1 Correction

2 Corrections

24

28

26
210
Runtime (seconds)

212 24

26

28

210

212

26

28

210

212

214

Runtime (seconds)

Runtime (seconds)

3 Corrections

4 Corrections

5 Corrections

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

−1

10

−2

10

−3

10

e
c
n
e
r
e
f
e
r
o
t
e
c
n
a
t
s
i
d
)
(cid:126)x
d
(

2
L

26

210

212
28
Runtime (seconds)

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

214

Figure 4.59: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid-IIc (red and blue) discrete ordinates methods based on the LS-IDC5 integrator using TN
quadratures applied to the hohlraum test problem with various numbers of correction iterations. Hybrid
methods using angular quadratures that differ in order by a factor of 4 are shown in red and other hybrid
methods are shown in blue.

219

124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8124816322-14-28-18-416-116-216-832-132-232-432-164-18-216-432-8vation holds for all two-grid methods regardless of the reconstruction procedure that is used. However,

two-grid methods using angular quadratures that differ in order by a factor of eight appear to be a supe-

rior choice in many cases: two-grid methods using angular quadratures that differ in order by a factor

of eight generally require between 20 and 40 percent less solve time that the equivalent method using

angular quadratures that differ in order by a factor of four. In a few cases – for the hybrid-IIb IDC3 and

LS-IDC3 methods using TN quadratures – the two-grid methods using angular quadratures that differ

in order by a factor of eight yield a small increase in the solution error, but if one additional correc-

tion iteration is performed (i.e., three correction iterations in total) then this deviation in accuracy is

not longer present. This is not an issue for the hybrid-IIc methods using Nyström reconstruction proce-

dures, and it seems that the hybrid-IIc methods may in fact support further coarsening the resolution of

the quadrature that is applied to the collided component. However, doing so reduces overall solve time

by only a very small amount (less than 10 percent), because the uncollided portion begins to dominate

the solve time. Since other problems may be more sensitive to further reducing the resolution of the

collided quadrature, we conclude that reductions beyond a factor of eight difference in quadrature order

are unlikely to be worthwhile.

Each figure in the remaining set is designed to compare four types of methods based on IDC or LS-

IDC integrators: (i) standard discrete ordinates methods, (ii) hybrid-Ib discrete ordinates methods using

angular quadratures that differ in order by a factor of four, (iii) hybrid-IIb discrete ordinates methods

using angular quadratures that differ in order by a factor of eight, and (iii) hybrid-IIc discrete ordinates

methods using angular quadratures that differ in order by a factor of eight. All of the methods shown in

each figure are constructed using the same time-integration scheme and angular quadrature types. Re-

sults are again presented for IDC and LS-IDC integrators using various numbers of correction iterations.

Results are shown for methods based on the IDC3 scheme in figures 4.60 and 4.61, for methods based

on the IDC5 scheme in figures 4.62 and 4.63, for methods based on the LS-IDC3 scheme in figures 4.64

and 4.65, and for methods based on the LS-IDC5 scheme in figures 4.66 and 4.67.

In each case, the two-grid methods (hybrid-IIb and hybrid-IIc) offer significant improvements in

computational efficiency relative to both the standard discrete ordinates methods and the hybrid-Ib

methods. It is also clear that the hybrid-IIc methods are more expensive than an equivalent hybrid-

IIb method. For example, the S32,4 hybrid-IIb methods using TN quadratures require between one-

thirteenth and one-sixteenth the solve time of the corresponding standard discrete ordinates method

220

using a TN quadrature of order 32, while the S32,4 hybrid-IIc methods require between one-eighth

and one-tenth the solve time of the S32 method. However, the time savings offered by the hybrid-IIb

methods do not come without sacrifice. First, whereas the hybrid-IIc methods offer excellent computa-

tional efficiency regardless of the number of correction iterations that are performed, the efficacy of the

hybrid-IIb methods depends much more strongly on the iteration count. Because of this, the hybrid-IIb

methods may be much less effective when applied to other types of problems. In addition, the Nyström

reconstruction procedures used by the hybrid-IIc methods are much more flexible than the quadrature-

specific reconstruction procedures used by the hybrid-IIb methods. In particular, the implementation

of the hybrid-IIc methods is much simpler since it does not depend in any way on the choice of angular

quadrature.

221

Prediction

1 Correction

2 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

−1

10

−2

10

e
c
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s
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)
(cid:126)x
d
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24

28

26
210
Runtime (seconds)

212

24

26

28

210

212

214

26

Runtime (seconds)

3 Corrections

4 Corrections

28

212
Runtime (seconds)

210

214

−1

10

−2

10

e
c
n
e
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f
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r
o
t
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c
n
a
t
s
i
d
)
(cid:126)x
d
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26

210

28
212
Runtime (seconds)

214

26

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

210

28
Runtime (seconds)

212

214

Figure 4.60: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid (blue, teal, and green) discrete ordinates methods based on the IDC3 integrator using CL
quadratures applied to the hohlraum test problem with various numbers of correction iterations.

222

2481632648-216-432-864-1616-232-464-88-216-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-2 248163264 8-216-432-864-16 16-232-464-88-2 16-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-2 248163264 8-216-432-864-16 16-232-464-88-2 16-232-464-88-2Prediction

1 Correction

2 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

−1

10

−2

10

−3

10

e
c
n
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f
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c
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)
(cid:126)x
d
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28

26
210
Runtime (seconds)

212

24

26

28

210

212

214

26

Runtime (seconds)

3 Corrections

4 Corrections

28

212
Runtime (seconds)

210

214

−1

10

−2

10

−3

10

e
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)
(cid:126)x
d
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26

210

28
212
Runtime (seconds)

214

26

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

210

28
Runtime (seconds)

212

214

Figure 4.61: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid (red, pink, and orange) discrete ordinates methods based on the IDC3 integrator using TN
quadratures applied to the hohlraum test problem with various numbers of correction iterations.

223

124816324-18-216-432-88-116-232-44-18-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1 12481632 4-18-216-432-8 8-116-232-44-1 8-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1 12481632 4-18-216-432-8 8-116-232-44-1 8-116-232-44-1−1

10

−2

10

−1

10

−2

10

e
c
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(cid:126)x
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e
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(cid:126)x
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26

Prediction

1 Correction

2 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

26

210
Runtime (seconds)

28

212

24

28

26
Runtime (seconds)

210

212

25

29

27
Runtime (seconds)

211

3 Corrections

4 Corrections

5 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

28

212
Runtime (seconds)

210

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

213

214

Figure 4.62: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid (blue, teal, and green) discrete ordinates methods based on the IDC5 integrator using CL
quadratures applied to the hohlraum test problem with various numbers of correction iterations.

224

2481632648-216-432-864-1616-232-464-88-216-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-2 248163264 8-216-432-864-16 16-232-464-88-2 16-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-2 248163264 8-216-432-864-16 16-232-464-88-2 16-232-464-88-2−1

10

−2

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−3

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26

Prediction

1 Correction

2 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

26

210
Runtime (seconds)

28

212

24

28

26
Runtime (seconds)

210

212

25

29

27
Runtime (seconds)

211

3 Corrections

4 Corrections

5 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

28

212
Runtime (seconds)

210

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

213

214

Figure 4.63: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid (red, pink, and orange) discrete ordinates methods based on the IDC5 integrator using TN
quadratures applied to the hohlraum test problem with various numbers of correction iterations.

225

124816324-18-216-432-88-116-232-44-18-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1 12481632 4-18-216-432-8 8-116-232-44-1 8-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1 12481632 4-18-216-432-8 8-116-232-44-1 8-116-232-44-1Prediction

1 Correction

2 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

−1

10

−2

10

e
c
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28

26
210
Runtime (seconds)

212

24

26

28

210

212

214

26

Runtime (seconds)

3 Corrections

4 Corrections

28

212
Runtime (seconds)

210

214

−1

10

−2

10

e
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210

28
212
Runtime (seconds)

214

26

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

210

28
Runtime (seconds)

212

214

Figure 4.64: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid (blue, teal, and green) discrete ordinates methods based on the LS-IDC3 integrator using CL
quadratures applied to the hohlraum test problem with various numbers of correction iterations.

226

2481632648-216-432-864-1616-232-464-88-216-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-2 248163264 8-216-432-864-16 16-232-464-88-2 16-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-2 248163264 8-216-432-864-16 16-232-464-88-2 16-232-464-88-2Prediction

1 Correction

2 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

−1

10

−2

10

−3

10

e
c
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210
Runtime (seconds)

212

24

26

28

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212

214

26

Runtime (seconds)

3 Corrections

4 Corrections

28

212
Runtime (seconds)

210

214

−1

10

−2

10

−3

10

e
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)
(cid:126)x
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210

28
212
Runtime (seconds)

214

26

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

210

28
Runtime (seconds)

212

214

Figure 4.65: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid (red, pink, and orange) discrete ordinates methods based on the LS-IDC3 integrator using TN
quadratures applied to the hohlraum test problem with various numbers of correction iterations.

227

124816324-18-216-432-88-116-232-44-18-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1 12481632 4-18-216-432-8 8-116-232-44-1 8-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1 12481632 4-18-216-432-8 8-116-232-44-1 8-116-232-44-1−1

10

−2

10

−1

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−2

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e
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26

Prediction

1 Correction

2 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

26

210
Runtime (seconds)

28

212

24

28

26
Runtime (seconds)

210

212

25

29

27
Runtime (seconds)

211

3 Corrections

4 Corrections

5 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

28

212
Runtime (seconds)

210

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

213

214

Figure 4.66: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black)
and hybrid (blue, teal, and green) discrete ordinates methods based on the LS-IDC5 integrator using CL
quadratures applied to the hohlraum test problem with various numbers of correction iterations.

