ON SUPERCONVERGENT DISCONTINUOUS GALERKIN METHODS
FOR SCHR ¨ODINGER EQUATIONS AND SPARSE GRID CENTRAL

DISCONTINUOUS GALERKIN METHOD

By

Anqi Chen

A DISSERTATION

Submitted to

Michigan State University

in partial fulﬁllment of the requirements

for the degree of

Applied Mathematics – Doctor of Philosophy

2019

ABSTRACT

ON SUPERCONVERGENT DISCONTINUOUS GALERKIN METHODS FOR

SCHR ¨ODINGER EQUATIONS AND SPARSE GRID CENTRAL DISCONTINUOUS

GALERKIN METHOD

By

Anqi Chen

In this thesis, we design and analyze a discontinuous Galerkin (DG) method for one-

dimensional Schr¨odinger equations under a general class of numerical ﬂuxes, and another

eﬃcient DG method for high-dimensional hyperbolic equations.

In the ﬁrst DG method, we develop an ultra-weak discontinuous Galerkin (UWDG)

method to solve the one-dimensional nonlinear Schr¨odinger equation. Stability conditions

and error estimates are derived for the scheme with a general class of numerical ﬂuxes. The

error estimates are based on detailed analysis of the projection operator associated with each

individual ﬂux choice. Depending on the parameters, we ﬁnd out that in some cases, the

projection can be deﬁned element-wise, facilitating analysis. In most cases, the projection
is global, and its analysis depends on the resulting 2 × 2 block-circulant matrix structures.

For a large class of parameter choices, optimal a priori L2 error estimates can be obtained.

Numerical examples are provided verifying theoretical results.

In addition to the stability and error analysis, we analyze the superconvergence properties

of the UWDG method for one-dimensional linear Schr¨odinger equation with various choices

of ﬂux parameters. Depending on the ﬂux choices and if the polynomial degree k is even or
odd, we prove 2k or (2k − 1)-th order superconvergence rate for cell averages and numerical
ﬂux of the function, as well as (2k−1) or (2k−2)-th order for numerical ﬂux of the derivative.

In addition, we prove superconvergence of (k + 2) or (k + 3)-th order of the UWDG solution

towards a special projection. At a class of special points, the function values and the ﬁrst and

second order derivatives of the UWDG solution are superconvergent with order k +2, k +1, k,

respectively. The proof relies on the correction function techniques initiated in [12], and

applied to [10] for direct DG (DDG) methods for diﬀusion problems. By negative norm

estimates, we apply the post-processing technique and show that the accuracy of our scheme

can be enhanced to order 2k. Theoretical results are veriﬁed by numerical experiments.

In the second DG method, we develop sparse grid central discontinuous Galerkin (CDG)

scheme for linear hyperbolic systems with variable coeﬃcients in high dimensions. The

scheme combines the CDG framework with the sparse grid approach, with the aim of break-

ing the curse of dimensionality. A new hierarchical representation of piecewise polynomials

on the dual mesh is introduced and analyzed, resulting in a sparse ﬁnite element space that

can be used for non-periodic problems. Theoretical results, such as L2 stability and error

estimates are obtained for scalar problems. CFL conditions are studied numerically com-

paring discontinuous Galerkin (DG), CDG, sparse grid DG and sparse grid CDG methods.

Numerical results including scalar linear equations, acoustic and elastic waves are provided.

Copyright by
ANQI CHEN
2019

ACKNOWLEDGMENTS

I would like to express my great appreciation to my advisor, Professor Yingda Cheng,

for her invaluable guidance, inspiration, encouragement and patience during my ﬁve-year

PhD study. Professor Cheng taught me how to think and express my ideas rigorously like a

mathematician. She was always there whenever I needed advices, and she always provided

me with kind and thoughtful help. I could never ﬁnish this thesis without her help.

I would also like to thank all my committee members, Professor Andrew Christlieb,

Professor Chichia Chiu and Professor Jianliang Qian for their guidance and advices.

I would also like to thank my friends, Zhanjing Tao, Wei Guo, Yan Jiang, Menglun Wang,

Xin Yang, Kedi Wu, Zixuan Cang, Xiao Feng, Qinfeng Gao, Chao Song and Ruochuan Zhang

for their friendships and many precious moments.

Last but not least, I am grateful to my family. My father Weibing Chen and mother

Manzhen Deng always encouraged me to pursue what I wanted. I can’t say thanks enough

for their support in my life. And I am grateful to my girlfriend Bixi Zhang for her love and

support during the ﬁnal stage of my PhD.

v

TABLE OF CONTENTS

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

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

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 UWDG method for one-dimensional Schr¨odinger equations . . . . . . . . . .
1.3 Superconvergence analysis of DG methods . . . . . . . . . . . . . . . . . . .
1.4 Sparse-grid DG methods for high-dimensional PDEs . . . . . . . . . . . . . .

1
1
2
4
5

Chapter 2 An UWDG method for Schr¨odinger equation in one dimension 7
8
15
16
20
27
32
32
39

2.1 Numerical scheme and stability . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Projection P (cid:63)
h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Local projection condition . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 P (cid:63)
h properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Numerical results of the projection operator P (cid:63)
h . . . . . . . . . . . .
2.4.2 Numerical results of the DG scheme . . . . . . . . . . . . . . . . . . .

Chapter 3

Superconvergence analysis of UWDG method on linear Schr¨odinger
46
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.1 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.2 Projections
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Superconvergence properties . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
55
. . . . . . . . . . . . . .
56
. . . . . . . . . . . . . . .
3.4 Superconvergence of postprocessed solution . . . . . . . . . . . . . . . . . . .
68
73
3.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.1
3.3.2 Correction functions and the main results

Some intermediate superconvergence results

Chapter 4

4.1 Numerical Scheme

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Periodic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Discussions on implementations . . . . . . . . . . . . . . . . . . . . .
4.1.3 Discussions on CFL conditions . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Non-periodic problems . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Sparse grid central DG methods for linear hyperbolic systems 81
82
83
88
90
92
4.2 Stability and convergence
99
4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
System case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.3.1
4.3.2

vi

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

vii

LIST OF TABLES

Table 2.1: Notations for some frequently used quantities. Subscript j will be dropped
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

for uniform mesh.

21

Table 2.2: Interpretation of error estimate (2.29). . . . . . . . . . . . . . . . . . . . .

24

Table 2.3: Example 2.4.1. Error of local projection P (cid:63)

h u − u on a nonuniform mesh.
Flux parameters: α1 = 0.3, β1 = 0.4, β2 = 0.4. . . . . . . . . . . . . . . . .
h u − u on a nonuniform mesh.
Flux parameters: α1 = 0.3, β1 = 0.4/h, β2 = 0.4h. . . . . . . . . . . . . . .

Table 2.4: Example 2.4.1. Error of local projection P (cid:63)

Table 2.5: Example 2.4.1. Diﬀerence of local projection P (cid:63)

h : P (cid:63)
nonuniform mesh. Flux parameters: α1 = 0.5, β1 = 1, β2 = 0.

h with P 1

h u− P 1

h u on a
. . . . . . .

33

34

34

Table 2.6: Example 2.4.2. Error of global projection P (cid:63)

α1 = 0.25, ˜β1 = 1, ˜β2 = 1, p1 = −0.5, p2 = 2. (A1.1)

h u − u. Flux parameters:

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

35

Table 2.7: Example 2.4.2. Error of global projection P (cid:63)

8

2

+ k(k+1)

0, β1 = 1

h u− u. Flux parameters: α1 =
2h, β2 = h. (A1.5) . . . . . . . . . . . . . . . . . . . . . . . . . . .
h u− u. Flux parameters: α1 =
0.25, ˜β1 = k(k−1)
, ˜β2 = 1.0, p1 = −1, p2 = 2, 3. Note here
limh→0 λ1, λ2 = (−1)k. (A1.6.1) . . . . . . . . . . . . . . . . . . . . . . .
h u − u. Flux parameters:
2k(k−1), p1 = −2,−3, p2 = 1. Note here
α1 = 0.25, ˜β1 = 1, ˜β2 =
limh→0 λ1, λ2 = (−1)k+1. (A1.7.2) . . . . . . . . . . . . . . . . . . . . . .
h u − u. Flux parameters:
2k(k+1), p1 = −2,−3, p2 = 1. Note that
α1 = 0.25, ˜β1 = −1, ˜β2 =
limh→0 λ1, λ2 = 1. (A1.7.2) . . . . . . . . . . . . . . . . . . . . . . . . . .
h u − u. (Central ﬂux) Flux
parameters: α1 = 0, β1 = 0, β2 = 0. . . . . . . . . . . . . . . . . . . . . . .
h u − u. Flux parameters:

1

1

α1 = 0, β1 = 0, β2 = 1.

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

36

36

37

37

38

38

Table 2.9: Example 2.4.2. Error of global projection P (cid:63)

Table 2.8: Example 2.4.2. Error of global projection P (cid:63)

Table 2.10: Example 2.4.2. Error of global projection P (cid:63)

Table 2.11: Example 2.4.3. Error of global projection P (cid:63)

Table 2.12: Example 2.4.3. Error of global projection P (cid:63)

viii

1

Table 2.13: Example 2.4.4. Error of global projection P (cid:63)

and similar to A1.7.2 in Table 2.9): α1 = 0.25, ˜β1 = −1, ˜β2 =
−2,−3, p2 = 1. Note here limh→0 λ1, λ2 = (−1)k+1.

h u − u. Flux parameters (A3,
2k(k−1), p1 =
. . . . . . . . . . . .
Table 2.14: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation
(2.37) on a nonuniform mesh using ﬂux parameters (corresponding to Table
2.3) α1 = 0.3, β1 = β2 = 0.4, ending time Te = 0.3. . . . . . . . . . . . . .
Table 2.15: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation
(2.37) on a nonuniform mesh using ﬂux parameters (corresponding to Table
2.4) α1 = 0.3, β1 = 0.4h, β2 = 0.4/hj, ending time Te = 1. . . . . . . . . .
Table 2.16: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation
(2.37) using central ﬂux (corresponding to A2 in Table 2.11) α1 = β1 =
β2 = 0, ending time Te = 1. . . . . . . . . . . . . . . . . . . . . . . . . . .
Table 2.17: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation
(2.37) using ﬂux parameters (corresponding to A2 in Table 2.12): α1 =
β1 = 0, β2 = 1, ending time Te = 1. . . . . . . . . . . . . . . . . . . . . . .
Table 2.18: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation
(2.37) using ﬂux parameters (corresponding to A1.6.1 in Table 2.8): α1 =
0.25, ˜β1 = k(k−1)
, ˜β2 = 1.0, p1 = −1, p2 = 2, 3, ending time
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Te = 1.
Table 2.19: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation
(2.37) using ﬂux parameters (corresponding to A1.7.2 in Table 2.10): α1 =
2k(k+1), p1 = −2,−3, p2 = 1, ending time Te = 1. . . .
0.25, ˜β1 = −1, ˜β2 =

+ k(k+1)

2

8

1

Table 3.1: Example 3.5.1. Error table when using alternating ﬂux on nonuniform
mesh. Ending time Te = 1, x ∈ [0, 2π]. . . . . . . . . . . . . . . . . . . . .

Table 3.2: Example 3.5.1. Error table when using ﬂux parameters: α1 = 0.3, β1 =
. . .

h , β2 = 0.4h on nonuniform mesh. Ending time Te = 1, x ∈ [0, 2π].

0.4

Table 3.3: Example 3.5.1. Error table when using central ﬂux on uniform mesh. End-
ing time Te = 1, x ∈ [0, 2π]. . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.4: Example 3.5.1. Error table when using ﬂux parameters: α1 = 0.25, β1 =
h, β2 = 0 on uniform mesh. Ending time Te = 1, x ∈ [0, 2π]. . . . . .

2
h , 5

h, 9

Table 3.5: Example 3.5.1. Error table for intermediate quantities when using alter-
nating ﬂux on nonuniform mesh. Ending time Te = 1, x ∈ [0, 2π]. . . . . .

ix

39

42

42

43

43

44

44

75

76

77

78

79

Table 3.6: Example 3.5.1. Error table for intermediate quantities when using ﬂux
h , β2 = 0.4h on nonuniform mesh. Ending
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

parameters: α1 = 0.3, β1 = 0.4
time Te = 1, x ∈ [0, 2π].

Table 3.7: Example 3.5.1. Error table for intermediate quantities when using central
ﬂux on uniform mesh. Ending time Te = 1, x ∈ [0, 2π]. . . . . . . . . . . .

Table 3.8: Example 3.5.1. Postprocessing error table for the four sets of parameters.
Ending time Te = 1, uniform mesh on x ∈ [0, 2π]. The ﬁrst row below
labels the parameters by (˜α1, ˜β1, ˜β2). . . . . . . . . . . . . . . . . . . . . .

Table 4.1: CFL numbers of the DG method, CDG method, sparse grid DG method
and sparse grid CDG method with piecewise degree k polynomials, Runge-
Kutta method of order ν for Example 4.3.1 with d=2. The CFL numbers
of the sparse grid DG/CDG methods are measured with regard to the most
reﬁned mesh hN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 4.2: L2 errors and orders of accuracy for L2 projection operator ˜PD of (4.9)
onto ˆ˜Vk
N,D when d = 2 and d = 3. N is the number of mesh levels, k is the
polynomial order, d is the dimension. L2 order is calculated with respect
to hN .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

80

80

91

98

Table 4.3: L2 errors and orders of accuracy for Example 4.3.1 at T = 1 when d = 2,
T = 2/3 when d = 3, and T = 0.5 when d = 4. N denotes mesh level, hN
is the size of the smallest mesh in each direction, k is the polynomial order,
d is the dimension. L2 order is calculated with respect to hN .

. . . . . . 109

Table 4.4: L2 errors and orders of accuracy for Example 4.3.1 with Dirichlet boundary
condition on the inﬂow edges at T = 1 when d = 2 and T = 2/3 when d = 3.
N denotes mesh level, hN is the size of the smallest mesh on the primal
mesh in each direction, k is the polynomial order, d is the dimension. L2
order is calculated with respect to hN .

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

Table 4.5: L2 errors and orders of accuracy for Example 4.3.2 at T = 2π. N denotes
mesh level, hN is the size of the smallest mesh in each direction, k is the
polynomial order, d is the dimension. L2 order is calculated with respect
to hN .

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

Table 4.6: L2 errors and orders of accuracy for Example 4.3.3 at T = 1.5. N denotes
mesh level, hN is the size of the smallest mesh in each direction, k is the
polynomial order, d is the dimension. L2 order is calculated with respect
to hN . d = 2.

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

x

Table 4.7: L2 errors and orders of accuracy for Example 4.3.4 at T = 1. N denotes
mesh level, hN is the size of the smallest mesh in each direction, k is the
polynomial order, d is the dimension. L2 order is calculated with respect
to hN . d = 2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Table 4.8: L2 errors and orders of accuracy for Example 4.3.5 at T = 1. N denotes
mesh level, hN is the size of the smallest mesh in each direction, k is the
polynomial order, dimension d = 2. L2 order is calculated with respect to
hN .

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

Table 4.9: L2 errors and orders of accuracy for Example 4.3.6 at T = 1. N denotes
mesh level, hN is the size of the smallest mesh in each direction, k is the
polynomial order, d is the dimension. L2 order is calculated with respect
to hN . d = 3.