228

2481632648-216-432-864-1616-232-464-88-216-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-2 248163264 8-216-432-864-16 16-232-464-88-2 16-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-22481632648-216-432-864-1616-232-464-88-216-232-464-88-2 248163264 8-216-432-864-16 16-232-464-88-2 16-232-464-88-2−1

10

−2

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−3

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−1

10

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(cid:126)x
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26

Prediction

1 Correction

2 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

26

210
Runtime (seconds)

28

212

24

28

26
Runtime (seconds)

210

212

25

29

27
Runtime (seconds)

211

3 Corrections

4 Corrections

5 Corrections

Non-hybrid
Hybrid-Ib
Hybrid-IIb
Hybrid-IIc

28

212
Runtime (seconds)

210

214

26

210

28
212
Runtime (seconds)

214

26

210

28
Runtime (seconds)

212

213

214

Figure 4.67: Solve times in seconds versus L2 (d(cid:126)x) distance to reference solution for standard (black) and
hybrid (red, pink, and orange) discrete ordinates methods based on the LS-IDC5 integrator using TN
quadratures applied to the hohlraum test problem with various numbers of correction iterations.

229

124816324-18-216-432-88-116-232-44-18-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1 12481632 4-18-216-432-8 8-116-232-44-1 8-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1124816324-18-216-432-88-116-232-44-18-116-232-44-1 12481632 4-18-216-432-8 8-116-232-44-1 8-116-232-44-1CHAPTER 5

CONCLUSIONS AND FUTURE WORK

5.1 Low-storage integral deferred correction methods for scientific computing

In chapter 2, a simple modification of traditional integral deferred correction (IDC) time-integration

schemes was proposed that significantly reduced the storage requirements of the methods. These meth-

ods, which we refer to as low-storage IDC or LS-IDC methods, require storing only one copy of each stage

vector throughout the iteration process. This reduction was achieved by splitting the spectral integration

matrix into upper-triangular and lower-triangular parts, and applying the lower-triangular portion to the

most recent set of stages in the residual calculation instead of the previous set. This results in lower stor-

age requirements than other implicit methods of the same order of accuracy.

An analytic proof was given, showing that the LS-IDC methods converge with the same order of accu-

racy as traditional IDC methods for linear, constant-coefficient problems, and numerical investigations

were used to examine the linear stability properties of some LS-IDC methods. It was found that the low-

storage modification negatively impacts the stability properties of many LS-IDC methods, particularly

for methods using right-biased quadratures that include the right endpoint of the timestep interval but

not the left. This suggests that the stability of LS-IDC methods is insufficient for very stiff problems.

Many LS-IDC methods using Gauss-Lobatto quadratures of order up to 10 were found to have better

contraction properties in the strongly-stiff limit than IDC methods using the same quadratures, but were

found to have inferior stability behavior for values near the imaginary axis. These methods may be an

attractive option for certain non-stiff systems in which stability for values with large negative real part is

more important than for values near the imaginary axis.

Two test problems were used to demonstrate the efficacy of LS-IDC methods. For a simple nonlinear

ODE problem, LS-IDC methods were found to yield moderately larger errors than equivalent IDC meth-

ods using the same number of correction iterations, but similar accuracy when one additional correction

iteration was applied. When applied to a linear transport equation solved by the method of lines, LS-IDC

methods were found to yield similar or better accuracy than other methods without applying additional

correction iterations. These results suggest that even though the low-storage modification negatively

230

impacts the stability properties of many LS-IDC methods, these methods are still an attractive option

for solving high-dimensional partial differential equations that lack a significant degree of numerical

stiffness.

In general, the reduction in memory footprint offered by LS-IDC methods depends strongly on the

implicit solver that is used. The memory savings that were obtained when solving the transport equation

(1.2) relied on the fact that the solvers used for discrete ordinates methods – either the source iteration

method (3.13) or the GMRES formulation (3.14) and (3.15) – require storing only one angular flux vec-

tor throughout the iterative process. Similar implementations that rely on simple fixed-point methods

or solving systems of reduced dimension should derive comparable benefits from the low-storage mod-

ification. However, there is little apparent advantage to LS-IDC methods in cases where, for example,

Krylov methods are applied to the full system or Jacobian matrices must be constructed.

At this time it is unclear if the stability properties of LS-IDC methods can be improved. The modi-

fication of the traditional IDC approach used to produce the LS-IDC methods is similar to the work of

[184], in which linear algebraic arguments were used to tune the contractive properties of spectral de-

ferred correction methods to particular scenarios. An analogous technique could be applied to LS-IDC

methods to improve their stability properties by selecting a low-order DIRK scheme to replace the im-

plicit Euler steps. Since the low-order scheme only needs to provide a stable, consistent discretization,

the additional degrees of freedom that are present when multiple Euler steps are combined could be

augmented to improve the stability properties of the LS-IDC iteration. However, it would likely be nec-

essary to hand-tune these modifications to each desired quadrature, and it is not obvious what sort of

stability improvements might be attainable or how such modifications would impact other properties of

the methods.

5.2 High-order hybrid methods for radiation transport

In chapter 3, extensions of a collision-based hybrid method for time-dependent radiation transport

simulations were discussed. The hybrid methods were constructed by splitting the radiation flux into

collided and uncollided components to which low- and high-resolution discrete ordinates approxima-

tions were applied, respectively. A reconstruction procedure was defined to map the collided compo-

nent to the high-resolution angular quadrature between timesteps. The use of arbitrarily high-order nu-

merical approximations was emphasized, with particular attention paid to high-order time-integration

231

schemes. A range of fully-implicit time integrators were considered, including DIRK, STDG, IDC, and

LS-IDC methods of up to fifth-order accuracy, and multiple approaches for constructing hybrid meth-

ods were compared.

A sequence of convergence tests under a diffusive scaling was conducted in one-dimensional slab

geometry. While the hybrid methods exhibited convergence stagnation and order reduction in certain

scenarios, the overall accuracy of the hybrid approximations was found to be comparable to standard

discrete ordinates methods in many cases. The stagnation phenomenon that was observed is readily

explained: Since particles are moved to the collided distribution when they undergo collisions, the over-

all accuracy of the hybrid methods degrades to the accuracy of the low-resolution approximation used

for the collided component when problem conditions are such that the majority of particles experience

collisions over each timestep. However, a rigorous explanation for the observed order reduction phe-

nomenon is less straightforward. We believe that this phenomenon is the result of a truncation error in

time that is induced by the relabeling or reconstruction procedure that is applied. A rigorous study of the

mechanism by which this phenomenon develops is currently underway.

With two goals in mind, the hybrid methods were applied to a test problem in two-dimensional

x y-geometry consisting of a mock-up of a standard hohlraum configuration. The first goal was to inves-

tigate how the resolutions of the angular quadratures applied to the components of the hybrid methods

could be chosen to adequately balance solve time and solution accuracy. Hybrid methods using angular

quadratures with orders differing by a factor of four were found to consistently attain good efficiency

relative to standard discrete ordinates methods and other hybrid methods. The second goal was to com-

pare the computational efficiency of the hybrid methods with standard discrete ordinates methods. It

was discovered that replacing a standard discrete ordinates method using an angular quadrature of or-

der N with a hybrid discrete ordinates method using angular quadratures of order 2N and N /2 in the

uncollided and collided components, respectively, reduced solve time by a factor of 2 or more and in

many cases also increased solution accuracy by a factor of up to 2, depending on the type of angular

quadrature used. The suggested replacement was noted to have the disadvantage of increasing memory

consumption by a factor of approximately 4.25, which may make these hybrid methods undesirable in

applications where available system memory is limited.

Two types of angular quadratures were considered for multidimensional geometries. The first type

of quadratures, the Chebyshev-Legendre (CL) quadratures, are constructed using a tensor-product ap-

232

proach and are designed to satisfy large numbers of moment conditions1. The second type of quadra-

tures, the TN quadratures, are constructed using a geometric approach that more evenly distributes

quadrature nodes over the surface of the sphere. It was observed that the advantage in computational

efficiency derived from the hybrid approach was dependent on the angular quadrature that was used,

and that standard discrete ordinates methods using angular quadratures tailored towards the character-

istics of the problem under consideration could yield similar errors in the same amount of solve time as

hybrid methods using a less efficient choice of angular quadratures. Just as the choice of angular quadra-

ture can significantly impact the accuracy of standard discrete ordinates methods, judiciously selecting

the angular quadrature for the collided and uncollided components of the hybrid methods is a crucial

component of their overall efficiency.

The angular quadratures used in this work were selected based on the fact that (i) they permit an arbi-

trary level of refinement with strictly positive quadrature weights, and (ii) their construction is relatively

straightforward. Having demonstrated the feasibility of the hybrid approach, we believe that further

investigation into the choice of angular quadratures is warranted. Future explorations should include

more advanced quadratures, e.g. [6, 7, 107, 131, 132, 133], and should examine whether any benefit may

be derived from selecting angular quadratures for the components of the hybrid methods that are of dif-

ferent types. For example, using angular quadratures for the collided component that satisfy a certain

number of moment conditions may improve accuracy in highly collisional regimes, while it would be

more effective to emphasize other properties in the uncollided quadrature. Adaptive refinement strate-

gies could also be investigated, though we would hypothesize that an adaptive strategy would be far

more beneficial for the high-resolution quadrature than the low-resolution quadrature.

In chapter 4, two mechanisms for increasing the effectiveness of the hybrid methods were presented.

First, a novel reconstruction procedure for mapping between arbitrary discrete ordinates quadratures

within the context of these hybrid methods was described. This approach, called Nyström reconstruc-

tion, was shown to have a number of significant advantages over previous reconstruction methods. Re-

1More specifically, these quadratures were chosen because an arbitrary number of moment condi-
tions may be satisfied by simply increasing the order of the quadrature. It should be noted that these
quadratures are rather inefficient in the sense that the number of moment conditions satisfied by these
quadratures is relatively small compared to the number of nodes in each quadrature. More efficient
quadratures exist (e.g., [6, 131, 132, 133]) but they are generally much more difficult to construct, partic-
ularly for very high orders.

233

peating the convergence tests conducted previously, it was found that hybrid methods using Nyström

reconstruction procedures suffered from the same convergence stagnation phenomenon observed pre-

viously, but concerns over the order reduction phenomenon were significantly reduced. Hybrid methods

using a Gauss-Legendre quadrature of order 2 in the collided component had previously been observed

to converge with second-order accuracy when reconstruction procedures based on polynomial interpo-

lation were used, but fifth-order accuracy was observed when Nyström reconstruction procedures were
used. This suggests a superconvergence rate of order 2N + 1 for hybrid methods based on Nyström re-

constructions when a Gauss-Legendre quadrature of order N is used in the collided component.

When applied to the hohlraum test problem, the Nyström reconstruction procedures were found to

significantly increase the accuracy of the hybrid methods. Whereas pronounced ray effects were ob-

served in many of the solutions obtained from hybrid methods using other reconstruction procedures,

no visibly apparent ray effects due to the lower-resolution angular quadrature were observed in the scalar

flux distributions obtained from hybrid methods using Nyström reconstructions. For the previously rec-

ommended quadrature resolutions, the Nyström reconstruction procedures were found to increase the

solve time of the hybrid methods by up to 55 percent. However, the increased solve time was generally

outweighed by increased solution accuracy. The accuracy of hybrid methods using Nyström reconstruc-

tion procedures and angular quadratures of order N and N /4 for the uncollided and collided compo-

nents, respectively, was consistently found to be comparable to that of a standard discrete ordinates

method using an angular quadrature of order N . In each case, the hybrid methods required solve times

that were between 4 and 8 times smaller than the standard discrete ordinates methods, with larger re-

ductions observed when higher-resolution angular quadratures were used. By eliminating the need to

increase the resolution of the angular quadrature that is applied to the uncollided component of the

hybrid methods, the Nyström reconstruction procedures remedy the effective increase in memory con-

sumption required by other hybrid methods. The revised recommendations given for hybrid methods

using Nyström reconstruction – that angular quadratures of order N and N /4 be used for the uncol-

lided and collided components, respectively – increases overall memory consumption by approximately

6 percent compared to a standard discrete ordinates method using an angular quadrature of order N .

Aside from increasing solution accuracy and reducing effective memory requirements to acceptable

levels, the Nyström reconstruction procedures have a number of additional benefits. First, the Nys-

tröm reconstruction procedures are extremely flexible and eliminate the need to construct quadrature-

234

specific mapping procedures. The same reconstruction algorithm can be used in combination with any

pair of angular quadratures. This approach could also be used to combine angular discretizations other

than discrete ordinates methods. For example, a diffusion approximation, moment method, or finite

element method could all be applied to the collided component instead of a low-resolution discrete

ordinates approximation. In general, the Nyström reconstruction procedures only require that an ap-

proximation of the scalar flux distribution be constructed from the collided and uncollided components

in a manner that is consistent with the spatial discretization applied to the uncollided flux. This flexibil-

ity should greatly simplify future studies that examine the effectiveness of combining different angular

approximations within the context of these hybrid methods.

In certain applications (e.g., thermal radiative transfer) maintaining positivity in the scalar flux dis-

tribution is important not only because negative values are non-physical, but severe instabilities may

occur when the scalar flux contains negative values. While we did not specifically examine mechanisms

for maintaining positivity in this work, Nyström reconstruction procedures should be more well-behaved

in this regard than other reconstruction approaches. Some methods, including the reconstruction pro-

cedures discussed in section 3.3.2 that were based on polynomial interpolation or hyperinterpolation

steps, may introduce negative values into the reconstructed solution even if the initial values of the col-

lided component are strictly non-negative. So long as the spatial and angular discretizations applied to

the components of the hybrid methods yield positive representations of these components, the Nyström

reconstruction procedures should preserve this property. That is, the Nyström reconstruction proce-

dures should not introduce negative values into the solution where they do not already exists.

In the final portion of this work, it was demonstrated that hybrid methods based on IDC integrators

could be modified to iteratively improve the accuracy of the hybrid splitting in addition to the tempo-

ral accuracy of the time discretization. These methods, which were referred to as two-grid IDC meth-

ods, demonstrate the effectiveness of a combined defect correction approach based on IDC integrators.

When the convergence tests conducted previously were repeated for the two-grid methods, it was found

that the saturation and order reduction that was observed with previous hybrid methods were both sig-

nificantly reduced by the two-grid approach, and almost completely eliminated when Nyström recon-

struction procedures were used.

When applied to the hohlraum test problem, it was found that the defect correction approach al-

lowed for the resolution of the angular quadrature applied to the collided component to be further re-

235

duced without sacrificing solution accuracy. Specifically, two-grid methods using angular quadratures

that differ in order by a factor of eight were found to consistently attain good efficiency relative to stan-

dard discrete ordinates methods and other hybrid methods. Additionally, the accuracy of two-grid meth-

ods using angular quadratures of order N and N /8 for the uncollided and collided components, respec-

tively, was consistently found to be comparable to that of a standard discrete ordinates method using

an angular quadrature of order N . In each case, the two-grid methods required solve times that were

between 6 and 16 times smaller than the standard discrete ordinates methods, depending on the recon-

struction procedure and angular quadratures that were applied.

These results demonstrate that the two-grid approach is able to iteratively decrease the error of

the hybrid methods as intended, but several questions remain. In particular, it is not fully clear when

such methods would have an efficiency advantage over hybrid methods based on other time-integration

schemes. LS-IDC methods were demonstrated to have smaller memory requirements than other time

integrators of the same order, but the total number of stages that must be computed is higher than for

other methods. For example, the SDIRK3 and LS-IDC3 schemes require 3 and 6 total stages, respectively,

and the ARK8 and LS-IDC5 schemes require 7 and 15 implicit stages, respectively. It should be noted

that the computational cost associated with the stages of implicit IDC methods is not necessarily the

same as that of implicit DIRK methods. Since the previous set of provisional solutions provides a rather

good initial guess for the iterative solves required during subsequent correction iterations, the number

of solver iterations required for each stage can decrease significantly as additional correction iterations

are performed. Hence a one-to-one comparison of stage counts may not reliably quantify the differ-

ences in computational cost between time-integration schemes. A thorough comparison of the com-

putational efficiency of hybrid methods constructed using different time integration schemes should be

established, particularly for calculations in multidimensional geometries.

When Nyström reconstruction procedures were used, the performance of different hybrid methods

was rather consistent relative to standard discrete ordinates methods when applied to the hohlraum test

problem. However, the properties of the various hybrid methods provides some direction for the types of

problems in which the performance of the two-grid methods may be superior to that of hybrid methods

based on other time integrators. In general, the first-collision source technique works well for problems

that transition from optically thin to optically thick, but is ineffective for problems that transition from

optically thick to optically thin. The motivation for introducing a relabeling or reconstruction procedure

236

100

10
0

0

1

1

2

Radius

3

4

0

0

1

2

Radius

3

4

4

3

2

1

0

0

1

2

3

4

0

-1

-2

-3

-4

(a) Total cross section (σt).

(b) Emission source (q).

(c) Sample solution.

Figure 5.1: Layout and sample solution for the inhomogeneous sphere problem. Color scale of solution
is log10.

is to allow first-collision source methods to more accurately capture thick to thin transitions. The Nys-

tröm reconstruction procedures introduced in this work significantly increase the effectiveness of hybrid

methods based on the first-collision source technique in the thick to thin transition, particularly when

this transition is sharp, as is the case for the hohlraum test problem. A common problem used to test

radiation transport codes is the homogeneous sphere problem [1, 43, 163, 165, 172], which emphasizes a

thick to thin transition at a sharp material interface. Since hybrid methods using Nyström reconstruction

seem to perform very well when this transition is sharp, ongoing work to evaluate the hybrid methods

discussed in this work has focused on a modification of the homogeneous sphere problem in which the

thick to thin transition occurs more slowly. Some initial results for this modified problem, which we refer

to as the inhomogeneous sphere problem, are now briefly presented.

The inhomogeneous sphere problem consists of a spherically symmetric body of varying density

radiating into a vacuum. We consider a two-dimensional version of the problem on a spatial domain
of [−4,4]× [−4,4] with vacuum (zero outflow) boundary conditions on all sides. The parameters of the

problem and one quadrant of a reference solution are shown in figure 5.1. It is assumed that particle
collisions are composed of pure scattering; i.e., σs = σt. The convergence behavior of hybrid-Ic SDIRK3
methods and hybrid-IIc LS-IDC3 methods are compared in figure 5.2. All solutions are computed using

TN quadratures. As with previous convergence tests, we observe that (i) the saturation phenomenon is

an issue for the hybrid-Ic DIRK methods, and (ii) the two-grid defect correction methods reduce the im-

pact of this saturation phenomenon. Because the geometry of this problem is much more complex than

that of the one-dimensional convergence tests conducted in previous portions of this work, the accuracy

of the hybrid-IIc LS-IDC3 methods suffers when very coarse angular approximations are applied to the

237

e
c
n
e
r
e
f
e
r

.
s
v
r
o
r
r
e
)
(cid:126)x
d
(

2
L

−2

10

−3

10

−4

10

−5

10

−6

10

S16
S4
S2
S1
S16,4
S16,2
S16,1

27

26

28

29

210

Spatial cells

e
c
n
e
r
e
f
e
r

.
s
v
r
o
r
r
e
)
(cid:126)x
d
(

2
L

−2

10

−3

10

−4

10

−5

10

−6

10

S16
S4
S2
S1
S16,4
S16,2
S16,1

27

26

28

29

210

Spatial cells

(a) SDIRK3 hybrid-Ic.

(b) LS-IDC3 hybrid-IIc.

Figure 5.2: L2-convergence studies of hybrid-Ic SDIRK3 methods and hybrid-IIc LS-IDC3 methods ap-
plied to the inhomogeneous sphere problem with a final time of 8 and a CFL of 128. All solutions are
computed using TN quadratures.

collided component. However, we note that the S16,1 approximations use quadratures that differ more

significantly in their resolution than might be recommended: whereas the order 16 quadrature contains

256 nodes per octant, the order 1 quadrature contains only 1 quadrature node per octant.