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

xi

LIST OF FIGURES

Figure 2.1: A sketch to illustrate the diﬀerent cases parametrized by the values of p1, p2. 24
Figure 2.2: Example 2.4.5. Absolute diﬀerence of (cid:107)uh(t,·)(cid:107) with (cid:107)uh(0,·)(cid:107) with two
sets of parameters (α1, α2, β1, β2) = (0.25,−0.25, 1 − i, 1 + i) (denoted
by “imag”) and (α1, α2, β1, β2) = (0.25,−0.25, 1, 1) (denoted by “real”)
. . . . . . . . . . . . . . . .
when k = 2, N = 40, ending time Te = 100.

41

Figure 2.3: Example 2.4.7. Double soliton collision graphs at t = 0, 2.5, 5 and a
x − t plot of the numerical solution. N = 250, P 2 elements with periodic
boundary conditions on [-25,25]. Central ﬂux (α1 = β1 = β2 = 0) is used.

Figure 4.1:

Illustration of one-dimensional bases on diﬀerent levels for k = 0: non-
periodic problems. Diﬀerent colors represent diﬀerent bases.
. . . . . . .

45

99

Figure 4.2:

Illustration of one-dimensional bases on diﬀerent levels for k = 1: non-
periodic problems. Diﬀerent colors represent diﬀerent bases.

. . . . . . . 100

Figure 4.3: Example 4.3.1. The time evolution of the error of L2 norm of numerical
solutions uh and vh of the sparse grid CDG method with d = 2. (a) k=1,
(b) k=2, (c) k=3. N = 4, 5, 6.

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

Figure 4.4: L2 errors and associated CPU times of DG, CDG, sparse grid DG and
sparse grid CDG methods for Example 4.1 with initial condition (4.24) at
T = 1 for d=2. (a) k=1, (b) k=2, (c) k=3.

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

Figure 4.5: Example 4.3.3. Deformational ﬂow test. The contour plots of the numer-
ical solutions on primal mesh at t = T /2 (a, c, e) and t = T (b, d, f).
k = 1 (a, b), k = 2 (c, d), and k = 3 (e, f). N = 7.

. . . . . . . . . . . . 114

xii

Chapter 1

Introduction

1.1 Overview

The discontinuous Galerkin (DG) method is a class of ﬁnite element methods using com-

pletely discontinuous piecewise function space for test functions and numerical solution.

The ﬁrst DG method was introduced by Reed and Hill in [59]. A major development

of DG method is the Runge-Kutta DG (RKDG) framework introduced for solving hyper-

bolic conservation laws containing only ﬁrst order spatial derivatives in a series of papers

[25, 24, 22, 21, 26]. Because of the completely discontinuous basis, DG method has several

attractive properties.

It can be used on many types of meshes, even those with hanging

nodes. The method has h-p adaptivity and very high parallel eﬃciency.

A particular type of DG methods that is related to this thesis is central DG (CDG)

scheme. The CDG schemes [48, 50, 51] are a class of DG schemes on overlapping cells that

combine the idea of the central schemes [56, 45, 49] with the DG weak formulation. Such

methods are intrinsically Riemann solver free, therefore no costly ﬂux evaluations are needed

in the computation.

It is well known that the CDG schemes allow larger CFL numbers

than the standard DG methods except for piecewise constant approximations [50, 60]. This

compensates the increased cost caused by duplicate representation of the solution on the

dual mesh.

1

In this thesis, we will focus on the design of a new DG method for one-dimensional

Schr¨odinger equations and its error analysis, and superconvergence analysis, as well as a

new sparse-grid central DG method for high-dimensional hyperbolic equations to make the

simulation more eﬃcient and accurate.

1.2 UWDG method for one-dimensional Schr¨odinger

equations

In this section, we introduce the one-dimensional time-dependent nonlinear Schr¨odinger

(NLS) equation and review the numerical methods designed to solve this equation.

The NLS equation is written as follows:

iut + uxx + f (|u|2)u = 0,

(1.1)

where f (u) is a nonlinear real function and u is a complex function. The Schr¨odinger equation

is the fundamental equation in quantum mechanics, reaching out to many applications in ﬂuid

dynamics, nonlinear optics and plasma physics. It is also called Schr¨odinger wave equation

as it can describe how the wave functions of a physical system evolve over time. Many

numerical methods have been applied to solve NLS equations [14, 28, 42, 43, 58, 71, 74]. In

[14, 74], several important ﬁnite diﬀerence schemes are implemented, analyzed and compared.

In [58], the author introduced a pseudo-spectral method for general NLS equations. Many

ﬁnite element methods have been tested, such as quadratic B-spline for NLS in [28, 71] and

space-time DG method for nonlinear (cubic) Schr¨odinger equation in [42, 43].

Various types of DG schemes has been applied to solve Schr¨odinger equations and they

2

have diﬀerent discretization for the second order spatial derivative term. One group of such

methods is the so-called local DG (LDG) method invented in [25] for convection-diﬀusion

equations. The algorithm is based on introducing auxiliary variables and reformulating

the equation into its ﬁrst order form. In [80], an LDG method using alternating ﬂuxes is

developed with L2 stability and proved (k + 1

2)-th order of accuracy. Later in [81], Xu and

Shu proved optimal accuracy for both the solution and the auxiliary variables in the LDG

method for high order wave equations based on reﬁned energy estimates. In [47], the authors

presented an LDG method with exponential time diﬀerencing Runge-Kutta scheme and

investigated the energy conservation performance of the scheme. Another group of method

involves treating the second order spatial derivative directly in the weak formulations, such

as IPDG method [77, 30] and NIPG method [62, 63]. Those schemes enforce a penalty jump

term in the weak formulation, and they have been extensively applied to acoustic and elastic

wave propagation [37, 3, 61]. As for Schr¨odinger equations, the direct DG (DDG) method

was applied to Schr¨odinger equation in [52] and achieved energy conservation and optimal

accuracy.

In Chapter 2, we choose to discretize the second order spatial derivative term directly

using UWDG method, which can be traced backed to [13], and refer to those DG methods

[72] that rely on repeatedly applying integration by parts so all the spatial derivatives are

shifted from the solution to test function in the weak formulation. In [18], Cheng and Shu

developed UWDG methods for general time dependent problems with higher order spatial

derivatives. In [6], Bona et. al. proposed an UWDG scheme for generalized KdV equation

and performed error estimates.

We investigate a most general form of the numerical ﬂux functions that ensures stability

along with our ultra-weak formulation. To estimate the convergence rate of our scheme,

3

we introduce special projections associated diﬀerent ﬂux parameters, and proved detailed

estimates for the projections. With the results for the convergence rate of the projections, a

priori L2 error estimates are obtained. Numerical tests are provided to verify the theoretical

results for projection operators and solution convergence rates under various ﬂux parameters.

1.3 Superconvergence analysis of DG methods

The study of superconvergence is of importance because a posteriori error estimates can be

derived guiding adaptive calculations. For superconvergence of DG methods, many results

exist in the literature. We refer the readers to [2, 1] for ordinary diﬀerential equation re-

sults. In [19], Cheng and Shu proved that the DG and LDG solutions are (k + 3/2)-th order

superconvergent towards projections of exact solutions of hyperbolic conservation laws and

convection-diﬀusion equations using specially designed test functions when piecewise poly-

nomials of degree k are used. For linear hyperbolic problems, in [82], Yang and Shu proved

that, under suitable initial discretization, the DG solutions of linear hyperbolic systems are

convergent with optimal (k + 2)-th order at Radau points. More recently, in [12], Cao et al

proved the (2k + 1)-th superconvergence rate for cell average and DG numerical ﬂuxes by

introducing a locally deﬁned correction function. The correction function also helps simplify

the proof for point wise (k + 2)-th superconvergence rate at Radau points and prove the

derivative of DG solution has (k + 1)-th superconvergence rate at so-called “left Radau”

points. Then this technique has been extended to prove the superconvergence of DG so-

lutions for linear and nonlinear hyperbolic PDEs in [11, 9], DDG method for convection

diﬀusion equations [10] and LDG method for linear Schr¨odinger equations [84]. Overall, for

equations with higher order spatial derivatives, the same type of correction functions can

4

be used for the LDG method which is based on a reformulation into a ﬁrst order system of

equations. For DDG method, new correction functions are needed treating the second order

derivative directly [10].

Another type of superconvergence of DG methods is achieved by postprocessing the

solution by convolution with a kernel function, which is a linear combination of B-spline

functions. For linear hyperbolic systems, [23] provided a framework for constructing such

postprocessor and proving the superconvergence of the postprocessed DG solutions. Through

the analysis of negative norm estimates and divided diﬀerence estimates, they showed that

the postprocessed solution is superconvergent at a rate of 2k + 1. More recently, in [41, 54]

the analysis are extended to scalar nonlinear hyperbolic equations.

In Chapter 3, we study the superconvergence properties of the UWDG methods for linear

Schr¨odinger equation with scale invariant ﬂux parameters. Such choice include all commonly

used ﬂuxes, e.g. alternating, central, DDG and interior penalty DG (IPDG) ﬂuxes. Using the

correction function idea and negative norm estiamtes, we are able to prove superconvergence

rate for solution and its derivatives at certain points, for cell averages and numerical ﬂuxes,

for solution towards a special projection, and for the postprocessed solution.

1.4 Sparse-grid DG methods for high-dimensional

PDEs

It has been a challenging problem for numerical simulations in high dimensions due to the

so-called curse of dimensionality, which means the cost of computing and storing an approx-

imation with a prescribed accuracy increases exponentially on dimension d. Many discretiza-

tion techniques and computation techniques have been developed to alleviate the problem

5

to some extent. Among them, the sparse grid method has been a successful tool. It was

originally developed to solve PDEs [83, 33] based on tensor product hierarchical basis rep-
resentation. The method can reduce the full grid discretization complexity from O(h−d) to
O(h−1| log2 h|d−1), where h is the uniform mesh size in each dimension, and only slightly

deteriorate the accuracy. In recent years, sparse grid techniques have been incorporated in

collocation methods for high-dimensional stochastic diﬀerential equations [79, 78, 57, 53],

ﬁnite element methods [83, 8, 66], ﬁnite diﬀerence methods [34, 36], ﬁnite volume methods

[40], and spectral methods [35, 32, 68, 69] for high-dimensional PDEs.

Recently, our research group initiated a line of research on the development of sparse

grid DG methods [76, 38, 39]. The sparse grid DG methods use the sparse ﬁnite element

space, which has multidimensional multiwavelet bases constructed by tensor products from

one-dimensional wavelet basis, in the DG framework to treat high-dimensional problems.
The methods has been proven to reduce the degrees of freedom of O(h−d) in the standard
full grid approximation space to O(h−1| log2 h|d−1) and remain a L2 convergence rate of
O(hk+1/2| log2 h|d) for transport equations and a convergence rate of O(hk| log2 h|d) in the

energy norm for elliptic equations.

In Chapter 4, we incorporate sparse grid DG method with central DG scheme to de-

velop a class of conservative numerical schemes with high computational eﬃciency for high-

dimensional hyperbolic equations. Our work consists of construction of the sparse ﬁnite

element space , L2 stability and error estimates, and numerical validation of the scheme.

6

Chapter 2

An UWDG method for Schr¨odinger

equation in one dimension

In this chapter, we develop and analyze a new ultra-weak discontinuous Galerkin (UWDG)

method for solving one-dimensional nonlinear Schr¨odinger (NLS) equations (1.1). The

method solves the equation without introducing any auxiliary variables or rewriting the

equation into a larger system.

The focus of this chapter is on the investigation of a most general form of the numerical

ﬂux functions that ensures stability along with our ultra-weak formulation. The ﬂuxes

under consideration include the alternating ﬂuxes, and also the ﬂuxes considered in [52], and

therefore allows for ﬂexibility for the design of the schemes. The analysis in this chapter

relies on a detailed analysis of a special projection associated with diﬀerent ﬂux parameters,

whose dependence on mesh size can be freely enforced. Under certain ﬂux parameters, the

projection can be deﬁned locally. For other ﬂux parameters, the projection is global and the
projection analysis is based on a block-circulant matrix with 2×2 blocks. Our analysis reveals

that under a large class of parameter choices, the UWDG method is optimally convergent

in L2 norm, which is veriﬁed by extensive numerical tests for both the projection operators

and the numerical schemes for (1.1).

The remainder of this chapter is organized as follows. In Section 2.1, we introduce the

7

UWDG method with general ﬂux deﬁnitions for one-dimensional nonlinear Schr¨odinger equa-

tions and study its stability properties. We introduce a new projection operator and analyze

its properties in Section 2.2, which is later used in Section 2.3 to obtain the convergence

results of the schemes. The main body of this chapter, the error estimates, is contained

in Section 2.3. Numerical validations are provided in Section 2.4. Some technical details,

including proof of most lemmas are collected in Appendix.

The major contents of this chapter has been published in [16].

2.1 Numerical scheme and stability

In this subsection, we formulate and discuss stability results of a DG scheme for one-

dimensional NLS equation (1.1) on interval I = [a, b] with initial condition u(x, 0) = u0(x)

and periodic boundary conditions. Here f (u) is a given real function. Our method can be

deﬁned for general boundary conditions, but the error analysis will require slightly diﬀerent

tools, and therefore we only consider periodic boundary conditions in this chapter.

To facilitate the discussion, ﬁrst we introduce some notations and deﬁnitions. For a 1-D

interval I = [a, b], the usual DG meshes are deﬁned as:

and

a = x 1
2

< x 3
2

< ··· < x

N + 1
2

= b,

Ij = (x

j− 1
2

, x

j+ 1
2

),

xj =

1
2

(x

j− 1
2

+ x

),

j+ 1
2

hj = x

j+ 1
2

− x

,

j− 1
2

h = max

j

hj,

8

with mesh regularity requirement

< σ, σ is ﬁxed during mesh reﬁnement.

h

min hj

Denote ZN = 1, 2,··· , N . The approximation space is deﬁned as:

h = {vh : vh|Ij
V k

∈ P k(Ij), ∀j ∈ ZN},

h , we use (vh)−
j− 1
2

meaning vh is a piecewise polynomial of x with degree up to k on each cell Ij. For a function
vh ∈ V k
Ij−1 and the right cell Ij respectively. The jump and average are deﬁned as [vh] = v+
and {vh} = 1

to refer to the value of vh at x

from the left cell

h − v−

and (vh)+

h ) at cell interfaces.

h + v−

2(v+

j− 1
2

j− 1
2

h

Throughout this chapter, we use the standard Sobolev norm notations (cid:107) · (cid:107)W s,p(I)

and broken Sobolev space on mesh IN . We denote (cid:107)v(cid:107)2
Hs(Ij ) and
(cid:107)v(cid:107)W s,∞(IN ) = maxj (cid:107)v(cid:107)W s,∞(Ij ). In Section 3.4, we consider negative norms and the def-
=(cid:80)N
inition is (cid:107)v(cid:107)
. Additionally, we denote by (cid:107)v(cid:107)L2(∂IN ) the
Hl(I)
broken L2 norm on cell interfaces, i.e., (cid:107)v(cid:107)2
L2(∂Ij )
(v−
)2. We also denote (cid:107)·(cid:107) = (cid:107)·(cid:107)L2(I) = (cid:107)·(cid:107)L2(IN ) to shorten the notation.

I v(x)Φ(x)dx
(cid:107)Φ(cid:107)

, where(cid:107)v(cid:107)2

= supΦ∈C∞

j=1 (cid:107)v(cid:107)2

j=1 (cid:107)v(cid:107)2

L2(∂IN )

)2 + (v+
x

H−l(I)

L2(∂Ij )

0 (I)

(cid:82)

=

x
j+ 1
2
Lastly, we recall inverse inequalities, ∀vh ∈ V k
h ,

j− 1
2

Hs(IN ) = (cid:80)N

(cid:107)(vh)x(cid:107)L2(Ij ) ≤ Ch−1

j (cid:107)vh(cid:107)L2(Ij ),

− 1
2(cid:107)vh(cid:107)L2(Ij ),
(cid:107)vh(cid:107)L2(∂Ij ) ≤ Ch
− 1
(cid:107)vh(cid:107)L∞(Ij ) ≤ Ch
2(cid:107)vh(cid:107)L2(Ij ),

(2.1)

and trace inequalities

(cid:107)v(cid:107)2

L2(∂Ij )

≤ Ch−1

j (cid:107)v(cid:107)2

L2(Ij )

,

(2.2)

here and below C is a constant independent of the function u and the mesh size h.

In this chapter, we consider a DG scheme motivated by [18] and based on integration by

9

parts twice, or the so-called ultra-weak formulation. In particular, we look for the unique
function uh = uh(t) ∈ V k

h , t ∈ (0, T ], such that

(cid:90)

Ij

(cid:90)

i

(uh)tvhdx +

uh(vh)xxdx − ˆuh(vh)−
x |
(cid:90)
j+ 1
2

Ij

+(cid:94)(uh)xv−
h |
j+ 1
2

− (cid:94)(uh)xv+
h |
j− 1
2

+

Ij

x |
+ ˆuh(vh)+
j− 1
2

f (|uh|2)uhvhdx = 0

(2.3)

holds for all vh ∈ V k

h and all j = 1, ··· , N . Here, we require k ≥ 1, because k = 0 yields an

inconsistent scheme. Notice that (2.3) can be written equivalently in a weak formulation by

performing another integration by parts back as:

(cid:90)

i

Ij

(cid:90)

Ij

(uh)tvhdx −

(uh)x(vh)xdx + (u−

h − ˆuh)(vh)−
x |
j+ 1
2
− (cid:94)(uh)xv+
h |
j− 1
2

+

+ (ˆuh − u+
(cid:90)

x |
h )(vh)+
j− 1
2

f (|uh|2)uhvhdx = 0

(2.4)

Ij

+(cid:94)(uh)xv−
h |
j+ 1
2

The “hat” and“tilde” terms are the numerical ﬂuxes we pick for u and ux at cell bound-

aries, which are single valued functions deﬁned as:

(cid:94)(uh)x = {(uh)x} + α1[(uh)x] + β1[uh],

ˆuh = {uh} + α2[uh] + β2[(uh)x],

(2.5)

where α1, α2, β1, β2 are prescribed complex parameters. They may depend on the mesh

parameter h. Commonly used ﬂuxes such as the central ﬂux (by setting α1 = α2 = β1 =
β2 = 0) and alternating ﬂuxes (by setting α1 = −α2 = ± 1

2 , β1 = β2 = 0) belong to this

ﬂux family. The direct DG scheme considered in [52] is a special case of our method when
α1 = −α2, β1 = c
framework as α1 = α2 = β2 = 0, β1 = c

h, β2 = 0, c > 0, α1 ∈ R. The IPDG method can also be casted in this

h, c > 0.

10

We write the scheme (2.3) in following short-hand notation:

(cid:90)

Ij

aj(uh, vh) − i

holds for all vh ∈ V k

h , where

f (|uh|2)uhvhdx = 0,

∀j ∈ ZN

(2.6)

(cid:90)

Ij

(uh)tvhdx − iAj(uh, vh),

aj(uh, vh) =

with

Aj(uh, vh) =

(cid:90)

Ij

uh(vh)xxdx − ˆuh(vh)−
x |
j+ 1
2

x |
+ ˆuh(vh)+
j− 1
2

+ (cid:94)(uh)xv−
h |
j+ 1
2

− (cid:94)(uh)xv+
h |
j− 1
2

as the UWDG spatial discretization for the second order derivative term.

Using periodic boundary condition, we sum up on j for (2.6) and get

(cid:90)

I

a(uh, vh) − i

f (|uh|2)uhvhdx = 0,

(2.7)

where

a(uh, vh) =

A(uh, vh) =

(cid:90)
(uh)tvhdx − iA(uh, vh),
N(cid:88)

(cid:90)

I

Aj(uh, vh) =

uh(vh)xxdx +

I

j=1

N(cid:88)

j=1

(cid:16)
ˆuh[(vh)x] − (cid:94)(uh)x[vh]

(cid:17)(cid:12)(cid:12)j+ 1

2

.

The following theorem contains the results on semi-discrete L2 stability.

Theorem 2.1.1. (Stability) For u, v ∈ H2(IN ) satisfying periodic boundary condition, we

have A(u, v) = A(v, u).

11

The solution of semi-discrete UWDG scheme (2.3) using numerical ﬂuxes (2.5) satisﬁes

L2 stability condition

if

d
dt

(cid:90)

I

|uh|2dx ≤ 0,

Imβ2 ≥ 0, Imβ1 ≤ 0, |α1 + α2|2 ≤ −4Imβ1Imβ2.

(2.8)

In particular, when all parameters α1, α2, β1, β2 are restricted to be real, this condition

amounts to

without any requirement on β1, β2.

α1 + α2 = 0

(2.9)

Proof. From integration by parts, we have

A(u, v) = −

uxvxdx +

Similarly, A(v, u) = −(cid:82)

I

I uxvxdx +(cid:80)N

(cid:90)

N(cid:88)
(ˆu[vx] − [uvx] −(cid:102)ux[v])(cid:12)(cid:12)j+ 1
j=1 (ˆv[ux] − [vux] − (cid:101)vx[u])(cid:12)(cid:12)j+ 1

j=1

.

2

2

,∀j ∈ ZN

j+ 1
2

. Plugging in the deﬁ-

nition of the numerical ﬂuxes in (2.5), we have at x

ˆu[vx] − [uvx] −(cid:102)ux[v] =(cid:0){u} − α1[u] + β2[ux](cid:1)[vx] − ({u}[vx] + [u]{vx})

−(cid:0){ux} + α1[ux] + β1[u](cid:1)[v]
= [ux](cid:0){v} − α1[v] + β2[vx](cid:1) − [u](cid:0){vx} + α1[vx] + β1[v](cid:1)
−(cid:0)[ux]{v} + {ux}[v](cid:1)
= [ux]ˆv − [u](cid:101)vx − [uxv],

12

then the proof for A(u, v) = A(v, u) is complete.

From integration by parts, we have, for ∀vh ∈ V k
N(cid:88)

(cid:90)

(cid:90)

h

(uh)tvhdx + i

(uh)x(vh)xdx + i

a(uh, vh) =

I

I

j=1

([uh(vh)x] − ˆuh[(vh)x] + (cid:94)(uh)x[vh])|

.

j+ 1
2

(cid:90)

I

A(uh, vh) = −

N(cid:88)

(uh)x(vh)xdx +

j=1

(ˆuh[(vh)x] − [uh(vh)x] − (cid:94)(uh)x[vh])|

j+ 1
2

Taking vh = ¯uh in (2.7) and compute its conjugate as well, we get

(cid:90)

(cid:90)

I

f (|uh|2)|uh|2dx

0 = i

I

f (|uh|2)|uh|2dx + i
(cid:90)

= a(uh, ¯uh) + a(uh, ¯uh)

=

d
dt

I

|uh|2dx − iA(uh, uh) + iA(uh, uh).

.

(2.10)

−iA(uh, uh) + iA(uh, uh) = −2Im

N(cid:88)

([uh(¯uh)x] − ˆuh[(¯uh)x] + (cid:94)(uh)x[¯uh])|

j=1

j+ 1
2

(cid:16){uh}[(¯uh)x] + [uh]{(¯uh)x} −(cid:0){uh} + α2[uh] + β2[(uh)x](cid:1)[(¯uh)x]
(cid:0) − β2|[(uh)x]|2 + β1|[uh]|2 + α1[(uh)x][¯uh] − α2[uh][(¯uh)x](cid:1)|

(cid:17)|

j+ 1
2

j+ 1
2

+ ({(uh)x} + α1[(uh)x] + β1[uh])[¯uh]

= −2Im

= −2Im

N(cid:88)

j=1

N(cid:88)
N(cid:88)

j=1

= 2Im

(β2|[(uh)x]|2 − β1|[uh]|2 − (α1 + α2)[¯uh][(uh)x])|

j+ 1
2

j=1

13

Plug it back into (2.10):

(cid:90)

I

d
dt

N(cid:88)

j=1

|uh|2dx +

2Imβ2|[(uh)x]|2 − 2Imβ1|[uh]|2 − 2Im{(α1 + α2)[¯uh][(uh)x]}|

= 0.

j+ 1
2

(2.11)

If the stability condition (2.8) is satisﬁed, we have

If all parameters are real and (2.9) is satisﬁed, then (2.11) further yields:

d
dt

|uh|2dx ≤ 0.

(cid:90)

(cid:90)

I

I

d
dt

which implies energy conservation.

|uh|2dx = 0,

(2.12)

For simplicity of the discussion, in the contents below, we will only consider real param-

eters, i.e. when α1, α2, β1, β2 are real and α1 + α2 = 0. This property of our scheme is

consistent with the energy conservation property of Schr¨odinger equations. It is essential to

have a symmetric A(uh, vh) for designing a ﬁnite element scheme which is energy-preserving

for Schr¨odinger equations.

Now the numerical ﬂuxes are deﬁned by three parameters as,

(cid:94)(uh)x = {(uh)x} + α1[(uh)x] + β1[uh],

ˆuh = {uh} − α1[uh] + β2[(uh)x], α1, β1, β2 ∈ R.

(2.13)

14

Note we can rewrite the ﬂux deﬁnition in a matrix form

 ˆuh

(cid:94)(uh)x

 = G

 u−

h

 + H

 u+

h

 , G =

 1

(uh)−

x

(uh)+
x

 , H =

 1

2 − α1
β1

 ,

β2

1
2 + α1

2 + α1 −β2
2 − α1
−β1

1

(2.14)
where I2 denotes the 2 × 2 identity matrix. Note that G + H = I2, det G = det H =
−(α2

4) and GH = −(det G)I2.

1 + β1β2 − 1

2.2 Projection P (cid:63)
h

In this section, we perform detailed studies of a projection operator that is key to the analysis

of the UWDG scheme.

Deﬁnition 2.2.1. For the UWDG scheme with ﬂux choice (2.13), we deﬁne the associated

h for any periodic function u ∈ W 1,∞(I) to be the unique polynomial

projection operator P (cid:63)
h u ∈ V k
P (cid:63)

h (when k ≥ 1) satisfying
(cid:90)

(cid:90)

Ij

P (cid:63)
h u vhdx =

Ij

(cid:100)P (cid:63)
h u = {P (cid:63)
(cid:94)(P (cid:63)
h u)x = {(P (cid:63)

h u} − α1[P (cid:63)
h u)x} + α1[(P (cid:63)

h u] + β2[(P (cid:63)

h u)x] = u

h u)x] + β1[P (cid:63)

h u] = ux

u vhdx

∀vh ∈ P k−2(Ij),

(2.15a)

at x

at x

,

,

j+ 1
2

j+ 1
2

(2.15b)

(2.15c)

for all j. When k = 1, only conditions (2.15b)-(2.15c) are needed.

This deﬁnition is to ensure u − (cid:100)P (cid:63)

h u = 0 and ux − (cid:94)(P (cid:63)

h u)x = 0 at cell boundaries, which

will be used in error estimates for the scheme. In the following, we analyze the projection

when the parameter choice reduces it to a local projection in Section 2.2.1, and then we

15

consider the more general global projection in Section 2.2.2.

We can write (2.15b)-(2.15c) in vector form as

 (cid:100)P (cid:63)

h u
(cid:94)(P (cid:63)
h u)x

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x


= G

j+ 1
2

 P (cid:63)

h u

(P (cid:63)

h u)x

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


 P (cid:63)

h u

(P (cid:63)

h u)x

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


+

x

j+ 1
2

 u

ux

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x


=

j+ 1
2

.

(2.16)

−

x

j+ 1
2

+ H

2.2.1 Local projection condition

In general, the projection P (cid:63)

h is globally deﬁned, and its existence, uniqueness and ap-

proximation properties are quite complicated mathematically. However, with some special

parameter choices, P (cid:63)

h can be reduced to a local projection, meaning that it can be solved

element-wise, and hence the analysis can be greatly simpliﬁed.

For example, with the alternating ﬂuxes α1 = ± 1
h for parameter choice α1 = 1
P 1
h and P 2
as: for each cell Ij, we ﬁnd the unique polynomial of degree k, P 1

h deﬁned below. P (cid:63)

h = P 1

2 , β1 = β2 = 0, P (cid:63)

h can be reduced to

2 , β1 = β2 = 0 is formulated

h u, satisfying

(cid:90)

Ij

(cid:90)

P 1
h u vhdx =
Ij
h u)− = u
(P 1

(P 1

h u)+

x = ux

u vhdx

∀vh ∈ P k−2(Ij),

at x

j+ 1
2

at x

j− 1
2

,

.

(2.17a)

(2.17b)

(2.17c)

When k = 1, only conditions (2.17b)-(2.17c) are needed.

Similarly, we can deﬁne P (cid:63)

h = P 2

h for parameter choice α1 = − 1

2 , β1 = β2 = 0 as: for

16

each cell Ij, we ﬁnd the unique polynomial of degree k, P 2

h u, satisfying

(cid:90)

Ij

(cid:90)

Ij

P 2
h u vhdx =

u vhdx

∀vh ∈ P k−2(Ij),

h u)+ = u
(P 2
h u)−
(P 2

x = ux

at x

j− 1
2

at x

j+ 1
2

,

.

(2.18a)

(2.18b)

(2.18c)

When k = 1, only conditions (2.18b)-(2.18c) are needed.

Similar local projections have been introduced and considered in [18]. It is obvious that

h u can be solved element-wise, and their existence, uniqueness are straightforward.

P 1
h u, P 2
From a standard scaling argument by Bramble-Hilbert lemma in [20], P 1
following error estimates: let u ∈ W k+1,p(Ij)(p = 2,∞), then

h and P 2

h have the

(cid:107)u − P ν

(cid:107)ux − P ν

h u(cid:107)Lp(Ij ) ≤ Chk+1
h ux(cid:107)Lp(Ij ) ≤ Chk

j

j|u|

|u|

W k+1,p(Ij )

W k+1,p(Ij )

p = 2,∞, ν = 1, 2,

p = 2,∞, ν = 1, 2,

,

,

(2.19)

where here and below, C is a generic constant that is independent of the mesh size hj, the

parameters α1, β1, β2 and the function u, but may take diﬀerent value in each occurrence.

Naturally, the immediate question is that if there are other parameter choices such that

P (cid:63)
h can be reduced to a local projection. The following lemma addresses this issue.

Lemma 2.2.1 (The condition for reduction to a local projection). If α2

1 + β1β2 = 1

4, P (cid:63)

h is

17

a local projection. Moreover, (2.15b) and (2.15c) is equivalent to

 P (cid:63)

h u

(P (cid:63)

h u)x

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


G

−

x

j+ 1
2

+ H

 P (cid:63)

h u

(P (cid:63)

h u)x

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


= G

+

x

j− 1
2

 u

ux

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x


+ H

j+ 1
2

 u

ux

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x


j− 1
2

.