There are many directions in which the two-grid hybrid methods may be extended in future work.

First, the convergence of the two-grid methods using IDC integrators must be formally established. Ad-

ditionally, there are many practical issues that must be resolved before these methods can be considered

for widespread use. One issue of particular importance is preserving positivity in the solution. At present,

we are not aware of any work that examines enforcing positivity constraints when IDC time integrators

are used. IDC methods attain high-order accuracy through the use of residual terms in the correction

equations that act as fictitious sources. Since these terms are purely mathematical in nature, they may

be either positive or negative, depending on the properties of the solution and the high-order quadra-

ture that is used to evaluate the residual. It is not clear how positivity may be enforced while preserving

high-order accuracy when negative source terms are present.

It was noted in section 4.1 that the two-grid iteration on which the two-grid hybrid methods are

based does not converge in a satisfactory manner. Indeed, the numerical results in section 4.4 showed

that the convergence of the two-grid iterations is generally not monotonic. However, other approaches

to two-grid methods exist that do converge monotonically, and their convergence is also generally much

faster [17, Section 6.2.2]. Future work should investigate whether additional benefits may be derived

238

from incorporating these methods into the hybrid approach.

239

BIBLIOGRAPHY

240

BIBLIOGRAPHY

[1]

[2]

E. ABDIKAMALOV, A. BURROWS, C. D. OTT, F. LÖFFLER, E. O’CONNOR, J. C. DOLENCE, AND
E. SCHNETTER, A new Monte Carlo method for time-dependent neutrino radiation transport, The
Astrophysical Journal, 755 (2012), p. 111, doi:10.1088/0004-637x/755/2/111.

I. K. ABU-SHUMAYS, Angular quadratures for improved transport computations, Transport Theory
and Statistical Physics, 30 (2001), pp. 169–204, doi:10.1081/TT-100105367.

[3] M. L. ADAMS, Discontinuous finite element transport solutions in thick diffusive problems, Nuclear

Science and Engineering, 137 (2001), pp. 298–333, doi:10.13182/nse00-41.

[4] M. L. ADAMS AND E. W. LARSEN, Fast iterative methods for discrete-ordinates particle transport
calculations, Progress in Nuclear Energy, 40 (2002), pp. 3–159, doi:10.1016/S0149-1970(01)00023-
3.

[5] M. P. ADAMS, M. L. ADAMS, W. D. HAWKINS, T. SMITH, L. RAUCHWERGER, N. M. AMATO, T. S.
BAILEY, AND R. D. FALGOUT, Provably optimal parallel transport sweeps on regular grids, in Inter-
national Conference on Mathematics and Computational Methods Applied to Nuclear Science &
Engineering (M&C 2013), Sun Valley, Idaho, USA, May 5–9, 2013, American Nuclear Society.

[6]

[7]

[8]

[9]

C. AHRENS AND G. BEYLKIN, Rotationally invariant quadratures for the sphere, Proceedings of
the Royal Society A: Mathematical, Physical and Engineering Sciences, 465 (2009), pp. 3103–3125,
doi:10.1098/rspa.2009.0104.

C. D. AHRENS, Highly efficient, exact quadratures for three-dimensional discrete ordinates transport
calculations, Nuclear Science and Engineering, 170 (2012), pp. 98–101, doi:10.13182/nse10-69tn.

R. E. ALCOUFFE, A first collision source method for coupling Monte Carlo and discrete ordinates
for localized source problems, in Monte-Carlo Methods and Applications in Neutronics, Pho-
tonics and Statistical Physics, R. Alcouffe, R. Dautray, A. Forster, G. Ledanois, and B. Mercier,
eds., vol. 240 of Lecture Notes in Physics, Springer, Berlin, Heidelberg, 1985, pp. 352–366,
doi:10.1007/BFb0049062.

R. E. ALCOUFFE, R. D. O’DELL, AND F. W. BRINKEY, JR., A first-collision source method that sat-
isfies discrete SN transport balance, Nuclear Science and Engineering, 105 (1990), pp. 198–203,
doi:10.13182/NSE90-A23749.

[10] R. ALEXANDER, Diagonally implicit Runge-Kutta methods for stiff O.D.E.’s, SIAM Journal on Nu-

merical Analysis, 14 (1977), pp. 1006–1021, doi:10.1137/0714068.

[11] P. ALLU AND S. MAZUMDER, Hybrid ballistic-diffusive solution to the frequency-dependent phonon
Boltzmann transport equation, International Journal of Heat and Mass Transfer, 100 (2016),
pp. 165–177, doi:10.1016/j.ijheatmasstransfer.2016.04.049.

[12] D. N. ARNOLD, An interior penalty finite element method with discontinuous elements, SIAM Jour-

nal on Numerical Analysis, 19 (1982), pp. 742–760, doi:10.1137/0719052.

[13] D. N. ARNOLD, F. BREZZI, B. COCKBURN, AND D. MARINI, Unified analysis of discontinuous
Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 39 (2002), pp. 1749–
1779, doi:10.1137/S0036142901384162.

241

[14] T. ASAOKA, Y. NAKAHARA, AND K. SAITO, Multiple collision method for neutron transport prob-
lems, Journal of Nuclear Energy. Parts A/B. Reactor Science and Technology, 18 (1964), pp. 665–701,
doi:10.1016/0368-3230(64)90122-3.

[15] U. M. ASHER AND L. R. PETZOLD, Computer Methods for Ordinary Differential Equations and

Differential-Algebraic Equations, SIAM, 1998.

[16] K. ATKINSON AND W. HAN, Spherical Harmonics and Approximations on the Unit Sphere: An
Introduction, vol. 2044 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 2012,
doi:10.1007/978-3-642-25983-8.

[17] K. E. ATKINSON, The Numerical Solution of Integral Equations of the Second Kind, vol. 4 of Cam-
bridge Monographs on Applied and Computational Mathematics, Cambridge University Press,
Cambridge, United Kingdom, 1997.

[18] W. AUZINGER, H. HOFSTÄTTER, W. KREUZER, AND E. WEINMÜLLER, Modified defect correc-
tion algorithms for ODEs part I: General theory, Numerical Algorithms, 36 (2004), pp. 135–155,
doi:10.1023/B:NUMA.0000033129.73715.7f.

[19] W. AUZINGER, H. HOFSTÄTTER, W. KREUZER, AND E. WEINMÜLLER, Modified defect correction al-
gorithms for ODEs part II: Stiff initial value problems, Numerical Algorithms, 40 (2005), pp. 285–
303, doi:10.1007/s11075-005-5327-4.

[20] M. BADRI, P. JOLIVET, B. ROUSSEAU, AND Y. FAVENNEC, High performance computation of radiative
transfer equation using the finite element method, Journal of Computational Physics, 360 (2018),
pp. 74–92, doi:10.1016/j.jcp.2018.01.027.

[21] T. S. BAILEY AND R. D. FALGOUT, Analysis of massively parallel discrete-ordinates transport sweep
algorithms with collisions, in International Conference on Mathematics, Computational Methods,
and Reactor Physics (M&C 2009), Saratoga Springs, New York, USA, May 3–7, 2009, American Nu-
clear Society.

[22] F. BASSI AND S. REBAY, A high-order accurate discontinuous finite element method for the numerical
solution of the compressible Navier-Stokes equations, Journal of Computational Physics, 131 (1997),
pp. 267–279, doi:10.1006/jcph.1996.5572.

[23] H. BERNINGER, E. FRÉNOD, M. GANDER, M. LIEBENDÖRFER, AND J. MICHAUD, Derivation
of the isotropic diffusion source approximation (IDSA) for supernova neutrino transport by
asymptotic expansions, SIAM Journal on Mathematical Analysis, 45 (2013), pp. 3229–3265,
doi:10.1137/12089243x.

[24] H. BERNINGER, E. FRÉNOD, M. J. GANDER, M. LIEBENDÖRFER, J. MICHAUD, AND N. VASSET, A
mathematical description of the IDSA for supernova neutrino transport, its discretization and a
comparison with a finite volume scheme for Boltzmann’s equation, ESAIM: Proceedings, 38 (2012),
pp. 163–182, doi:10.1051/proc/201238009.

[25] K. BÖHMER, Discrete Newton methods and iterated defect corrections, Numerische Mathematik, 37

(1981), pp. 167–192, doi:10.1007/bf01398251.

[26] K. BÖHMER, P. W. HEMKER, AND H. J. STETTER, The defect correction approach, in Defect Cor-
rection Methods: Theory and Applications, H. Böhmer and H. Stetter, eds., vol. 5 of Computing
Supplementa, Springer, Vienna, Vienna, 1984, pp. 1–32, doi:10.1007/978-3-7091-7023-6_1.

242

[27] C. BÖRGERS, E. W. LARSEN, AND M. L. ADAMS, The asymptotic diffusion limit of a linear discon-
tinuous discretization of a two-dimensional linear transport equation, Journal of Computational
Physics, 98 (1992), pp. 285–300, doi:10.1016/0021-9991(92)90143-m.

[28]

S. BOSCARINO, L. PARESCHI, AND G. RUSSO, Implicit-explicit Runge-Kutta schemes for hyperbolic
systems and kinetic equations in the diffusion limit, SIAM Journal on Scientific Computing, 35
(2013), pp. A22–A51, doi:10.1137/110842855.

[29]

J. P. BOYD, Chebyshev and Fourier Spectral Methods, Dover Publications, Inc., second ed., 2001.

[30] F. BREZZI, G. MANZINI, D. MARINI, P. PIETRA, AND A. RUSSO, Discontinuous Galerkin approx-
imations for elliptic problems, Numerical Methods for Partial Differential Equations, 16 (2000),
pp. 365–378, doi:10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y.

[31] T. A. BRUNNER, Forms of approximate radiation transport, Tech. Report SAND2002-1778, Sandia

National Laboratories, Albuquerque, New Mexico, 2002.