(2.20)

Proof. The deﬁnition (2.15a) provides k − 1 linearly independent equations for solving P (cid:63)
h u

on each cell. If (2.15b) and (2.15c) can be locally decoupled, P (cid:63)

assumption α2

1 + β1β2 = 1

4, if β1 = β2 = 0, then α1 = ± 1

2 and P (cid:63)

h is a local projection. By
h u = P 1

h , and (2.20)

h or P 2

holds. The rest of the cases are

• if β1 (cid:54)= 0, left multiply (2.16) by a matrix, we have

β1

1
2 + α1
2 − α1)

β1 −( 1



 u

ux

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x


 0
β1

0

β1 −( 1

2 − α1)

0



1
2 + α1

0

h u

 P (cid:63)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

 P (cid:63)

h u)x

(P (cid:63)

(P (cid:63)

h u

h u)x

+

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


−

x

j+ 1
2

,

x

j+ 1
2

j+ 1
2

=

+

which implies the following decoupled relations

h u)+ +
(P (cid:63)
h u)− − 1
(P (cid:63)

1
2 + α1
β1
2 − α1
β1

h u)+
(P (cid:63)
h u)−
(P (cid:63)

x = u +
x = u − 1

1
2 + α1
β1
2 − α1
β1

ux at x

ux at x

,

.

j− 1
2

j+ 1
2

(2.21)

18

• if β2 (cid:54)= 0, by similar linear transformation, we have

 1

2 − α1
−( 1
2 + α1) β2

β2



 u

ux

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x



 1

j+ 1
2

=

+

0

h u

 P (cid:63)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

 P (cid:63)

h u)x

(P (cid:63)

(P (cid:63)

h u

h u)x

+



(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


−

x

j+ 1
2

,

x

j+ 1
2

0

−( 1

2 + α1) β2

2 − α1 β2

0

0

which implies

(P (cid:63)
h u)+
x +
x − 1
h u)−
(P (cid:63)

1

2 − α1
β2

2 + α1

β2

(P (cid:63)
h u)+ = ux +
h u)− = ux − 1
(P (cid:63)

1

2 − α1
β2

2 + α1

β2

u at x

u at x

,

.

j− 1
2

j+ 1
2

(2.22)

(2.21), (2.22) are the desired decoupled conditions on each cell Ij, and it’s easy to verify that

(2.21), (2.22) are equivalent to (2.20). Therefore the proof is complete.

This lemma implies that for any parameter satisfying α2

1+β1β2 = 1

4, P (cid:63)

h is locally deﬁned.

We remark that this condition turns out to be the same as the optimally convergent numerical

ﬂux families in [17] for two-way wave equations, although they arise in diﬀerent contexts.

Unfortunately, for the general deﬁnition of P (cid:63)

h , unlike P 1

h and P 2

h , we cannot directly use the

Bramble-Hilbert lemma and the standard scaling argument to obtain optimal approximation

property, since the second and third relations in (2.21) and (2.22) may break the scaling.

19

2.2.2 P (cid:63)

h properties

Before moving on to detailed discussion on P (cid:63)
h , we introduce some notations to facilitate the
discussion. We deﬁne the Legendre expansion of a function u ∈ L2(I) on cell Ij as follows,

∞(cid:88)

m=0

u|Ij

=

uj,mLj,m(x),

(2.23)

where Lj,m(x) := Lm(ξ), ξ =
degree m on [−1, 1].

x−xj
hj /2 , and Lm(·) is the standard Legendre polynomial of

In what follows, we write Lj,m(x) as Lj,m, and Lm(ξ) as Lm for

notational convenience. We can compute uj,m using orthogonality of Legendre polynomials

(cid:90) 1

2m + 1

and Rodrigues’ formula,

uj,m =

=

=

2m + 1

hj

2m + 1

2

2m + 1

2

(cid:90)

Ij
1

2mm!
(−1)l
2mm!

(cid:90) 1
(cid:90) 1

−1

−1

u(x)Lj,mdx =

ˆuj(ξ)Lmdξ

2

−1
dξm (ξ2 − 1)mdξ

d

ˆuj(ξ)

d
dξl

ˆuj(ξ)

d

dξm−l

(ξ2 − 1)mdξ,

(2.24)

where ˆuj(ξ) = u(x(ξ)) is deﬁned as the function u|Ij
[−1, 1]. By Holder’s inequality, if u ∈ W l,p(I),

transformed to the reference domain

(cid:12)(cid:12)uj,m

(cid:12)(cid:12) ≤ Ch

l− 1
p
j

|u|

W l,p(Ij )

0 ≤ l ≤ m.

,

(2.25)

The L2 projection P 0

h is closely related to uj,m. By orthogonality of Legendre polynomi-

als, we have

P 0
h u =

k(cid:88)

uj,mLj,m.

We collect some frequently used notations in Table 2.1 for quick reference.

m=0

20

Table 2.1: Notations for some frequently used quantities. Subscript j will be dropped for
uniform mesh.

β1 +

β2
h2
j

1 + β1β2 + 1
4)

Notation

G

Γj

L−

j,m

Aj
Q
Mj,m

(cid:21)


)

1

(cid:20) 1
Deﬁnition
2 + α1 −β2
−β1
2 − α1
 Lj,m(x
k2(k2 − 1) − 2k2
(α2
hj

j+ 1
2
2
d
dx Lj,m(x
j+ 1
hj
j,k−1, L−
G[L−
2
j,k]
−A−1B
(Aj + Bj)−1(GL−

)

j,m + HL+

j,m)

− 2k
hj

Deﬁnition
β2

(cid:20) 1
(cid:21)
2 − α1
1
2 + α1
β1

 Lj,m(x
1 + β1β2 − 1
(α2
4)

j− 1
2
2
d
dx Lj,m(x
j− 1
hj
2
H[L+
j,k−1, L+
j,k]
Ql(I2 − QN )−1

)

)

Notation

H

Λj

L+

j,m

Bj
rl

Next, we will write the explicit formula of P (cid:63)

h in order to get a clear view of the existence

and uniqueness condition, as well as the error estimates. Suppose

k(cid:88)

m=0

P (cid:63)
h u =

´uj,mLj,m.

By the deﬁnition (2.15a), ´uj,m = uj,m, m ≤ k − 2, i.e.,
k(cid:88)

k−2(cid:88)

P (cid:63)
h u =

m=0

uj,mLj,m +

´uj,mLj,m.

m=k−1

In what follows, we analyze the existence and uniqueness of P (cid:63)

h , i.e., the existence and

uniqueness of ´uj,k−1, ´uj,k based on the following assumptions on parameters:

4 and Γj (cid:54)= 0.

• A0. (Local projection) α2

1 + β1β2 = 1

• A1. (Global projection) uniform mesh (hj = h,∀j), α2

(cid:12)(cid:12)(cid:12) > 1.
(cid:12)(cid:12)(cid:12) Γ
(cid:12)(cid:12)(cid:12) = 1.
(cid:1)N (cid:54)= 1. If N is odd, and if k is odd, we require Γ = −Λ; if k is even,

(Global projection) uniform mesh (hj = h,∀j), α2

1 + β1β2 (cid:54)= 1
4,

(cid:0)(−1)k+1 Γ

1 + β1β2 (cid:54)= 1

(cid:12)(cid:12)(cid:12) Γ

• A2.

4 and

Λ

Λ

Λ

21

we require Γ = Λ.

(Global projection) uniform mesh (hj = h,∀j), α2

1 + β1β2 (cid:54)= 1
4,

• A3.

(cid:32)

(−1)k+1 Γ

Λ +

(cid:114)(cid:16) Γ

Λ

(cid:33)N
(cid:17)2 − 1

(cid:54)= 1.

(cid:12)(cid:12)(cid:12) Γ

Λ

(cid:12)(cid:12)(cid:12) < 1,

Lemma 2.2.2 (P (cid:63)

h existence, uniqueness and formula). If any of the assumptions above is

satisﬁed, P (cid:63)

h exists and is uniquely deﬁned. Furthermore, if assumption A0 is satisﬁed, then

´uj,k−1

 =

uj,k−1

 +

´uj,k

uj,k

∞(cid:88)

m=k+1

uj,mMj,m.

(2.26)

If any of the assumptions A1/A2/A3 is satisﬁed, then

´uj,k−1

 =

uj,k−1

 +

´uj,k

uj,k

∞(cid:88)

(cid:0)uj,mV1,m +

m=k+1

N−1(cid:88)

l=0

(cid:1),

uj+l,mrlV2,m

(2.27)

k−1, L−
where V1,m = [L+
when j + l ≥ N , and rl is deﬁned in Table 2.1.

m, V2,m = [L−

k ]−1L+

k−1, L+

k ]−1L−

m−[L+

k−1, L+

k ]−1L+

m, uj+l = uj+l−N

Proof. The proof of this lemma can be found in Appendix.

Lemma 2.2.3. Suppose any of the assumptions A0/A1/A2/A3 holds and u satisﬁes the
condition in Deﬁnition 2.2.1. For p = 2,∞, if assumption A0 is satisﬁed,

(cid:18)

(cid:19)

|β1|,

| 1
2 +α1|

h

| 1
2−α1|
,
minj |Γj|

h

|β2|
h2

,

.

(2.28)

(cid:107)P (cid:63)

h u − u(cid:107)Lp(I) ≤ Chk+1|u|

W k+1,∞(I)

max

22

If assumption A1 is satisﬁed,

(cid:107)P (cid:63)

h u − u(cid:107)Lp(I) ≤ Chk+1|u|

W k+1,∞(I)

(1 +

(cid:107)Q1(cid:107)∞
|1 − |λ1|| +

(cid:107)I2 − Q1(cid:107)∞
|1 − |λ2||

),

(2.29)

where λ1, λ2 are deﬁned in (49), Q1 is deﬁned in (56) and (57).

If assumption A2 is satisﬁed,

(cid:107)P (cid:63)

h u − u(cid:107)Lp(I) ≤ Chk+1(cid:107)u(cid:107)

W k+4,∞(I)

(1 +

(cid:107)Q2(cid:107)∞
|Γ|

),

(2.30)

where Q2 is deﬁned in (59).

If assumption A3 is satisﬁed, and assuming

(cid:12)(cid:12)(cid:12)1 − λN

1

(cid:12)(cid:12)(cid:12) = O(hδ(cid:48)

), |1 − λ1| = O(hδ/2) with

0 ≤ δ ≤ 2,

(cid:107)P (cid:63)

h u − u(cid:107)Lp(I) ≤ Chk+1(cid:107)u(cid:107)

W k+3,∞(I)

(1 + h−δ(cid:48)−δ/2((cid:107)Q1(cid:107)∞ + (cid:107)I2 − Q1(cid:107)∞)).

(2.31)

Proof. Proof is given in Appendix.

Above estimates provides error bound that can be computed once the parameters are

given, yet its dependence on the mesh size h is not fully revealed, particularly when the

parameters α1, β1, β2 also have h-dependence. To clarify such relations, next we will inter-

pret (2.29) when considering the following common choice of parameters, where α1 has no

dependence on h, β1 = ˜β1hp1, β2 = ˜β2hp2, ˜β1, ˜β2 are nonzero constants that do not depend
on h. If indeed β1 or β2 is zero, it is equivalent to let p1, p2 → +∞ in the discussions below.

We will discuss whether the parameter choice yields optimal (k + 1)-th order accuracy.

To distinguish diﬀerent cases, we illustrate the choice of parameters p1, p2 in Figure 2.1.

23

Figure 2.1: A sketch to illustrate the diﬀerent cases parametrized by the values of p1, p2.
For example, A1.1 means p1 > −1, p2 > 1, A1.5 means p1 = −1, p2 = 1 and A1.7.1 means
p1 > −1, p2 = 1. The main results are summarized in Table 2.2.

Table 2.2: Interpretation of error estimate (2.29).

1

2

If k = 1 and p2 < 1, then

P (cid:63)
h is suboptimal and is (k + p2)-th order accurate,

3 else

4

5

6

7

8

9 end

if limh→0 |λ1, λ2| = 1 with |λ1, λ2| = 1 + O(hδ/2), then

h is suboptimal and is (k + 1 − δ)-th order accurate,
P (cid:63)

else

end

P (cid:63)
h has optimal (k + 1)-th order error estimates.

The main reason of order reduction for k = 1, p2 < 1 in line 2 of Table 2.2 is that the

term such as

1|λ1|−1(cid:107)Q1(cid:107)∞ is of O(hp2−1) instead of O(1), and this will cause (1 − p2)-th

order reduction. The situation happens for A1.3, A1.4 and A1.6.2 when k = 1.

24

−3−2−10p1         −1012          p2A1.1A1.2A1.3A1.4A1.5A1.6.1A1.6.2A1.7.1A1.7.21

1

1

1−|λ1|,

1−|λ1|,

(cid:12)(cid:12)(cid:12) Γ

in (2.29). The fractions

By deﬁnition of λ1, λ2 in (49), we know that

The main reason of order reduction in line 5 is because of the terms such as

|1
1−|λ2|
1−|λ2| cannot be bounded by a constant if limh→0 |λ2| = 1.

(cid:12)(cid:12)(cid:12) = 1 + O(hδ), δ > 0, then |λ1, λ2| = 1 + O(hδ/2), then

(cid:12)(cid:12)(cid:12) → 1 ⇔ |λ1, λ2| → 1. More precisely, if
1−|λ2| = O(h−δ/2). The
relation Γ2 − Λ2 = (b1 − b2)(b1 + b2) + c2
2 also indicates that there is some cancellation of
leading terms in b1 − b2 or b1 + b2, making (cid:107)Q1(cid:107)∞ ∼ O(h−δ/2), multiplying these factors
together will result in δ-th order reduction in the error estimation of P (cid:63)

(cid:12)(cid:12)(cid:12) Γ

1−|λ1|,

Λ

Λ

1

1

h . Note that b1, b2, c2

and Q1 are deﬁned in (47), (48), (45) and (56).

Then we look at what parameter choices make

(cid:12)(cid:12)(cid:12) Γ

Λ

(cid:12)(cid:12)(cid:12) → 1. Since

k +

1 +

β1+

k2(k2−1)

h2
Λ

β2− k2

h

β1− 1

h

Λ

k > 1,

k = 1,

Γ
Λ

=

we have

Λ

Λ

(cid:12)(cid:12)(cid:12) Γ

(cid:12)(cid:12)(cid:12)(cid:12) 1

2 +2α2
1
2−2α2
1
1

(cid:12)(cid:12)(cid:12) →

1k(k ∓ 1),

(cid:12)(cid:12)(cid:12)(cid:12) = 1.
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)k +
(cid:12)(cid:12)(cid:12) Γ
(cid:12)(cid:12)(cid:12) →
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)k +
(cid:12)(cid:12)(cid:12) Γ
(cid:12)(cid:12)(cid:12) →
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)k +
(cid:12)(cid:12)(cid:12) →
(cid:12)(cid:12)(cid:12) Γ
(cid:12)(cid:12)(cid:12) → 1.
(cid:12)(cid:12)(cid:12) →(cid:12)(cid:12)(cid:12)k +

β1
Λ

2α2
1
k(k±1),

Λ

Λ

2

,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → 1.
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → 1.
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → 1.