[32] B. G. CARLSON, Tables of equal weight quadrature EQn over the unit sphere, Tech. Report LA-4734,

Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1971.

[33] B. G. CARLSON AND C. E. LEE, Mechanical quadrature and the transport equation, Tech. Report

LAMS-2573, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1961.

[34] E. CARSON AND N. J. HIGHAM, Accelerating the solution of linear systems by iterative refine-
ment in three precisions, SIAM Journal on Scientific Computing, 40 (2018), pp. A817–A847,
doi:10.1137/17m1140819.

[35]

J. CASH, A variable order deferred correction algorithm for the numerical solution of nonlinear two
point boundary value problems, Computers & Mathematics with Applications, 9 (1983), pp. 257–
265, doi:10.1016/0898-1221(83)90129-3.

[36] P. CASTILLO, B. COCKBURN, I. PERUGIA, AND D. SCHÖTZAU, An a priori error analysis of the lo-
cal discontinuous Galerkin method for elliptic problems, SIAM Journal on Numerical Analysis, 38
(2000), pp. 1676–1706, doi:10.1137/S0036142900371003.

[37] R. O. CASTRO AND J. P. TRELLES, Spatial and angular finite element method for radiative transfer
in participating media, Journal of Quantitative Spectroscopy and Radiative Transfer, 157 (2015),
pp. 81–105, doi:10.1016/j.jqsrt.2015.02.008.

[38] C. CERCIGNANI AND D. H. SATTINGER, Scaling Limits and Models in Physical Processes, Birkhäuser

Basel, 1998, doi:10.1007/978-3-0348-8810-3.

[39] G. CHEN, Ballistic-diffusive heat-conduction equations, Physical Review Letters, 86 (2001),

pp. 2297–2300, doi:10.1103/physrevlett.86.2297.

[40] G. CHEN, Ballistic-diffusive equations for transient heat conduction from nano to macroscales,

Journal of Heat Transfer, 124 (2002), p. 320, doi:10.1115/1.1447938.

[41] G.-Q. CHEN, C. D. LEVERMORE, AND T.-P. LIU, Hyperbolic conservation laws with stiff relaxation
terms and entropy, Communications on Pure and Applied Mathematics, 47 (1994), pp. 787–830,
doi:10.1002/cpa.3160470602.

243

[42] Y. CHENG AND C.-W. SHU, A discontinuous Galerkin finite element method for time dependent par-
tial differential equations with higher order derivatives, Mathematics of Computation, 77 (2008),
pp. 699–730, doi:10.1090/S0025-5718-07-02045-5.

[43] P. CHIDYAGWAI, M. FRANK, F. SCHNEIDER, AND B. SEIBOLD, A comparative study of limiting strate-
gies in discontinuous Galerkin schemes for the M1 model of radiation transport, Journal of Compu-
tational and Applied Mathematics, 342 (2018), pp. 399–418, doi:10.1016/j.cam.2018.04.017.

[44] A. CHRISTLIEB, W. GUO, M. MORTON, AND J.-M. QIU, A high order time splitting method based
on integral deferred correction for semi-Lagrangian Vlasov simulations, Journal of Computational
Physics, 267 (2014), pp. 7–27, doi:10.1016/j.jcp.2014.02.012.

[45] A. CHRISTLIEB, M. MORTON, B. ONG, AND J. QIU, Semi-implicit integral deferred correction con-
structed with additive Runge-Kutta methods, Communications in Mathematical Sciences, 9 (2011),
pp. 879–902, doi:10.4310/CMS.2011.v9.n3.a10.

[46] A. CHRISTLIEB AND B. ONG, Implicit parallel time integrators, Journal of Scientific Computing, 49

(2011), pp. 167–179, doi:10.1007/s10915-010-9452-4.

[47] A. CHRISTLIEB, B. ONG, AND J.-M. QIU, Integral deferred correction methods constructed with
high order Runge-Kutta integrators, Mathematics of Computation, 79 (2010), pp. 761–783,
doi:10.1090/S0025-5718-09-02276-5.

[48] A. J. CHRISTLIEB, R. D. HAYNES, AND B. W. ONG, A parallel space-time algorithm, SIAM Journal on

Scientific Computing, 34 (2012), pp. C233–C248, doi:10.1137/110843484.

[49] A. J. CHRISTLIEB, Y. LIU, AND Z. XU, High order operator splitting methods based on an inte-
gral deferred correction framework, Journal of Computational Physics, 294 (2015), pp. 224–242,
doi:10.1016/j.jcp.2015.03.032.

[50] A. J. CHRISTLIEB, C. B. MACDONALD, AND B. W. ONG, Parallel high-order integrators, SIAM Journal

on Scientific Computing, 32 (2010), pp. 818–835, doi:10.1137/09075740X.

[51] A. J. CHRISTLIEB, B. W. ONG, AND J.-M. QIU, Comments on high-order integrators embedded
within integral deferred correction methods, Communications in Applied Mathematics and Com-
putational Science, 4 (2009), pp. 27–56, doi:10.2140/camcos.2009.4.27.

[52] M. CLEVELAND AND A. WOLLABER, Corrected implicit Monte Carlo, Journal of Computational

Physics, 359 (2018), pp. 20–44, doi:10.1016/j.jcp.2017.12.038.

[53] B. COCKBURN, S. HOU, AND C.-W. SHU, The Runge-Kutta local projection discontinuous Galerkin
finite element method for conservation laws IV: The multidimensional case, Mathematics of Com-
putation, 54 (1990), pp. 545–581, doi:10.1090/S0025-5718-1990-1010597-0.

[54] B. COCKBURN, G. KANSCHAT, I. PERUGIA, AND D. SCHÖTZAU, Superconvergence of the local dis-
continuous Galerkin method for elliptic problems on Cartesian grids, SIAM Journal on Numerical
Analysis, 39 (2001), pp. 264–285, doi:10.1137/S0036142900371544.

[55] B. COCKBURN, S.-Y. LIN, AND C.-W. SHU, TVB Runge-Kutta local projection discontinuous
Galerkin finite element method for conservation laws III: One-dimensional systems, Journal of Com-
putational Physics, 84 (1989), pp. 90–113, doi:10.1016/0021-9991(89)90183-6.

244

[56] B. COCKBURN AND C. SHU, The local discontinuous Galerkin method for time-dependent
convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), pp. 2440–2463,
doi:10.1137/S0036142997316712.

[57] B. COCKBURN AND C.-W. SHU, TVB Runge-Kutta local projection discontinuous Galerkin finite ele-
ment method for conservation laws II: General framework, Mathematics of Computation, 52 (1989),
pp. 411–435, doi:10.1090/S0025-5718-1989-0983311-4.

[58] B. COCKBURN AND C.-W. SHU, The Runge-Kutta local projection P 1-discontinuous-Galerkin fi-
nite element method for scalar conservation laws, ESIAM: Mathematical Modeling and Nu-
merical Analysis – Modèlisation Mathématique et Analyse Numérique, 25 (1991), pp. 337–361,
doi:10.1051/m2an/1991250303371.

[59] B. COCKBURN AND C.-W. SHU, The Runge-Kutta discontinuous Galerkin method for conservation
laws V: Multidimensional systems, Journal of Computational Physics, 141 (1998), pp. 199–224,
doi:10.1006/jcph.1998.5892.

[60]

J. C. COMPTON AND C. J. CLOUSE, Tiling models for spatial decomposition in AMTRAN, in Pro-
ceedings of Joint Russian-American Five-Laboratory Conference on Computational Mathemat-
ics/Physics, Vienna, Austria, June 19–23, 2005.

[61] M. M. CROCKATT, A. J. CHRISTLIEB, C. K. GARRETT, AND C. D. HAUCK, An arbitrary-order, fully im-
plicit, hybrid kinetic solver for linear radiative transport using integral deferred correction, Journal
of Computational Physics, 346 (2017), pp. 212–241, doi:10.1016/j.jcp.2017.06.017.

[62] B. DAVISON, Neutron Transport Theory, Clarendon Press, Oxford, England, 1957.

[63] T. DEAKIN, S. MCINTOSH-SMITH, M. MARTINEAU, AND W. GAUDIN, An improved parallelism
scheme for deterministic discrete ordinates transport, The International Journal of High Perfor-
mance Computing Applications, (2016), pp. 1–14, doi:10.1177/1094342016668978.

[64] M. DELFOUR, W. HAGER, AND F. TROCHU, Discontinuous Galerkin methods for ordinary differ-
ential equations, Mathematics of Computation, 36 (1981), pp. 455–455, doi:10.1090/s0025-5718-
1981-0606506-0.

[65]

[66]

J. D. DENSMORE, Interface methods for hybrid Monte Carlo-diffusion radiation-transport simula-
tions, Annals of Nuclear Energy, 33 (2006), pp. 343–353, doi:10.1016/j.anucene.2005.11.007.

J. D. DENSMORE, T. J. URBATSCH, T. M. EVANS, AND M. W. BUKSAS, A hybrid transport-diffusion
method for Monte Carlo radiative-transfer simulations, Journal of Computational Physics, 222
(2007), pp. 485–503, doi:10.1016/j.jcp.2006.07.031.

[67] P. DEUFLHARD AND A. HOHMANN, Numerical Analysis in Modern Scientific Computing, Springer

New York, 2003, doi:10.1007/978-0-387-21584-6.

[68]

J. DOUGLAS, JR. AND T. DUPONT, Interior penalty procedures for elliptic and parabolic Galerkin
methods, in Computing Methods in Applied Sciences, vol. 58 of Lecture Nodes in Physics, Springer,
Berlin, Heidelberg, 1976, pp. 207–216, doi:10.1007/BFb0120591.

[69]

J. J. DUDERSTADT AND W. R. MARTIN, Transport Theory, John Wiley & Sons, Inc., 1979.

245

[70] A. DUTT, L. GREENGARD, AND V. ROKHLIN, Spectral deferred correction methods

for
40 (2000), pp. 241–266,

ordinary differential equations, Bit Numerical Mathematics,
doi:10.1023/a:1022338906936.