− k2
h

β1− k2

h

Λ

k2(k2−1)β2

h2
Λ

k2(k2−1)β2

h2

Λ

1. A1.1 (p1 > −1, p2 > 1) with k = 1, α1 = 0,

2. A1.6.1 (p1 = −1, p2 > 1) ˜β1 = k(k±1)

2 + 2α2

3. A1.6.2 (p1 = −1, p2 < 1) with k > 1, ˜β1 = k(k±1)

4. A1.7.1 (p1 > −1, p2 = 1) ˜β2 =

1

2k(k∓1) +

5. A1.7.2 (p1 < −1, p2 = 1) ˜β2 =

1

2k(k±1) ,

(cid:12)(cid:12)(cid:12) Γ

Λ

25

Remark 2.2.1. We only considered T given by (51) (when Q1 is given by (56)) in the

discussion above. By Appendix, we can conclude that under the parameter conditions in
assumption A1, (b1 + b2)(b1 − b2) = 0 only can happen if p1 = −1, p2 = 1 with (29) or (30).

This is A1.5, for which we always have optimal error estimate.

Remark 2.2.2. Through numerical tests, we found that (2.29) is mostly sharp with two ex-
ceptions. When limh→0 |λ1, λ2| = 1, the estimates show that there will be order reduction for
error of P (cid:63)
h , while in numerical experiments (see e.g. Tables 2.8, 2.9), such order reduction
is observed only when limh→0 λ1, λ2 = 1 but not −1. We believe when limh→0 λ1, λ2 = −1, a

reﬁned estimate can be obtained similar to (2.30) under assumption A1. We have not carried

out this estimate in this thesis.

Another example we ﬁnd for which (2.29) is not sharp is k = 2, p1 = −2 or −3, p2 = 1,
(α1, ˜β1, ˜β2) = (0.25,−1, 1
2k(k+1) and λ1, λ2 →
1+O(h−(1+p1)/2). The theoretical results predict accuracy order of (k +2+p1) but numerical

12), where parameters belong to A1.7.2, ˜β2 =

1

experiments in Table 2.10 show the order to be (k + 3 + p1). Our estimations can’t resolve

this one order diﬀerence. This special parameter may trigger a cancellation we didn’t capture

in analysis. We will improve this estimate in our future work.

Remark 2.2.3. In most cases, (2.30) yields optimal accuracy order, except when k = 1, α1 =

0, β1 = 0, β2 = O(hp2), p2 < 1, where the P (cid:63)
(cid:107)Q2(cid:107)∞

h2 β2+ 1

h is only (k + p2)-th order accurate because
∼ O(hp2−1) in (2.30). This is veriﬁed numerically in

|−−4

|

2h

|Λ| =

|b1+b2|

|Λ| =

1
2h

Table 2.12.

Remark 2.2.4. If δ/2 > 1, we can show δ/2 = δ(cid:48) + 1. This is because |1− λ1| = |1− eiθ| =
1 | = |1 − eiN θ| = 2| sin(N θ/2)|. When δ/2 > 1, one can assert that
2| sin(θ/2)|, and |1 − λN
1 | ∼ N θ, i.e. δ/2 = δ(cid:48) + 1. With this condition, we notice that (2.31)
|1 − λ1| ∼ θ,|1 − λN

26

yields an reduction of δ-th order in convergence rate by checking the order of each term. This

order reduction is consistent with numerical experiments in Example 2.4.4.

Now we can summarize the estimation of P (cid:63)

h for some frequently used ﬂux parameters.

For IPDG scheme with α1 = β2 = 0, β1 = c/h, and DDG scheme discussed in [52] with

α1 = constant, β1 = c/h, β2 = 0, and the more general scale invariant parameter choice

α1 = constant, β1 = c/h, β2 = ch, P (cid:63)

h always have optimal error estimates. For those

parameters, we can show that the eigenvalues λ1, λ2 are always constants independent of h,

therefore, by Lemma 2.2.3, we will have optimal convergence rate. Corresponding numerical

results are shown in Tables 2.4 and 2.7.

For a natural parameter choice where α1, β1, β2 are all real constants, if β2 (cid:54)= 0, then P (cid:63)

h

has ﬁrst order convergence rate when k = 1 and optimal convergence rate when k > 1 by

Lemma 2.2.3. Corresponding numerical results are shown in Tables 2.3 and 2.12. Lastly, for

central ﬂux α1 = α2 = β1 = β2 = 0, this parameter choice satisﬁes assumption A3 when

k = 1 and assumption A2 when k > 1, thus we can verify that P (cid:63)

h has optimal convergence

rate. Corresponding numerical results are shown in Table 2.11.

2.3 Error estimates

In this section, we will derive error estimates of the DG scheme (2.3) for the model NLS

equation (1.1). We will focus on the impact of the choice of the parameters α1, β1, β2 on the

accuracy of the scheme. The error estimates rely on the projection error estimates to obtain

convergence result.

Theorem 2.3.1. Assume that the exact solution u and the nonlinear term f (|u|2) of (1.1)
are suﬃciently smooth with bounded derivatives for any time t ∈ (0, Te] and that the nu-

27

merical ﬂux parameters in (2.13) satisfy the existence conditions of P (cid:63)
h in Lemma 2.2.2.
h u has at least ﬁrst order convergence rate in L2 and L∞
Furthermore, assume h = u − P (cid:63)
norm from the results in Lemma 2.2.3. With periodic boundary conditions solution space V k
h
(k ≥ 1), the following error estimation holds for uh, which is the numerical solution of (2.3)

with ﬂux (2.13):

(cid:107)u − uh(cid:107)L2(I) ≤ C(cid:63) ((cid:107)(u − uh)|t=0(cid:107) + (cid:107)(h)t(cid:107) + (cid:107)h(cid:107)) ,

(2.32)

where C(cid:63) depends on k,(cid:107)f(cid:107)W 2,∞, u as well as ﬁnal time Te, but not on h.

Proof. When P (cid:63)

h exists, we can decompose the error into two parts.

e = u − uh = u − P (cid:63)

h u + P (cid:63)

h u − uh := h + ζh.

By Galerkin orthogonality, ∀vh ∈ V k
h ,

(cid:90)

f (|u|2)uvhdx + i

0 = a(e, vh) − i
= a(h, vh) + a(ζh, vh) − i

I

(cid:90)

I

(cid:90)

I

f (|uh|2)uhvhdx

(cid:90)

I

f (|u|2)uvhdx + i

f (|uh|2)uhvhdx.

By letting vh = ζh and taking conjugate of above equation, we have

a(ζh, ζh) + a(ζh, ζh)
= − a(h, ζh) − a(h, ζh) − 2

(cid:90)

I

f (|u|2)Im(uζh)dx + 2

(cid:90)

I

(2.33)

f (|uh|2)Im(uhζh)dx.

28

By Taylor expansion

f (|uh|2) = f (|u|2) + f(cid:48)(|u|2)E +

ˆf(cid:48)(cid:48)E2,

1
2

where ˆf(cid:48)(cid:48) = f(cid:48)(cid:48)(c), c is a value between |uh|2 and |u|2. E = |uh|2 − |u|2 = −2Re(eu) + |e|2.

Therefore, the nonlinear part becomes

(cid:90)

(cid:90)
(cid:90)

=

f (|u|2)Im(uζh)dx −

f (|uh|2)Im(cid:0)eζh

f (|uh|2)Im(uhζh)dx

(cid:1) +(cid:0)f (|u|2) − f (|uh|2)(cid:1)Im(uζh)dx

I

I

I

=N1 + N2 + N3,

where

N1 =

N2 =

N3 =

(cid:90)
(cid:90)
(cid:90)

I

I

I

f (|u|2)Im(cid:0)eζh
(cid:1) − f(cid:48)(|u|2)EIm(uζh)dx,
(cid:1) − 1
f(cid:48)(|u|2)EIm(cid:0)eζh
ˆf(cid:48)(cid:48)E2Im(cid:0)eζh
(cid:1)dx,

ˆf(cid:48)(cid:48)E2Im(uζh)dx,

2

1
2

will be estimated separately as follows.

• N1 and N2 terms.

Since eζh = hζh + |ζh|2,(cid:12)(cid:12)EIm(uζh)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(−2Re(eu) + |e|2)Im(uζh)

(cid:12)(cid:12)(cid:12) ≤ C((cid:107)u(cid:107)2

L∞(I) +

29

(cid:107)u(cid:107)L∞(I)(cid:107)e(cid:107)L∞(I))(|h|2 + |ζh|2), we have

(cid:16)

|N1| ≤ C(cid:107)f(cid:107)W 1,∞
1 + (cid:107)u(cid:107)2
|N2| ≤ C(cid:107)f(cid:107)W 2,∞(cid:107)E(cid:107)L∞(I)

(cid:16)
L∞(I) + (cid:107)u(cid:107)L∞(I)(cid:107)e(cid:107)L∞(I)

1 + (cid:107)u(cid:107)2

L∞(I) + (cid:107)u(cid:107)L∞(I)(cid:107)e(cid:107)L∞(I)

(cid:17)

((cid:107)h(cid:107)2 + (cid:107)ζh(cid:107)2),

(cid:17)

((cid:107)h(cid:107)2 + (cid:107)ζh(cid:107)2).

• N3 term.

|N3| ≤ C(cid:107)f(cid:48)(cid:48)(cid:107)L∞(cid:107)E(cid:107)2

L∞(I)((cid:107)h(cid:107)2 + (cid:107)ζh(cid:107)2).

To conduct a proper estimate for the nonlinear part, we would like to make an a priori

assumption that, for h small enough,

(cid:107)e(cid:107) = (cid:107)u − uh(cid:107) ≤ h0.5.

(2.34)

h , (cid:107)h(cid:107)Lp(I) ≤ C1h, p = 2,∞, thus (cid:107)ζh(cid:107) ≤ C1h0.5 and (cid:107)ζh(cid:107)L∞(I) ≤
By our assumption on P (cid:63)
C1 by inverse inequality, then (cid:107)e(cid:107)L∞(I) ≤ C1, (cid:107)E(cid:107)L∞(I) ≤ C1. Here and below, C1 is a

generic constant that has no dependence on h, but may depend on u according to the lemma

used to estimate h.

Therefore, we get the estimate:

|N1| + |N2| + |N3| ≤ C1((cid:107)h(cid:107)2 + (cid:107)ζh(cid:107)2),

(2.35)

where C1 depends on (cid:107)f(cid:107)W 2,∞ and u.

30

For linear part of the right hand side in (2.33), we have

a(h, ζh) + a(h, ζh) =

(cid:90)
((cid:98)h[(ζh)x] −(cid:103)(h)x[ζh])|

(h)(ζh)xxdx

I

j+ 1
2

I

(cid:90)
(cid:90)
(h)(ζh)xxdx − i
N(cid:88)
(cid:90)

(h)tζh + (h)tζhdx − i
N(cid:88)
((cid:98)h[(ζh)x] −(cid:103)(h)x[ζh])|
Re(cid:0)(h)tζh

(cid:1)dx.

j=1

j=1

I

+ i

+ i

= 2

I

,

j+ 1
2

The last equality holds because of the deﬁnition of P (cid:63)

h u. For the left hand side of (2.33), by

similar computation in stability analysis we have

a(ζh, ζh) + a(ζh, ζh) =

Combine these two equations with (2.35):

(cid:90)

I

d
dt

|ζh|2dx.

(2.36)

(cid:107)ζh(cid:107)2 ≤ (cid:107)(h)t(cid:107)2 + (cid:107)ζh(cid:107)2 + C1((cid:107)h(cid:107)2 + (cid:107)ζh(cid:107)2).

d
dt

Assuming ut, u have suﬃcient smoothness, then by Gronwall’s inequality, we can get:

(cid:16)(cid:107)ζh|t=0(cid:107)2

L2(I)

+ (cid:107)(h)t(cid:107)2 + (cid:107)(h)(cid:107)2(cid:17)

,

(cid:107)ζh(cid:107)2 ≤ C1

and we obtain (2.32).

To complete the proof, we shall justify the a priori assumption. To be more precise, we

consider h0, s.t., ∀h < h0, C(cid:63)h ≤ 1
2 h0.5, where C(cid:63) is deﬁned in (2.32), dependent on Te, but
not on h. Suppose ∃ t∗ = sup{t : (cid:107)u(t∗)− uh(t∗)(cid:107) ≤ h0.5}, we would have (cid:107)u(t∗)− uh(t∗)(cid:107) =

31

h0.5 by continuity if t∗ is ﬁnite. By (2.32), we obtain (cid:107)e(cid:107) ≤ C(cid:63)h ≤ 1
2 h0.5 if t∗ ≤ Te, which
contradicts the deﬁnition of t∗. Therefore, t∗ > Te and the a priori assumption is justiﬁed.

Remark 2.3.1. If f is a constant function, we can prove the same error estimates without
using the a prixori assumption. Therefore, the assumption that h = u − P (cid:63)
ﬁrst order convergence rate in L2 and L∞ norm is no longer needed.

h u has at least

Moreover, the estimates for (cid:107)h(cid:107) has been established in Lemma 2.2.3. In other words, the

error of the DG scheme (2.3) has the same accuracy as P (cid:63)

h u, as long as P (cid:63)

h u is well-deﬁned

and the numerical initial condition is chosen suﬃciently accurate.

2.4 Numerical results

In this subsection, we present numerical experiments to validate our theoretical results.

Particularly, in Section 2.4.1, we provide numerical validations of convergence rate for the

projection P (cid:63)

h as discussed in Lemma 2.2.3 with focus on the dependence of the errors

on parameters α1, β1, β2 . Section 2.4.2 illustrates the energy conservation property and

validates theoretical convergence rate of DG scheme for NLS equation (1.1).

2.4.1 Numerical results of the projection operator P (cid:63)
h

Example 2.4.1. In this example, we focus on local projection where α2
4, and
verify the conclusions in Lemma 2.2.1 by considering a smooth test function u = cos(x), x ∈

1 + β1β2 = 1

[0, 2π] on a nonuniform mesh and k = 1, 2, 3 for various sets of parameters (α1, β1, β2). The

nonuniform mesh is generated by perturbing the nodes of a uniform mesh of N cells by at

most 10%.

32

We ﬁrst consider two sets of parameters (α1, β1, β2) = (0.3, 0.4, 0.4) and (α1, β1, β2) =

(0.3, 0.4/h, 0.4h). The results with (α1, β1, β2) = (0.3, 0.4, 0.4) are listed in Table 2.3. By

plugging in the parameters into (2.28), we have that when k = 1, the projection has sub-

optimal ﬁrst order convergence rate, while for k > 1, optimal (k + 1)-th order convergence
rate should be achieved. For k = 1, Γj = β1 − 1

h , which does not depend on β2 any more.

This technical diﬀerence cause the discrepancy of the convergence order between k = 1 and

k > 1 in Table 2.3. Results in Table 2.3 agree well with the theoretical prediction. On the

other hand, when we choose parameters (α1, β1, β2) = (0.3, 0.4/h, 0.4h), by Lemma 2.2.3, we
should observe optimal convergence rate for all k ≥ 1, and this is veriﬁed by the numerical

results in Table 2.4.

In [16], we also proved that P (cid:63)

2 and β2/Γj ∼
o(1). We choose the parameters as (α1, β1, β2) = (0.5, 1, 0) to verify this claim, i.e., the

h when β2 = 0, α1 = ± 1

h is superclose to P 1

diﬀerence between P (cid:63)

h and P 1

h can have convergence rates higher than k + 1. The results are

listed in Table 2.5. The diﬀerence of the two projections is indeed of (k + 2)-th order for any
k ≥ 1 in all norms.

Table 2.3: Example 2.4.1. Error of local projection P (cid:63)
parameters: α1 = 0.3, β1 = 0.4, β2 = 0.4.

h u − u on a nonuniform mesh. Flux

N
160
320
640
1280
160
320
640
1280
160
320
640
1280

L1 error
1.98E-02
9.98E-03
5.01E-03
2.51E-03
2.18E-06
2.71E-07
3.37E-08
4.19E-09
2.82E-09
1.76E-10
1.10E-11
6.86E-13

P 1

P 2

P 3

-

-

0.99
0.99
1.00

order L2 error
1.56E-02
7.87E-03
3.95E-03
1.98E-03
1.91E-06
2.39E-07
2.97E-08
3.69E-09
2.45E-09
1.53E-10
9.50E-12
5.93E-13

3.01
3.01
3.01

4.00
4.00
4.00

-

-

-

0.99
0.99
1.00

order L∞ error
1.81E-02
9.20E-03
4.55E-03
2.27E-03
3.73E-06
5.14E-07
6.71E-08
7.99E-09
5.67E-09
3.76E-10
2.25E-11
1.46E-12

3.00
3.01
3.01

4.00
4.01
4.00

-

order

-

0.97
1.02
1.00

-

2.86
2.94
3.07

-

3.92
4.06
3.95

33

Table 2.4: Example 2.4.1. Error of local projection P (cid:63)
parameters: α1 = 0.3, β1 = 0.4/h, β2 = 0.4h.

h u − u on a nonuniform mesh. Flux

N
160
320
640
1280
160
320
640
1280
160
320
640
1280

L1 error
3.42E-04
8.55E-05
2.14E-05
5.34E-06
6.36E-06
8.17E-07
1.02E-07
1.27E-08
3.32E-09
2.08E-10
1.30E-11
8.09E-13

P 1

P 2

P 3

-

-

2.00
2.00
2.00

order L2 error
3.50E-04
8.75E-05
2.19E-05
5.47E-06
6.06E-06
7.99E-07
1.00E-07
1.24E-08
2.93E-09
1.83E-10
1.14E-11
7.12E-13

4.00
4.00
4.00

2.96
3.00
3.01

-

-

-

2.00
2.00
2.00

order L∞ error
8.62E-04
2.21E-04
5.45E-05
1.36E-05
2.06E-05
3.09E-06
4.51E-07
5.12E-08
7.58E-09
5.08E-10
3.04E-11
1.99E-12

4.00
4.01
4.00

2.92
2.99
3.02

-

order

-

1.96
2.02
2.00

-

2.73
2.78
3.14

-

3.90
4.06
3.93

Table 2.5: Example 2.4.1. Diﬀerence of local projection P (cid:63)
nonuniform mesh. Flux parameters: α1 = 0.5, β1 = 1, β2 = 0.

h with P 1

h : P (cid:63)

h u − P 1

h u on a

N
160
320
640
1280
160
320
640
1280
160
320
640
1280

L1 error
2.09E-05
2.56E-06
3.17E-07
3.94E-08
5.00E-09
3.14E-10
1.96E-11
1.22E-12
2.91E-12
9.11E-14
2.84E-15
8.84E-17

P 1

P 2

P 3

-

-

3.03
3.01
3.01

order L2 error
1.96E-05
2.40E-06
2.96E-07
3.67E-08
5.05E-09
3.21E-10
2.00E-11
1.24E-12
3.38E-12
1.06E-13
3.27E-15
1.02E-16

3.99
4.00
4.01

5.00
5.00
5.00

-

-

-

3.03
3.02
3.01

order L∞ error
4.66E-05
5.99E-06
7.17E-07
9.11E-08
1.82E-08
1.28E-09
8.56E-11
5.02E-12
1.40E-11
4.72E-13
1.40E-14
4.63E-16

3.98
4.00
4.01

5.00
5.01
5.00

-

order

-

2.96
3.06
2.98

-

3.83
3.90
4.09

-

4.89
5.08
4.92

Example 2.4.2. In this example, we consider global projection when the parameter choices

satisfy assumption A1. We consider a smooth test function u = ecos(x) on [0, 2π] with a

uniform mesh of size h = 2π/N and k = 1, 2, 3 for various sets of parameters (α1, β1, β2).

We ﬁrst test the situation when limh→0 |λ1, λ2| (cid:54)= 1 by setting the parameters (α1, ˜β1, ˜β2)

= (0.25, 1, 1), p1 = −0.5, p2 = 2. Another example is (α1, β1, β2) = (0, 1

2h , h), for which the

eigenvalues λ1, λ2 are constant dependent on k but not h. These two parameter choices

belong to A1.1 and A1.5, respectively. The numerical results shown in Tables 2.6 and 2.7

34

verify the optimal (k + 1)-th order convergence rate predicted by (2.29).

ters (α1, ˜β1, ˜β2) = (0.25, k(k−1)

Then we test the situation when limh→0 |λ1, λ2| = 1 by using two sets of parame-
, 1), p1 = −1, p2 = 2, 3, and (α1, ˜β1, ˜β2) =
k(k−1), 1), p1 = −2,−3, p2 = 1. The ﬁrst set of parameters belongs to A1.6.1 and
(0.25,
we can verify that limh→0 λ1, λ2 = (−1)k. (2.29) and Algorithm 2.2 imply (k + 2 − p2)-th

+ k(k+1)

2

2

8

convergence order. The numerical results listed in Table 2.8 show that the expected order
reduction only happens when limh→0 λ1, λ2 = 1, but not for limh→0 λ1, λ2 = −1. The second
set of parameters belongs to A1.7.2 and we can verify that limh→0 λ1, λ2 = (−1)k+1. (2.29)

and Algorithm 2.2 imply (k + 2 + p1)-th convergence order. The numerical results listed in

Table 2.9 also show that order reduction is only observed when limh→0 λ1, λ2 = 1.

Lastly, we test (α1, ˜β1, ˜β2) = (0.25,−1, 1

12) with k = 2, p1 = −2,−3, p2 = 1, where our
theoretical results predict accuracy order of (k + 2 + p1), but numerical experiments show

the order to be (k + 3 + p1) in Table 2.10. This is one of the exceptions that (2.29) is not

sharp and has been commented in Remark 2.2.2.

Table 2.6: Example 2.4.2. Error of global projection P (cid:63)
0.25, ˜β1 = 1, ˜β2 = 1, p1 = −0.5, p2 = 2. (A1.1)

h u − u. Flux parameters: α1 =

N
160
320
640
1280
160
320
640
1280
320
640
1280
2560

L1 error
0.10E-03
0.26E-04
0.67E-05
0.17E-05
0.63E-06
0.88E-07
0.11E-07
0.14E-08
0.64E-10
0.45E-11
0.29E-12
0.19E-13

P 1

P 2

P 3

-

-

1.93
1.98
1.99

order L2 error
0.69E-03
0.18E-03
0.46E-04
0.12E-04
0.52E-05
0.71E-06
0.91E-07
0.11E-07
0.49E-09
0.35E-10
0.23E-11
0.15E-12

2.85
2.95
2.99

3.82
3.93
3.97

-

-

-

1.93
1.97
1.99

order L∞ error
0.89E-03
0.23E-03
0.58E-04
0.15E-04
0.87E-05
0.11E-05
0.14E-06
0.17E-07
0.72E-09
0.52E-10
0.34E-11
0.22E-12

2.88
2.97
2.99

3.80
3.91
3.96

-

order

-

1.94
1.98
2.00

-

2.95
3.00
3.01

-

3.79
3.92
3.96

Example 2.4.3. In this example, we consider global projection when the parameter choices

35

Table 2.7: Example 2.4.2. Error of global projection P (cid:63)
1
2h, β2 = h. (A1.5)

h u−u. Flux parameters: α1 = 0, β1 =

N
320
640
1280
2560
320
640
1280
2560
320
640
1280
2560

L1 error
0.11E-03
0.28E-04
0.70E-05
0.18E-05
0.11E-06
0.14E-07
0.18E-08
0.22E-09
0.38E-10
0.24E-11
0.15E-12
0.92E-14

P 1

P 2

P 3

-

-

2.00
2.00
2.00

order L2 error
0.63E-03
0.16E-03
0.39E-04
0.98E-05
0.71E-06
0.89E-07
0.11E-07
0.14E-08
0.25E-09
0.16E-10
0.99E-12
0.62E-13

3.00
3.00
3.00

4.00
4.00
4.00

-

-

-

2.00
2.00
2.00

order L∞ error
0.38E-03
0.95E-04
0.24E-04
0.60E-05
0.62E-06
0.77E-07
0.96E-08
0.12E-08
0.22E-09
0.14E-10
0.86E-12
0.54E-13

4.00
4.00
4.00

3.00
3.00
3.00

-

order

-

2.00
2.00
2.00

-

3.00
3.00
3.00

-

4.00
4.00
3.99

Table 2.8: Example 2.4.2. Error of global projection P (cid:63)
0.25, ˜β1 = k(k−1)

h u − u. Flux parameters: α1 =
, ˜β2 = 1.0, p1 = −1, p2 = 2, 3. Note here limh→0 λ1, λ2 = (−1)k.

2 + k(k+1)

8

(A1.6.1)

P 1
p2 = 2
˜β1 = 1
4

P 2
p2 = 2
˜β1 = 7
4

P 2
p2 = 3
˜β1 = 7
4

P 3
p2 = 2
˜β1 = 9
2

N
640
1280
2560
5120
640
1280
2560
5120
640
1280
2560
5120
320
640
1280
2560

L1 error
0.75E-05
0.19E-05
0.48E-06
0.12E-06
0.15E-06
0.39E-07
0.98E-08
0.25E-08
0.14E-04
0.71E-05
0.35E-05
0.18E-05
0.12E-09
0.78E-11
0.49E-12
0.31E-13

-

-

1.97
1.99
1.99

1.94
1.97
1.98

order L2 error
0.52E-04
0.13E-04
0.34E-05
0.84E-06
0.12E-05
0.32E-06
0.82E-07
0.21E-07
0.12E-03
0.58E-04
0.29E-04
0.15E-04
0.95E-09
0.60E-10
0.38E-11
0.24E-12

1.00
1.00
1.00

3.99
3.99
4.00

-

-

-

1.97
1.98
1.99

1.93
1.97
1.98

order L∞ error
0.66E-04
0.17E-04
0.42E-05
0.11E-05
0.23E-05
0.61E-06
0.16E-06
0.39E-07
0.21E-03
0.11E-03
0.54E-04
0.27E-04
0.20E-08
0.13E-09
0.80E-11
0.51E-12

1.00
1.00
1.00

3.99
3.99
3.99

-

-

order

-

1.97
1.99
1.99

-

1.94
1.97
1.99

-

1.00
1.00
1.00

-

3.99
3.99
3.97

are similar to central ﬂuxes, and satisfy assumptions A1 and A2, for smooth function u =

ecos(x) on [0, 2π] with a uniform mesh of size h = 2π/N and k = 1, 2, 3.

For central ﬂux (α1, β1, β2) = (0, 0, 0), Γ = − k2

|Γ|
|Λ| = k > 1, ﬂux
parameters satisfy to assumption A1, and if k = 1, Γ = −Λ and ﬂux parameters satisfy

2h . If k > 1,

2h , Λ = k

36

Table 2.9: Example 2.4.2. Error of global projection P (cid:63)
˜β1 = 1, ˜β2 =

2k(k−1), p1 = −2,−3, p2 = 1. Note here limh→0 λ1, λ2 = (−1)k+1. (A1.7.2)

h u − u. Flux parameters: α1 = 0.25,

1

P 2
p1 = −3
˜β2 = 1
4

P 3
p1 = −2
˜β2 = 1
12

P 3
p1 = −3
˜β2 = 1
12

N
320
640
1280
2560
320
640
1280
2560
320
640
1280
2560

L1 error
0.28E-07
0.35E-08
0.44E-09
0.55E-10
0.70E-08
0.94E-09
0.12E-09
0.15E-10
0.16E-06
0.40E-07
0.10E-07
0.25E-08

-

-

3.00
3.00
3.00

order L2 error
0.21E-06
0.27E-07
0.33E-08
0.41E-09
0.57E-07
0.77E-08
0.99E-09
0.13E-09
0.13E-05
0.32E-06
0.79E-07
0.20E-07

2.90
2.95
2.98

2.00
2.00
2.00

-

-

-

3.00
3.00
3.00

order L∞ error
0.24E-06
0.31E-07
0.38E-08
0.48E-09
0.12E-06
0.16E-07
0.20E-08
0.26E-09
0.24E-05
0.61E-06
0.15E-06
0.38E-07

2.00
2.00
2.00

2.90
2.95
2.98

-

order

-

3.00
3.00
3.00

-

2.91
2.95
2.98

-

2.00
2.00
2.00

Table 2.10: Example 2.4.2. Error of global projection P (cid:63)
0.25, ˜β1 = −1, ˜β2 =

2k(k+1), p1 = −2,−3, p2 = 1. Note that limh→0 λ1, λ2 = 1. (A1.7.2)

h u − u. Flux parameters: α1 =

1

N
320
640
1280
2560
320
640
1280
2560

L1 error
0.72E-07
0.90E-08
0.11E-08
0.14E-09
0.80E-06
0.20E-06
0.50E-07
0.13E-07

order L2 error
0.56E-06
2.99
2.99
0.71E-07
0.89E-08
3.00
0.11E-08
3.00
2.01
0.63E-05
0.16E-05
2.00
0.39E-06
2.00
2.00
0.98E-07

order L∞ error
0.94E-06
2.98
2.99
0.12E-06
0.15E-07
3.00
0.19E-08
3.00
2.01
0.12E-04
0.30E-05
2.00
0.75E-06
2.00
2.00
0.19E-06

order
2.97
2.99
2.99
3.00
2.01
2.00
2.00
2.00

P 2
p1 = −2
˜β2 = 1
12

P 2
p1 = −3
˜β2 = 1
12

to assumption A2. We conclude that P (cid:63)

h exists and is unique for k = 1 when N is odd

and k > 1 for arbitrary N. P (cid:63)

h has optimal error estimates as proved in Lemma 2.2.3. Our

numerical test in Table 2.11 demonstrates optimal convergence rate for all k.

A similar ﬂux is (α1, β1, β2) = (0, 0, 1). When k = 1, this ﬂux parameter set satisﬁes

assumption A2 and (2.30) yields ﬁrst order convergence rate as discussed in Remark 2.2.3.

When k = 2, 3, similar to central ﬂux, this parameter choice satisﬁes assumption A1, showing

optimal convergence rate. The numerical test in Table 2.12 veriﬁes the theoretical results.