[71] T. ENDO AND A. YAMAMOTO, Development of new solid angle quadrature sets to satisfy even- and
odd-moment conditions, Journal of Nuclear Science and Technology, 44 (2007), pp. 1249–1258,
doi:10.1080/18811248.2007.9711368.

[72] A. FILIMON, M. DUMBSER,

AND C.-D. MUNZ, Discontinuous Galerkin schemes with
defect corrections based on reconstruction, Computers & Fluids, 62 (2012), pp. 71–85,
doi:10.1016/j.compfluid.2012.03.005.

[73] A. FILIMON AND C.-D. MUNZ, The application of iterated defect corrections based on WENO re-
construction, in Management and Minimization of Uncertainties and Errors in Numerical Aero-
dynamics, B. Eisfeld, H. Barnewitz, W. Fritz, and F. Thiele, eds., vol. 122 of Notes on Numerical
Fluid Mechanics and Multidisciplinary Design, Springer, Berlin, Heidelberg, 2013, pp. 129–154,
doi:10.1007/978-3-642-36185-2_6.

[74] W. A. FIVELAND, Discrete ordinate methods for radiative heat transfer in isotropically and aniso-

tropically scattering media, Journal of Heat Transfer, 109 (1987), p. 809, doi:10.1115/1.3248167.

[75]

J. FLECK AND J. CUMMINGS, An implicit Monte Carlo scheme for calculating time and frequency
dependent nonlinear radiation transport, Journal of Computational Physics, 8 (1971), pp. 313–342,
doi:10.1016/0021-9991(71)90015-5.

[76] R. FRANK AND C. W. UEBERHUBER, Iterated defect correction for the efficient solution of stiff systems

of ordinary differential equations, BIT, 17 (1977), pp. 146–159, doi:10.1007/bf01932286.

[77] R. FRANK AND C. W. UEBERHUBER, Collocation and iterated defect correction, in Numerical Treat-
ment of Differential Equations, R. Bulirsch, R. Grigorieff, and J. Schröder, eds., vol. 631 of Lecture
Notes in Mathematics, Springer, Berlin, Heidelberg, 1978, pp. 19–34, doi:10.1007/bfb0067461.

[78] B. GANAPOL, Multiple collision solution of the linearized Vlasov equation, Progress in Nuclear En-

ergy, 8 (1981), pp. 151–161, doi:10.1016/0149-1970(81)90009-3.

[79] B. D. GANAPOL, The multiple collision solution of Kramer’s equation for Brownian motion, Trans-

port Theory and Statistical Physics, 11 (1982), pp. 217–231, doi:10.1080/00411458208245742.

[80] B. D. GANAPOL, Exact time-dependent moments for the monoenergetic neutron transport
equation with isotropic scattering, Nuclear Science and Engineering, 89 (1985), pp. 256–260,
doi:10.13182/nse85-a17546.

[81] B. D. GANAPOL, R. S. BAKER, J. A. DAHL, AND R. E. ALCOUFFE, Homogeneous infinite media time-
dependent analytical benchmarks, Tech. Report LA-UR-01-1854, Los Alamos National Laboratory,
Los Alamos, New Mexico, 2001.

[82] B. D. GANAPOL AND V. C. BOFFI, The multiple collision method in solving the Boltz-
mann equation for time-dependent test particle transport, Meccanica, 15 (1980), pp. 75–81,
doi:10.1007/bf02133788.

[83] B. D. GANAPOL AND L. M. GROSSMAN, The collided flux expansion method for time-
dependent neutron transport, Nuclear Science and Engineering, 52 (1973), pp. 454–460,
doi:10.13182/nse73-a23312.

246

[84] B. D. GANAPOL, P. W. MCKENTY, AND K. L. PEDDICORD, The generation of time-dependent neutron
transport solutions in infinite media, Nuclear Science and Engineering, 64 (1977), pp. 317–331,
doi:10.13182/nse77-a27373.

[85] C. K. GARRETT AND C. D. HAUCK, A comparison of moment closures for linear kinetic transport
equations: The line source benchmark, Transport Theory and Statistical Physics, 42 (2013), pp. 203–
235, doi:10.1080/00411450.2014.910226.

[86] N. GENTILE, Implicit Monte Carlo diffusion– an acceleration method for Monte Carlo time-
dependent radiative transfer simulations, Journal of Computational Physics, 172 (2001), pp. 543–
571, doi:10.1006/jcph.2001.6836.

[87] N. GENTILE,

Including the effects of

Monte Carlo algorithm,
doi:10.1016/j.jcp.2011.03.029.

in the implicit
Journal of Computational Physics, 230 (2011), pp. 5100–5114,

temperature-dependent opacities

[88] L. M. GOMES AND P. N. STEVENS, Use of Monte Carlo procedures to estimate first and second col-
lision sources for discrete ordinates transport codes, Nuclear Science and Engineering, 114 (1993),
pp. 228–237, doi:10.13182/nse93-a24036.

[89] A. N. GORBAN AND I. V. KARLIN, Invariant Manifolds for Physical and Chemical Kinetics, Springer,

Berlin, Heidelberg, 2005, doi:10.1007/b98103.

[90] G. J. HABETLER AND B. J. MATKOWSKY, Uniform asymptotic expansions in transport theory with
small mean free paths, and the diffusion approximation, Journal of Mathematical Physics, 16
(1975), pp. 846–854, doi:10.1063/1.522618.

[91] E. HAIRER, S. P. NØRSETT, AND G. WANNER, Solving Ordinary Differential Equations I: Nonstiff
Problems, vol. 8 of Springer Series in Computational Mathematics, Springer, Berlin, Heidelberg,
second ed., 1993, doi:10.1007/978-3-540-78862-1.

[92] E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II: Stiff and Differential-
Algebraic Problems, vol. 14 of Springer Series in Computational Mathematics, Springer, Berlin,
Heidelberg, second ed., 1996, doi:10.1007/978-3-642-05221-7.

[93] E. HAIRER AND G. WANNER, Stiff differential equations solved by Radau methods, Journal of Com-
putational and Applied Mathematics, 111 (1999), pp. 93–111, doi:10.1016/S0377-0427(99)00134-X.

[94] T. HANGSTROM AND R. ZHOU, On the spectral deferred correction of splitting methods for initial
value problems, Communications in Applied Mathematics and Computational Science, 1 (2006),
pp. 169–205, doi:10.2140/camcos.2006.1.169.

[95] C. HAUCK AND R. MCCLARREN, Positive PN closures, SIAM Journal on Scientific Computing, 32

(2010), pp. 2603–2626, doi:10.1137/090764918.

[96] C. D. HAUCK, High-order entropy-based closures for linear transport in slab geometry, Communi-

cations in Mathematical Sciences, 9 (2011), pp. 187–205, doi:10.4310/cms.2011.v9.n1.a9.

[97] C. D. HAUCK AND R. G. MCCLARREN, A collision-based hybrid method for time dependent, lin-
ear, kinetic transport equations, Multiscale Modeling and Simulation, 11 (2013), pp. 1197–1227,
doi:10.1137/110846610.

247

[98] H. V. HENDERSON AND S. R. SEARLE, On deriving the inverse of a sum of matrices, SIAM Review, 23

(1981), pp. 53–60, doi:10.1137/1023004.

[99] A. HENNINK AND D. LATHOUWERS, A discontinuous Galerkin method for the mono-energetic
Fokker-Planck equation based on a spherical interior penalty formulation, Journal of Computa-
tional and Applied Mathematics, 330 (2018), pp. 253–267, doi:10.1016/j.cam.2017.08.019.

[100] N.

J. HIGHAM,

Accuracy
doi:10.1137/1.9780898718027.

and

Stability

of Numerical Algorithms,

SIAM,

2002,

[101] A. HOISIE, O. LUBECK, AND H. WASSERMAN, Performance and scalability analysis of
teraflop-scale parallel architectures using multidimensional wavefront applications, The In-
ternational Journal of High Performance Computing Applications, 14 (2000), pp. 330–346,
doi:10.1177/109434200001400405.

[102] Y.-C. HUA AND B.-Y. CAO, Ballistic-diffusive heat conduction in multiply-constrained nanostruc-
tures, International Journal of Thermal Sciences, 101 (2016), pp. 126–132, doi:10.1016/j.ijther-
malsci.2015.10.037.

[103] Y.-C. HUA AND B.-Y. CAO, The effective thermal conductivity of ballistic-diffusive heat conduction
in nanostructures with internal heat source, International Journal of Heat and Mass Transfer, 92
(2016), pp. 995–1003, doi:10.1016/j.ijheatmasstransfer.2015.09.068.

[104] J. HUANG, J. JIA, AND M. MINION, Accelerating the convergence of spectral deferred correction meth-

ods, Journal of Computational Physics, 214 (2006), pp. 633–656, doi:10.1016/j.jcp.2005.10.004.

[105] J. HUANG, J. JIA, AND M. MINION, Arbitrary order Krylov deferred correction methods for
differential algebraic equations, Journal of Computational Physics, 221 (2007), pp. 739–760,
doi:10.1016/j.jcp.2006.06.040.

[106] M. JANKOWSKI AND H. WO ´ZNIAKOWSKI, Iterative refinement implies numerical stability, BIT, 17

(1977), pp. 303–311, doi:10.1007/bf01932150.

[107] J. J. JARRELL, An adaptive angular discretization method for neutral-particle transport in three-

dimensional geometries, PhD thesis, Texas A&M University, 2010.

[108] C. JOHNSON AND J. PITKÄRANTA, An analysis of the discontinuous Galerkin method for a scalar

hyperbolic equation, Mathematics of Computation, 46 (1986), pp. 1–26, doi:10.2307/2008211.

[109] G. KANSCHAT, Solution of radiative transfer problems with finite elements, in Numerical Methods
in Multidimensional Radiative Transfer, G. Kanschat, E. Meinköhn, R. Rannacher, and R. Wehrse,
eds., Springer, Berlin, Heidelberg, 2009, pp. 49–98, doi:10.1007/978-3-540-85369-5_5.