Example 2.4.4. In this example, we consider global projection when the parameter choices

37

Table 2.11: Example 2.4.3. Error of global projection P (cid:63)
eters: α1 = 0, β1 = 0, β2 = 0.

h u − u. (Central ﬂux) Flux param-

N
93
279
837
2511
160
320
640
1280
160
320
640
1280

L1 error
0.12E-03
0.13E-04
0.15E-05
0.17E-06
0.11E-05
0.14E-06
0.17E-07
0.22E-08
0.11E-08
0.68E-10
0.42E-11
0.27E-12

P 1

P 2

P 3

-

-

2.00
2.00
2.00

order L2 error
0.74E-03
0.82E-04
0.91E-05
0.10E-05
0.85E-05
0.11E-05
0.13E-06
0.17E-07
0.83E-08
0.52E-09
0.32E-10
0.20E-11

4.00
4.00
4.00

3.00
3.00
3.00

-

-

-

2.00
2.00
2.00

order L∞ error
0.55E-03
0.61E-04
0.68E-05
0.76E-06
0.10E-04
0.13E-05
0.16E-06
0.20E-07
0.11E-07
0.68E-09
0.42E-10
0.26E-11

4.00
4.00
4.00

3.00
3.00
3.00

-

order

-

2.00
2.00
2.00

-

2.99
3.00
3.00

-

4.00
4.00
4.00

Table 2.12: Example 2.4.3. Error of global projection P (cid:63)
0, β1 = 0, β2 = 1.

h u − u. Flux parameters: α1 =

N
93
279
837
2511
160
320
640
1280
2560
160
320
640
1280

L1 error
0.21E-01
0.72E-02
0.24E-02
0.80E-03
0.11E-05
0.14E-06
0.17E-07
0.22E-08
0.27E-09
0.27E-08
0.17E-09
0.11E-10
0.66E-12

order

-

1.00
1.00
1.00

-

3.00
3.00
3.00
3.00

-

4.00
4.00
4.00

L2 error
0.12E+00
0.40E-01
0.13E-01
0.44E-02
0.86E-05
0.11E-05
0.13E-06
0.17E-07
0.21E-08
0.23E-07
0.14E-08
0.89E-10
0.55E-11

P 1

P 2

P 3

-

-

1.00
1.00
1.00

order L∞ error
0.68E-01
0.23E-01
0.75E-02
0.25E-02
0.10E-04
0.13E-05
0.16E-06
0.20E-07
0.25E-08
0.36E-07
0.22E-08
0.14E-09
0.87E-11

3.00
3.00
3.00
3.00

4.00
4.00
4.00

-

order

-

1.00
1.00
1.00

-

3.00
3.00
3.00
3.00

-

4.00
4.00
4.00

satisfy assumption A3 for the smooth function u = ecos(x) on [0, 2π] with uniform mesh size

h = 2π/N and k = 1, 2, 3.

1

(0.25,−1,

An example of A3 is shown in Table 2.13, where the parameters are (α1, ˜β1, ˜β2) =
2k(k−1)), p1 = −2,−3, p2 = 1, similar to the parameters in Table 2.9. The
asymptotic behavior of λ1, λ2 when h approaches 0 is indeed similar to Table 2.9, that is,
|λ1, λ2| = 1 + O(h−(p1+1)/2) and limh→0 λ1, λ2 = (−1)k+1. Same as previous examples,

38

order reductions are only observed when limh→0 λ1, λ2 = 1, that is for k = 3.

We performed more numerical results under assumption A3, and all are similar to those

of A1 as long as the eigenvalues λ1, λ2 are approaching 1 at the same rate. Hence, we will

not show more examples under assumption A3.

-

P 2
p1 = −3
˜β2 = 1
4

Table 2.13: Example 2.4.4. Error of global projection P (cid:63)
similar to A1.7.2 in Table 2.9): α1 = 0.25, ˜β1 = −1, ˜β2 =
Note here limh→0 λ1, λ2 = (−1)k+1.
L1 error
0.28E-07
0.35E-08
0.44E-09
0.55E-10
0.70E-08
0.94E-09
0.12E-09
0.15E-10
0.16E-06
0.40E-07
0.10E-07
0.25E-08

order L2 error
0.21E-06
0.27E-07
0.33E-08
0.41E-09
0.57E-07
0.77E-08
0.99E-09
0.13E-09
0.13E-05
0.32E-06
0.79E-07
0.20E-07

N
320
640
1280
2560
320
640
1280
2560
320
640
1280
2560

P 3
p1 = −2
˜β2 = 1
12

P 3
p1 = −3
˜β2 = 1
12

3.00
3.00
3.00

2.90
2.95
2.98

2.00
2.00
2.00

-

-

h u − u. Flux parameters (A3, and
2k(k−1) , p1 = −2,−3, p2 = 1.

1

-

-

3.00
3.00
3.00

order L∞ error
0.24E-06
0.31E-07
0.38E-08
0.48E-09
0.12E-06
0.16E-07
0.20E-08
0.26E-09
0.24E-05
0.61E-06
0.15E-06
0.38E-07

2.00
2.00
2.00

2.90
2.95
2.98

-

order

-

3.00
3.00
3.00

-

2.91
2.95
2.98

-

2.00
2.00
2.00

2.4.2 Numerical results of the DG scheme

In this subsection, we show the numerical results of the DG scheme applied to the NLS

equation. For the time discretization, we use third order IMEX Runge-Kutta method [5]

and ﬁx ∆t = 1/10000, which is small enough to guarantee that the spatial errors dominate.

To be more precise, we treat the DG discretization of linear term uxx implicitly and nonlinear
term f (|u|2)u explicitly.

Example 2.4.5. In this example, we verify the energy conservation property of our scheme

39

by considering the following linear equation

iut + uxx = 0,

with the progressive plane wave solution: u(x, t) = Aexp(i(x − t)), with A = 1.

We use L2 projection as the numerical initial condition.

In the discussion of stabil-
ity condition, we derive that when Imβ2 ≥ 0, Imβ1 ≤ 0,|α1 + α2|2 ≤ −4Imβ1Imβ2, our

scheme for Schr¨odinger equation is stable. Furthermore, when α1 + α2 = 0, β1, β2 are real

numbers, the scheme is energy conservative.

In this example, we compare two diﬀerent

parameter choices to verify the energy conservation property. The parameter choices are
(α1, α2, β1, β2) = (0.25,−0.25, 1 − i, 1 + i), and (α1, α2, β1, β2) = (0.25,−0.25, 1, 1) when

k = 2, N = 40, ending time T = 100. Both are numerically stable ﬂux parameters. For the

ﬁrst set of parameters, we expect energy decay due to the contributions from the imaginary

part of β1, β2 as in (2.11). For the second set of parameter, energy should be conserved.

In Fig. 2.2, we verify that as t increases from 0 to 100, the ﬂux with only real parameters
preserve (cid:107)uh(cid:107), while the ﬂux with complex numbers have much larger errors. More precisely,
for real parameters, (cid:107)uh(0,·)(cid:107)−(cid:107)uh(100,·)(cid:107) = 7.9E-09, for complex parameters, (cid:107)uh(0,·)(cid:107)−
(cid:107)uh(100,·)(cid:107) = 5.7E-04.

Example 2.4.6. Accuracy test for NLS equation

iut + uxx + |u|2u + |u|4u = 0,

(2.37)

which admits a progressive plane wave solution: u(x, t) = Aexp(i(cx − ωt)), where ω =
c2 − |A|2 − |A|4 with c = 1, A = 1.

40

Figure 2.2: Example 2.4.5. Absolute diﬀerence of (cid:107)uh(t,·)(cid:107) with (cid:107)uh(0,·)(cid:107) with two
sets of parameters (α1, α2, β1, β2) = (0.25,−0.25, 1 − i, 1 + i) (denoted by “imag”) and
(α1, α2, β1, β2) = (0.25,−0.25, 1, 1) (denoted by “real”) when k = 2, N = 40, ending time
Te = 100.

For numerical initial condition, P (cid:63)

h is used when applicable, otherwise standard L2 pro-

jection is applied. On uniform mesh, we use four sets of parameters. The numerical errors

and orders are shown in Tables 2.14 - 2.19, where corresponding projection results are listed

in Tables 2.3, 2.4, 2.11, 2.12, 2.8 and 2.10 respectively. Our numerical experiments show

that the order of convergence for the scheme is the same as the order of error estimates for

the projection P (cid:63)
h .

We would like to make some additional comments on Tables 2.16 and 2.17, whose pa-

rameter choices satisfy assumption A2 when k = 1. The existence of P (cid:63)

h requires N to be

odd for this parameter assumption. However, this assumption is not needed for the optimal

convergence rate of the numerical scheme for (2.37) as shown in Tables 2.16 and 2.17. Similar

comments have been made in [6].

41

Table 2.14: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation (2.37)
on a nonuniform mesh using ﬂux parameters (corresponding to Table 2.3) α1 = 0.3, β1 =
β2 = 0.4, ending time Te = 0.3.

N
40
80
160
320
640
40
80
160
320
40
80
160
320

L1 error
2.86E-02
1.26E-02
6.34E-03
3.18E-03
1.58E-03
2.22E-04
1.99E-05
3.17E-06
3.49E-07
1.54E-06
4.96E-08
2.81E-09
1.61E-10

P 1

P 2

P 3

-

-

1.18
1.00
1.00
1.01

order L2 error
2.48E-02
1.02E-02
4.99E-03
2.56E-03
1.27E-03
2.13E-04
2.13E-05
3.03E-06
3.34E-07
1.35E-06
4.36E-08
2.60E-09
1.57E-10

4.96
4.14
4.13

3.48
2.65
3.18

-

-

-

1.28
1.03
0.96
1.01

order L∞ error
3.92E-02
1.56E-02
6.77E-03
3.47E-03
1.85E-03
6.06E-04
7.28E-05
9.01E-06
1.23E-06
3.29E-06
1.29E-07
8.37E-09
7.68E-10

4.95
4.07
4.05

3.33
2.81
3.18

-

order

-

1.33
1.20
0.96
0.91

-

3.06
3.02
2.87

-

4.67
3.95
3.45

Table 2.15: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation (2.37)
on a nonuniform mesh using ﬂux parameters (corresponding to Table 2.4) α1 = 0.3, β1 =
0.4h, β2 = 0.4/hj, ending time Te = 1.

N
40
80
160
320
640
40
80
160
320
640
40
80
160
320

L1 error
7.47E-03
2.10E-03
4.82E-04
1.21E-04
3.12E-05
5.14E-04
6.81E-05
8.04E-06
9.53E-07
1.68E-07
1.30E-06
5.74E-08
4.44E-09
2.25E-10

P 1

P 2

P 3

-

-

1.83
2.12
1.99
1.96

order L2 error
6.50E-03
1.76E-03
4.18E-04
1.05E-04
2.71E-05
5.37E-04
7.00E-05
8.06E-06
9.75E-07
1.61E-07
1.25E-06
6.00E-08
4.12E-09
2.13E-10

2.92
3.08
3.08
2.50

4.51
3.69
4.30

-

-

-

1.89
2.07
1.99
1.95

order L∞ error
1.29E-02
4.22E-03
1.16E-03
2.87E-04
7.40E-05
1.74E-03
2.99E-04
3.58E-05
3.92E-06
4.90E-07
4.09E-06
2.60E-07
1.49E-08
9.65E-10

2.94
3.12
3.05
2.60

4.38
3.86
4.28

-

order

-

1.62
1.86
2.01
1.96

-

2.54
3.06
3.19
3.00

-

3.98
4.13
3.94

Example 2.4.7. A simulation for the NLS equation

iut + uxx + 2|u|2u = 0

(2.38)

42

Table 2.16: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation (2.37)
using central ﬂux (corresponding to A2 in Table 2.11) α1 = β1 = β2 = 0, ending time Te = 1.

N
40
80
160
320
640
40
80
160
320
640
40
80
160
320
640

L1 error
0.28E-02
0.71E-03
0.18E-03
0.45E-04
0.11E-04
0.13E-03
0.16E-04
0.21E-05
0.26E-06
0.32E-07
0.22E-06
0.16E-07
0.10E-08
0.62E-10
0.39E-11

P 1

P 2

P 3

-

-

2.00
2.00
2.00
2.00

order L2 error
0.22E-02
0.56E-03
0.14E-03
0.35E-04
0.88E-05
0.11E-03
0.14E-04
0.18E-05
0.22E-06
0.27E-07
0.18E-06
0.13E-07
0.79E-09
0.49E-10
0.31E-11

2.99
3.00
3.00
3.00

3.76
4.00
4.00
3.99

-

-

-

2.00
2.00
2.00
2.00

order L∞ error
0.27E-02
0.67E-03
0.17E-03
0.41E-04
0.10E-04
0.16E-03
0.20E-04
0.25E-05
0.31E-06
0.39E-07
0.24E-06
0.13E-07
0.84E-09
0.52E-10
0.33E-11

2.99
3.00
3.00
3.00

3.80
4.00
4.00
3.99

-

order

-

2.02
2.01
2.00
2.00

-

3.00
3.01
3.00
3.00

-

4.16
4.00
4.00
3.96

Table 2.17: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation (2.37)
using ﬂux parameters (corresponding to A2 in Table 2.12): α1 = β1 = 0, β2 = 1, ending
time Te = 1.

N
40
80
160
320
640
40
80
160
320
640
40
80
160
320
640

L1 error
0.17E+00
0.92E-01
0.48E-01
0.24E-01
0.12E-01
0.13E-03
0.16E-04
0.21E-05
0.26E-06
0.32E-07
0.68E-06
0.42E-07
0.26E-08
0.16E-09
0.10E-10

order

-

0.90
0.94
0.97
0.98

-

3.00
3.00
3.00
3.00

-

4.00
4.00
4.00
4.00

L2 error
0.13E+00
0.72E-01
0.38E-01
0.19E-01
0.97E-02
0.11E-03
0.14E-04
0.18E-05
0.22E-06
0.27E-07
0.56E-06
0.35E-07
0.22E-08
0.14E-09
0.85E-11

P 1

P 2

P 3

-

-

0.89
0.94
0.97
0.98

order L∞ error
0.14E+00
0.75E-01
0.38E-01
0.19E-01
0.98E-02
0.17E-03
0.20E-04
0.25E-05
0.31E-06
0.39E-07
0.83E-06
0.51E-07
0.32E-08
0.20E-09
0.13E-10

3.00
3.00
3.00
3.00

4.01
4.00
4.00
4.00

-

order

-

0.87
0.97
0.98
0.99

-

3.02
3.01
3.01
3.00

-

4.01
4.00
4.00
4.00

with double-soliton collision

u(x, t) = sech(x + 10− 4t) exp(i(2(x + 10)− 3t)) + sech(x− 10 + 4t) exp(i(−2(x− 10)− 3t)).