[110] C. A. KENNEDY AND M. H. CARPENTER, Additive Runge-Kutta schemes for convection-
diffusion-reaction equations, Applied Numerical Mathematics, 44 (2003), pp. 139–181,
doi:10.1016/s0168-9274(02)00138-1.

[111] C. A. KENNEDY, M. H. CARPENTER, AND R. LEWIS, Low-storage, explicit Runge-Kutta schemes for
the compressible Navier-Stokes equations, Applied Numerical Mathematics, 35 (2000), pp. 177–219,
doi:10.1016/s0168-9274(99)00141-5.

248

[112] D. I. KETCHESON, Highly efficient strong stability-preserving Runge-Kutta methods with low-
storage implementations, SIAM Journal on Scientific Computing, 30 (2008), pp. 2113–2136,
doi:10.1137/07070485x.

[113] D. I. KETCHESON, Runge-Kutta methods with minimum storage implementations, Journal of Com-

putational Physics, 229 (2010), pp. 1763–1773, doi:10.1016/j.jcp.2009.11.006.

[114] S. KHOLIN, Certain exact solutions of the nonstationary kinetic equation without taking retardation
into account, USSR Computational Mathematics and Mathematical Physics, 4 (1964), pp. 213–221,
doi:10.1016/0041-5553(64)90094-1.

[115] K. R. KOCH, R. S. BAKER, AND R. E. ALCOUFFE, Solution of the first-order form of three-dimensional
discrete ordinates equations on a massively parallel machine, Transactions of the Americal Nuclear
Society, 65 (1992), pp. 198–199.

[116] R. KOCH, W. KREBS, S. WITTIG, AND R. VISKANTA, Discrete ordinates quadrature schemes for mul-
tidimensional radiative transfer, Journal of Quantitative Spectroscopy & Radiative Transfer, 53
(1995), pp. 353–372, doi:10.1016/0022-4073(95)90012-8.

[117] J. KÓPHÁZI AND D. LATHOUWERS, A space-angle DGFEM approach for the Boltzmann radiation
transport equation with local angular refinement, Journal of Computational Physics, 297 (2015),
pp. 637–668, doi:10.1016/j.jcp.2015.05.031.

[118] A. KVÆRNØ, Singly diagonally implicit Runge-Kutta methods with an explicit first stage, BIT Nu-

merical Mathematics, 44 (2004), pp. 489–502, doi:10.1023/b:bitn.0000046811.70614.38.

[119] M. P. LAIU, C. D. HAUCK, R. G. MCCLARREN, D. P. O’LEARY, AND A. L. TITS, Positive filtered
PN moment closures for linear kinetic equations, SIAM Journal on Numerical Analysis, 54 (2016),
pp. 3214–3238, doi:10.1137/15m1052871.

[120] E. W. LARSEN AND J. B. KELLER, Asymptotic solution of neutron transport problems for small mean

free paths, Journal of Mathematical Physics, 15 (1974), pp. 75–81, doi:10.1063/1.1666510.

[121] E. W. LARSEN, A. KUMAR, AND J. E. MOREL, Properties of the implicitly time-differenced equa-
tions of thermal radiation transport, Journal of Computational Physics, 238 (2013), pp. 82–96,
doi:10.1016/j.jcp.2012.11.034.

[122] E. W. LARSEN AND W. F. MILLER JR., Convergence rates of spatial difference equations for the
discrete-ordinates neutron transport equations in slab geometry, Nuclear Science and Engineering,
73 (1980), pp. 76–83, doi:10.13182/NSE80-3.

[123] E. W. LARSEN AND J. E. MOREL, Advances in discrete-ordinates methodology, in Nuclear Computa-

tional Science, Springer, Dordrecht, Dec. 2009, pp. 1–84, doi:10.1007/978-90-481-3411-3_1.

[124] E. W. LARSEN AND P. NELSON, Finite-difference approximations and superconvergence for the
discrete-ordinate equations in slab geometry, SIAM Journal on Numerical Analysis, 19 (1982),
pp. 334–348, doi:10.1137/0719020.

[125] P. LASAINT AND P. A. RAVIART, On a finite element method for solving the neutron transport equa-
tion, in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed.,
Academic Press, Inc., 1974, pp. 89–123, doi:10.1016/B978-0-12-208350-1.50008-X.

249

[126] K. D. LATHROP, Ray effects in discrete ordinates equations, Nuclear Science and Engineering, 32

(1968), pp. 357–369, doi:10.13182/NSE68-4.

[127] K. D. LATHROP, Remedies for ray effects, Nuclear Science and Engineering, 45 (1971), pp. 255–268,

doi:10.13182/NSE45-03-255.

[128] K. D. LATHROP AND B. G. CARLSON, Discrete ordinates angular quadrature of the neutron transport
equation, Tech. Report LA-3186, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1964.

[129] C. Y. LAU, Adaptive discrete-ordinates quadratures based on discontinuous finite elements over

spherical quadrilaterals, ph.D., Texas A&M University, 2016.

[130] A. T. LAYTON AND M. L. MINION, Implications of the choice of quadrature nodes for Picard integral
deferred corrections methods for ordinary differential equations, BIT Numerical Mathematics, 45
(2005), pp. 341–373, doi:10.1007/s10543-005-0016-1.

[131] V. LEBEDEV, Values of the nodes and weights of ninth to seventeenth order Gauss-Markov quadrature
formulae invariant under the octahedron group with inversion, USSR Computational Mathematics
and Mathematical Physics, 15 (1975), pp. 44–51, doi:10.1016/0041-5553(75)90133-0.

[132] V. LEBEDEV, Quadratures on a sphere, USSR Computational Mathematics and Mathematical

Physics, 16 (1976), pp. 10–24, doi:10.1016/0041-5553(76)90100-2.

[133] V. I. LEBEDEV, Spherical quadrature formulas exact to orders 25–29, Siberian Mathematical Journal,

18 (1977), pp. 99–107, doi:10.1007/bf00966954.

[134] C. D. LEVERMORE AND G. C. POMRANING, A flux-limited diffusion theory, The Astrophysical Jour-

nal, 248 (1981), p. 321, doi:10.1086/159157.

[135] E. E. LEWIS AND W. F. MILLER JR, Computational Methods of Neutron Transport, Wiley, 1993.

[136] M. LIEBENDÖRFER, S. C. WHITEHOUSE, AND T. FISCHER, The isotropic diffusion source approx-
imation for supernova neutrino transport, The Astrophysical Journal, 698 (2009), pp. 1174–1190,
doi:10.1088/0004-637x/698/2/1174.

[137] R. G. MCCLARREN AND C. D. HAUCK, Robust and accurate filtered spherical harmonics ex-
pansions for radiative transfer, Journal of Computational Physics, 229 (2010), pp. 5597–5614,
doi:10.1016/j.jcp.2010.03.043.

[138] R. G. MCCLARREN, J. P. HOLLOWAY, AND T. A. BRUNNER, On solutions to the Pn equations
for thermal radiative transfer, Journal of Computational Physics, 227 (2008), pp. 2864–2885,
doi:10.1016/j.jcp.2007.11.027.

[139] R. G. MCCLARREN AND T. J. URBATSCH, A modified implicit Monte Carlo method for time-
dependent radiative transfer with adaptive material coupling, Journal of Computational Physics,
228 (2009), pp. 5669–5686, doi:10.1016/j.jcp.2009.04.028.

[140] C. D. MEYER, Matrix Analysis and Applied Linear Algebra, The SIAM, 2000.

[141] A. MEZZACAPPA AND O. MESSER, Neutrino transport in core collapse supernovae, Journal of Com-
putational and Applied Mathematics, 109 (1999), pp. 281–319, doi:10.1016/s0377-0427(99)00162-
4.

250

[142] J. MICHAUD, The IDSA and the homogeneous

Issues and possible improve-
ments, Discrete and Continuous Dynamical Systems - Series S, 9 (2016), pp. 1351–1375,
doi:10.3934/dcdss.2016054.

sphere:

[143] D. MIHALAS AND B. W. MIHALAS, Foundations of Radiation Hydrodynamics, Oxford University

Press, New York, 1984.

[144] M. L. MINION, Semi-implicit

for ordinary differ-
ential equations, Communications in Mathematical Sciences, 1 (2003), pp. 471–500,
doi:10.4310/CMS.2003.v1.n3.a6.

spectral deferred correction methods

[145] M. L. MINION, Semi-implicit projection methods for incompressible flow based on spectral de-
ferred corrections, Applied Numerical Mathematics, 48 (2004), pp. 369–387, doi:10.1016/j.ap-
num.2003.11.005.

[146] A. MITTAL AND S. MAZUMDER, Generalized ballistic-diffusive formulation and hybrid SN -PN solu-
tion of the Boltzmann transport equation for phonons for nonequilibrium heat conduction, Journal
of Heat Transfer, 133 (2011), p. 092402, doi:10.1115/1.4003961.

[147] A. MITTAL AND S. MAZUMDER, Hybrid discrete ordinates–spherical harmonics solution to the Boltz-
mann transport equation for phonons for non-equilibrium heat conduction, Journal of Computa-
tional Physics, 230 (2011), pp. 6977–7001, doi:10.1016/j.jcp.2011.05.024.

[148] M. F. MODEST, The method of spherical harmonics (PN -approximation), in Radiative Heat Trans-

fer, Elsevier, 2013, pp. 495–540, doi:10.1016/b978-0-12-386944-9.50016-9.

[149] J. MOREL, Diffusion-limit asymptotics of the transport equation, the P1/3 equations, and two flux-
limited diffusion theories, Journal of Quantitative Spectroscopy and Radiative Transfer, 65 (2000),
pp. 769–778, doi:10.1016/S0022-4073(99)00148-X.

[150] S. MOUSTAFA, I. DUTKA-MALEN, L. PLAGNE, A. PONÇOT, AND P. RAMET, Shared memory paral-
lelism for 3D Cartesian discrete ordinates solver, Annals of Nuclear Energy, 82 (2015), pp. 179–187,
doi:10.1016/j.anucene.2014.08.034.

[151] E. J. NYSTRÖM, Über die praktische auflösung von integralgleichungen mit anwendungen auf

randwertaufgaben, Acta Mathematica, 54 (1930), pp. 185–204, doi:10.1007/bf02547521.

[152] G. L. OLSON, L. H. AUER, AND M. L. HALL, Diffusion, P1, and other approximate forms of radia-
tion transport, Journal of Quantitative Spectroscopy and Radiative Transfer, 64 (2000), pp. 619–634,
doi:10.1016/s0022-4073(99)00150-8.

[153] K.-C. PAN, M. LIEBENDÖRFER, M. HEMPEL, AND F.-K. THIELEMANN, Two-dimensional core-
collapse supernova simulations with the isotropic diffusion source approximation for neutrino
transport, The Astrophysical Journal, 817 (2016), p. 72, doi:10.3847/0004-637x/817/1/72.

[154] N. PAPMEHL AND H. J. ZECH, The order of neutron-scattering method in plane geometry–
pp. 435–448,

one-velocity problems, Nuclear Science and Engineering,
doi:10.13182/nse72-a22435.

47 (1972),

[155] S. D. PAUTZ, An algorithm for parallel Sn sweeps on unstructured meshes, Nuclear Science and

Engineering, 140 (2002), pp. 111–136, doi:10.13182/NSE02-1.

251

[156] V. PEREYNA, Iterated deferred corrections for nonlinear boundary value problems, Numerische

Mathematik, 11 (1968), pp. 111–125, doi:10.1007/bf02165307.

[157] F. PETRINI, G. FOSSUM, J. FERNANDEZ, A. L. VARBANESCU, M. KISTLER, AND M. PERRONE,
Multicore surprises: Lessons learned from optimizing Sweep3D on the cell broadband en-
gine, in 2007 IEEE International Parallel and Distributed Processing Symposium, IEEE, 2007,
doi:10.1109/ipdps.2007.370252.

[158] P. PICCA, R. FURFARO, AND B. D. GANAPOL, A hybrid transport point-kinetic method for simulat-
ing source transients in subcritical systems, Annals of Nuclear Energy, 38 (2011), pp. 2680–2688,
doi:10.1016/j.anucene.2011.08.005.

[159] P. PICCA, R. FURFARO, AND B. D. GANAPOL, Derivation of a physically based hybrid technique for
the solution of source-driven time-dependent linear Boltzmann equations, Transport Theory and
Statistical Physics, 41 (2012), pp. 23–39, doi:10.1080/00411450.2012.671219.

[160] G. C. POMRANING, The Equations of Radiation Hydrodynamics, Pergamon Press, 1973.

[161] R. W. PREISENDORFER, Radiative Transfer on Discrete Spaces, Pergamon Press, 1965.

[162] W. QU, N. BRANDON, D. CHEN, J. HUANG, AND T. KRESS, A numerical framework for integrating
deferred correction methods to solve high order collocation formulations of ODEs, Journal of Scien-
tific Computing, 68 (2015), pp. 484–520, doi:10.1007/s10915-015-0146-9.

[163] D. RADICE, E. ABDIKAMALOV, L. REZZOLLA, AND C. D. OTT, A new spherical harmonics scheme for
multi-dimensional radiation transport I. static matter configurations, Journal of Computational
Physics, 242 (2013), pp. 648–669, doi:10.1016/j.jcp.2013.01.048.

[164] J. RAGUSA, J.-L. GUERMOND, AND G. KANSCHAT, A robust SN -DG-approximation for radiation
transport in optically thick and diffusive regimes, Journal of Computational Physics, 231 (2012),
pp. 1947–1962, doi:10.1016/j.jcp.2011.11.017.

[165] M. RAMPP AND H.-T. JANKA, Radiation hydrodynamics with neutrinos, Astronomy & Astrophysics,

396 (2002), pp. 361–392, doi:10.1051/0004-6361:20021398.

[166] W. H. REED AND T. R. HILL, Triangular mesh methods for the neutron transport equation, Tech.

Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1973.

[167] S. A. RUKOLAINE AND V. S. YUFEREV, Discrete ordinates quadrature schemes based on the angular
interpolation of radiation intensity, Journal of Quantitative Spectroscopy & Radiative Transfer, 69
(2001), pp. 257–275, doi:10.1016/S0022-4073(00)00079-0.

[168] Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal residual algorithm for solving non-
symmetric linear systems, SIAM Journal on Scientific and Statistical Computing, 7 (1986), pp. 856–
869, doi:10.1137/0907058.

[169] R. D. SKEEL, Iterative refinement implies numerical stability for Gaussian elimination, Mathemat-

ics of Computation, 35 (1980), pp. 817–817, doi:10.1090/s0025-5718-1980-0572859-4.

[170] I. SLOAN, Polynomial interpolation and hyperinterpolation over general regions, Journal of Approx-

imation Theory, 83 (1995), pp. 238–254, doi:10.1006/jath.1995.1119.

252

[171] I. H. SLOAN AND R. S. WOMERSLEY, Constructive polynomial approximation on the sphere, Journal

of Approximation Theory, 103 (2000), pp. 91–118, doi:10.1006/jath.1999.3426.

[172] J. SMIT, J. CERNOHORSKY, AND C. DULLEMOND, Hyperbolicity and critical points in two-moment

approximate radiative transfer, Astronomy & Astrophysics, 325 (1997), pp. 203–211.

[173] R. SPECK, D. RUPRECHT, M. EMMETT, M. MINION, M. BOLTEN, AND R. KRAUSE, A multi-
level spectral deferred correction method, BIT Numerical Mathematics, 55 (2015), pp. 843–867,
doi:10.1007/s10543-014-0517-x.

[174] R. SPECK, D. RUPRECHT, M. MINION, M. EMMET, AND R. KRAUSE, Inexact spectral deferred cor-
rections, in Domain Decomposition Methods in Science and Engineering XXII, T. Dickopf, M. J.
Gander, L. Halpern, R. Krause, and L. F. Pavarino, eds., vol. 104 of Lecture Notes in Computational
Science and Engineering, Springer International Publishing, 2016, doi:10.1007/978-3-319-18827-
0_39.

[175] C. SYROS, A generalization of the multiple collision method, Nukleonik, 8 (1966), pp. 467–471.

[176] C. SYROS, Solution of the neutron slowing down problem via a multiple collision approach, Nuclear

Science and Engineering, 28 (1967), pp. 203–214, doi:10.13182/nse67-a17470.

[177] C. SYROS, Time-dependent transport solution in plane geometry, Atomkernenergie, 16 (1970),

pp. 273–280.

[178] C. P. THURGOOD, A critical evaluation of the discrete ordinates method using HEART and TN

quadrature., PhD thesis, Queen’s University at Kingston, 1993.

[179] C. P. THURGOOD, A. POLLARD, AND H. A. BECKER, The TN quadrature set for the discrete ordinates

method, Journal of Heat Transfer, 117 (1995), p. 1068, doi:10.1115/1.2836285.

[180] C.-L. TIEN, A. MAJUMDAR, AND F. M. GERNER, eds., Microscale Energy Transport, Taylor & Francis,

1998.

[181] W.

J. WALTERS AND A. HAGHIGHAT, The adaptive collision source method for dis-
crete ordinates radiation transport, Annals of Nuclear Energy, 105 (2017), pp. 45–58,
doi:10.1016/j.anucene.2017.02.013.

[182] T. A. WAREING, J. M. MCGHEE, J. E. MOREL, AND S. D. PAUTZ, Discontinuous finite element SN
methods on three-dimensional unstructured grids, Nuclear Science and Engineering, 138 (2001),
pp. 256–268, doi:10.13182/nse138-256.

[183] J. S. WARSA, T. A. WAREING, AND J. E. MOREL, Krylov iterative methods and the degraded
effectiveness of diffusion synthetic acceleration for multidimensional SN calculations in prob-
lems with material discontinuities, Nuclear Science and Engineering, 147 (2004), pp. 218–248,
doi:10.13182/nse02-14.

[184] M. WEISER, Faster SDC convergence on non-equidistant grids by DIRK sweeps, BIT Numerical

Mathematics, 55 (2014), pp. 1219–1241, doi:10.1007/s10543-014-0540-y.

[185] G. WIDMER, R. HIPTMAIR, AND C. SCHWAB, Sparse adaptive finite elements for radiative transfer,

Journal of Computational Physics, 227 (2008), pp. 6071–6105, doi:10.1016/j.jcp.2008.02.025.

253

[186] A. B. WOLLABER, Four decades of implicit Monte Carlo, Journal of Computational and Theoretical

Transport, 45 (2016), pp. 1–70, doi:10.1080/23324309.2016.1138132.

[187] R. T. WOLLAEGER, D. R. VAN ROSSUM, C. GRAZIANI, S. M. COUCH, G. C. J. IV, D. Q. LAMB, AND
G. A. MOSES, Radiation transport for explosive outflows: A multigroup hybrid Monte Carlo method,
The Astrophysical Journal Supplement Series, 209 (2013), p. 36, doi:10.1088/0067-0049/209/2/36.

[188] R. YANG, G. CHEN, M. LAROCHE, AND Y. TAUR, Simulation of nanoscale multidimensional tran-
sient heat conduction problems using ballistic-diffusive equations and phonon Boltzmann equa-
tion, Journal of Heat Transfer, 127 (2005), p. 298, doi:10.1115/1.1857941.

[189] M. YAVUZ AND E. W. LARSEN, Spatial domain decomposition for neutron transport problems,
Transport Theory and Statistical Physics, 18 (1989), pp. 205–219, doi:10.1080/00411458908204321.

[190] W. ZHENG AND R. G. MCCLARREN, Moment closures based on minimizing the residual of the PN
angular expansion in radiation transport, Journal of Computational Physics, 314 (2016), pp. 682–
699, doi:10.1016/j.jcp.2016.03.030.

254