43

(2.39)

Table 2.18: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation (2.37)
using ﬂux parameters (corresponding to A1.6.1 in Table 2.8): α1 = 0.25, ˜β1 = k(k−1)
2 +

k(k+1)

8

-

-

1.77
1.93
1.86
1.93

P 2
p2 = 2
˜β1 = 7
4

P 1
p2 = 2
˜β1 = 1
4

, ˜β2 = 1.0, p1 = −1, p2 = 2, 3, ending time Te = 1.
order L2 error
0.37E-02
0.10E-02
0.25E-03
0.69E-04
0.18E-04
0.49E-04
0.73E-05
0.29E-05
0.92E-06
0.25E-06
0.34E-03
0.20E-03
0.11E-03
0.53E-04
0.27E-04
0.19E-05
0.38E-07
0.15E-08
0.90E-10
0.57E-11

L1 error
0.41E-02
0.12E-02
0.31E-03
0.87E-04
0.23E-04
0.49E-04
0.83E-05
0.31E-05
0.95E-06
0.26E-06
0.36E-03
0.21E-03
0.11E-03
0.56E-04
0.28E-04
0.19E-05
0.43E-07
0.15E-08
0.91E-10
0.58E-11

N
40
80
160
320
640
40
80
160
320
640
40
80
160
320
640
40
80
160
320
640

P 2
p2 = 3
˜β1 = 7
4

P 3
p2 = 2
˜β1 = 9
2

2.55
1.44
1.69
1.85

0.78
0.92
1.00
1.00

5.50
4.88
4.00
3.96

-

-

-

-

1.82
2.05
1.87
1.94

2.74
1.32
1.69
1.86

order L∞ error
0.72E-02
0.21E-02
0.39E-03
0.10E-03
0.26E-04
0.13E-03
0.14E-04
0.65E-05
0.20E-05
0.55E-06
0.74E-03
0.43E-03
0.23E-03
0.11E-03
0.58E-04
0.43E-05
0.84E-07
0.26E-08
0.17E-09
0.11E-10

0.76
0.92
1.00
1.00

5.65
4.68
4.02
3.99

-

-

order

-

1.80
2.39
1.94
1.97

-

3.23
1.12
1.70
1.87

-

0.77
0.92
0.99
1.00

-

5.66
5.00
3.94
3.98

Table 2.19: Example 2.4.6. Error in L1, L2 and L∞ norm for solving NLS equation (2.37)
using ﬂux parameters (corresponding to A1.7.2 in Table 2.10): α1 = 0.25, ˜β1 = −1, ˜β2 =
2k(k+1), p1 = −2,−3, p2 = 1, ending time Te = 1.

1

P 2
p1 = −2
˜β1 = 1
12

P 2
p1 = −3
˜β1 = 1
12

N
40
80
160
320
640
40
80
160
320
640

L1 error
0.60E-04
0.76E-05
0.96E-06
0.12E-06
0.15E-07
0.95E-04
0.21E-04
0.49E-05
0.12E-05
0.29E-06

-

2.99
3.00
3.00
3.00

order L2 error
0.54E-04
0.68E-05
0.85E-06
0.11E-06
0.13E-07
0.85E-04
0.18E-04
0.44E-05
0.11E-05
0.27E-06

2.22
2.08
2.02
2.02

-

-

2.98
3.00
3.00
3.00

order L∞ error
0.95E-04
0.12E-04
0.15E-05
0.19E-06
0.24E-07
0.15E-03
0.33E-04
0.79E-05
0.20E-05
0.48E-06

2.20
2.07
2.02
2.02

-

order

-

2.96
2.99
2.99
3.00

-

2.18
2.06
2.02
2.02

We use periodic boundary condition and L2 projection initialization to run the simulation

for double-soliton collision solution. The two waves propagate in opposite directions and

44

collide at t = 2.5, after that, the two waves separate. Such behaviors are accurately captured

by our numerical simulations, see Figure 2.3 for details.

Figure 2.3: Example 2.4.7. Double soliton collision graphs at t = 0, 2.5, 5 and a x− t plot of
the numerical solution. N = 250, P 2 elements with periodic boundary conditions on [-25,25].
Central ﬂux (α1 = β1 = β2 = 0) is used.

45

-20-1001020X00.20.40.60.8|uh|T = 0-20-1001020X00.511.5|uh|T = 2.5-20-1001020X00.20.40.60.8|uh|T = 5Chapter 3

Superconvergence analysis of UWDG

method on linear Schr¨odinger

equation

In this chapter, we study the superconvergence properties of the UWDG method on solving

the following linear Schr¨odinger equation:

iut + uxx = 0,

(x, t) ∈ I × (0, Te],

u(x, 0) = u0(x),

(3.1)

where I = [a, b] and periodic boundary condition. We consider solving the equation with scale

invariant ﬂux parameters. Such choice include all commonly used ﬂuxes, e.g. alternating,

central, DDG and interior penalty DG (IPDG) ﬂuxes.

We study the superconvergence property in two types. One type is the superconvergence

of cell averages, numerical ﬂuxes, solution at special points and superconvergence towards the

projection P (cid:63)
h in Chapter 2. Depending on the ﬂux choices and the evenness of oddness of the
polynomial degree k, we obtain 2k or (2k−1)-th order superconvergence rate for cell averages
and numerical ﬂux of the function, as well as (2k−1) or (2k−2)-th order for numerical ﬂux of

derivative. The proof relies on the correction function techniques for second order derivatives

46

applied to [10] for DDG methods for diﬀusion problems. We also prove the UWDG solution

is superconvergent with a rate of k + 3 to the projection P (cid:63)
h we introduced in Chapter 2
if k ≥ 3. At interior points whose locations are determined by roots of certain polynomials

associated with the ﬂux parameters, we show that the function values and the ﬁrst and

second order derivatives of the DG solution are superconvergent with order k + 2, k + 1, k,

respectively. Compared with [10] for solving diﬀusion problems, Schr¨odinger equation poses

unique challenges for superconvergence proof because of the lack of the dissipation mechanism

from the equation. One major highlight of our proof is that we introduce specially chosen

test functions in the error equation and show the superconvergence of the second derivative

and jump across the cell interfaces of the diﬀerence between numerical solution and projected

exact solution. This technique was originally proposed in [19] and is essential to elevate the

convergence order for our analysis.

Another type of superconvergence is by postprocessing the UWDG solution such that

the postprocessed solution is convergent faster than original solution. We introduce a dual

problem and prove (2k)-th order negative norm estimate. The order is one order less than

that in hyperbolic equations, due to the ultra-weak formulation which has boundary term of

the product of derivatives and function values. With the negative norm estimates and divided

diﬀerence estimates, we prove the (2k)-th order superconvergence rate for the postprocessed

solution.

The rest of this chapter is organized as follows. In Section 3.1, we recall the UWDG

scheme for linear Schr¨odinger equations and deﬁne some new notations. In Section 3.2, we

restate the projection results in Section 2.2 under scale invariant parameters and introduce

another related projection. Section 3.3 contains the superconvergence results of the UWDG

solution in various quantities. In Section 3.5, we provide numerical tests verifying theoretical

47

results. Some technical proof is provided in the Appendix.

The major contents of this chapter has been published in [15].

3.1 Numerical scheme

In this chapter, the semi-discrete UWDG scheme for solving linear Schr¨odinger equation is
deﬁned as follows: solve for the unique function uh = uh(t) ∈ V k

h , k ≥ 1, t ∈ (0, Te], such

that

aj(uh, vh) = 0,

∀j ∈ ZN

(3.2)

holds for all vh ∈ V k

h , where aj is deﬁned in (2.6) and the numerical ﬂuxes are deﬁned in

(2.14).

Some commonly used ﬂuxes take the following choices of parameters.

• central ﬂux, α1 = β1 = β2 = 0;

• alternating ﬂux, α1 = ± 1

2 , β1 = β2 = 0;
• IPDG like ﬂux, α1 = β2 = 0, β1 = ˜β1h−1;

• DDG like ﬂux, α1 = ˜α1, β2 = 0, β1 = ˜β1h−1;

• more generally, any scale invariant ﬂux, α1 = ˜α1, β1 = ˜β1h−1, β2 = ˜β2h;

where ˜α1, ˜β1, ˜β2 are prescribed constants independent of mesh size. In this chapter, we will

only consider scale invariant ﬂux choices.

Compared with discretization for diﬀusion equations, we don’t have any extra diﬀusion

term in (2.10) to help with the estimates. Therefore, superconvergence error estimates are

more challenging compared with [10].

48

To facilitate the discussion, we introduce notations that will be used in this chapter.

Similar to [12], we deﬁne operator D−1 for any integrable function v on Ij by

(cid:90) x

x

j− 1
2

(cid:90) ξ

−1

ˆv(ξ)dξ,

x ∈ Ij.

(3.3)

(cid:19)
(Lj,k − Lj,k−2)

,

(3.4a)

k ≥ 2,

(3.4b)

D−1v(x) =

2
hj

v(x)dx =

Using the property of Legendre polynomials, we have

(cid:0)Lj,k+1 − Lj,k−1
(cid:18) 1

(cid:1) ,

k ≥ 1.

2k + 1

2k + 3

(Lj,k+2 − Lj,k) − 1

2k − 1

D−1Lj,k =
D−2Lj,k =

1

2k + 1

1

where D−2 = D−1 ◦ D−1.

3.2 Projections

Under scale invariant ﬂux parameter assumption, we have more concise results for P (cid:63)

h . To

facilitate the superconvergence proof at special points, we introduce another projection op-

erator P

†
h in this section.

To shorten the notation, from here on we use two notations Cm and Cm,n to denote

W k+3+m,∞(I)

mesh independent constants. Cm may depend on |u|
and (cid:107)u(cid:107)
depend on |u|
sumption A3 and on (cid:107)u(cid:107)

for assumption A3, (cid:107)u(cid:107)

W k+1+m+2n,∞(I)

W k+4+m+4n,∞(I)

for assumption A2.

W k+1+m,∞(I)

for assumptions A0/A1,

W k+4+m,∞(I)

for assumption A2. Cm,n may

for assumptions A0/A1, on (cid:107)u(cid:107)

W k+3+m+3n,∞(I)

for as-

The deﬁnition of P (cid:63)

h is given in (2.15). When scale-invariant ﬂux parameters are used,

we have the following Lemma.

49

Lemma 3.2.1 (P (cid:63)

h under scale invariant ﬂux parameters). Suppose any of the assumptions

A0/A1/A2/A3 holds, u satisﬁes the condition in Deﬁnition 2.2.1, and scale-invariant ﬂux

parameters are used. We have the following estimates

(cid:12)(cid:12)´uj,m − uj,m

(cid:12)(cid:12) ≤ C0hk+1, m = k − 1, k,

(cid:107)u − P (cid:63)

h u(cid:107)Lν (IN ) ≤ C0hk+1,

ν = 2,∞.

(3.5)

In addition, if hj = hj+1,

(cid:12)(cid:12)´uj,m − uj,m − (´uj+1,m − uj+1,m)(cid:12)(cid:12) ≤ C1hk+2, m = k − 1, k.

(3.6)

Proof. When scale invariant parameters are used, when assumption A0 is satisﬁed, (3.5) is

a direct result of (2.29) in Lemma 2.2.3. On uniform mesh, under assumption A1/A2/A3,

the matrices Q1, Q2 and eigenvalues λ1, λ2 are constants independent of h. Thus, (3.5) is a

direct result of (2.30) and (2.31) in Lemma 2.2.3.

To prove the estimates for ´uj,m − uj,m − (´uj+1,m − uj+1,m), m = k − 1, k, we denote

´uj,k−1 − uj,k−1

´uj,k − uj,k

 −

´uj+1,k−1 − uj+1,k−1

´uj+1,k − uj+1,k

 .

Uj =

Plug (2.26), (2.27) in above formula. With the use of (33), Uj can be estimated in the same

way as Uj in the proof of Lemma 2.2.3, and then (3.6) is obtained. We omit the proof for

brevity.

With the optimal estimates of P (cid:63)

h u, we proved the optimal L2 error estimate of the DG

scheme in Theorem 2.3.1, which is restated in a more concise version below.

Theorem 3.2.2 (Theorem 2.3.1 under scale-invariant ﬂux parameters). Suppose any of the

50

assumptions A0/A1/A2/A3 holds, let the exact solution u of (3.1) be suﬃciently smooth,

satisfying periodic boundary condition and uh be the UWDG solution in (3.2), then

(cid:107)P (cid:63)

h u − uh(cid:107) ≤ C2hk+1,

(cid:107)u − uh(cid:107) ≤ C2hk+1.

(3.7)

Next, we introduce a local projection P

†
h as a variant of P (cid:63)

h and study its approximation

properties, especially the superconvergence property at a special set of points. Such super-

convergence estimates will help us reveal the superconvergence of UWDG solution at special

points. Similar ideas have been employed in [9] for proving the superconvergence at the

so-called generalized Radau points when using upwind-biased ﬂux for hyperbolic equations.

Deﬁnition 3.2.1. For DG scheme with ﬂux choice (2.14), we deﬁne a local projection op-

†
h for any periodic function u ∈ W 1,∞(I) to be the unique polynomial P

erator P
(when k ≥ 1) satisfying

 P

(P

†
hu
†
hu)x

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


−

x

j+ 1
2

+ H

G

(cid:90)
 P

(P

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


Ij
†
hu
†
hu)x

†
hu vhdx =

P

= G

+

x

j− 1
2

(cid:90)

Ij

u vhdx,

 u

ux

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x


∀vh ∈ P k−2

(Ij),

c

 u

ux

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x


j− 1
2

+ H

j+ 1
2

†
hu ∈ V k

h

(3.8a)

(3.8b)

for all j ∈ ZN . When k = 1, only condition (3.8b) is needed.

Projection P

†
h is always a local projection. Denote P

†
hu|Ij

=(cid:80)k

m=0 `uj,mLj,m, by (3.8a),

`uj,m = uj,m, m ≤ k − 2. The similarities in deﬁnition imply that P (cid:63)

h and P

to each other, as shown in the following lemma.

51

†
h are very close

Lemma 3.2.3. For periodic function u ∈ W 1,∞(I), if assumption A0 is satisﬁed, P (cid:63)
†
hu. If any of the assumptions A1/A2/A3 is satisﬁed, P
P
(−1)k+1 Γj
Λj

†
h exists and is uniquely deﬁned if

(cid:54)= 1 for all j ∈ ZN . Then,

h u =

(cid:107)u − P

†
hu(cid:107)Lν (Ij ) ≤ Chk+1|u|

W k+1,ν (Ij )

ν = 2,∞.

,

If any of the assumptions A1/A2/A3 is satisﬁed, we have

(cid:107)P (cid:63)

h u − P

†
hu(cid:107)Lν (IN ) ≤ C1hk+2,

ν = 2,∞.

(3.9)

(3.10)

Proof. When assumption A0 is satisﬁed, due to (2.20), P (cid:63)

h = P

†
h. The rest of the proof is

given in Appendix.

To analyze the superconvergence property at special points, we need to investigate the

expansion of the projection error of P

†
h on every cell Ij, if A0/A1/A2/A3,

(u − P

†
hu)|Ij

= [Lj,k−1, Lj,k]

uj,k−1 − `uj,k−1

uj,k − `uj,k

 +

∞(cid:88)

∞(cid:88)

uj,mRj,m,

uj,mLj,m =

m=k+1

m=k+1

where uj,m is deﬁned in (2.24) and

Rj,m = Lj,m − [Lj,k−1, Lj,k]Mm.

(3.11)

(3.12)

52

We write out the explicit expression of the leading term in expansions

Rj,k+1 = Lj,k+1 + bLj,k + cLj,k−1,

(3.13)

where

2k+1
hj

b = − 2α1
Γj + (−1)kΛj
c = − β1 − 2(k+1)2

hj

,

(α2

1 + β1β2 + 1

4) − (−1)k+1 2(k+1)
hj
Γj + (−1)kΛj

1 + β1β2 − 1
(α2
4)

−

β2
2 k(k + 2)(k + 1)2
hj
Γj + (−1)kΛj

to determine the location of superconvergent points.

dxs Rj,k+1, Ds =(cid:83)N

For s = 0, 1, 2, denote Ds

j as the roots of ds

j=1 Ds

j , then it follows from

(3.11) and (2.25) that, for x ∈ Ds
j ,

∞(cid:88)

m=k+2

∂s(u − P

†
hu)(x) =

uj,m

ds
dxs Rj,m ≤ Chk+ 3

2−s|u|

W k+2,s(Ij )

,

(3.14)

indicating superconvergence at those points. We state such superconvergence results in

Theorem 3.3.6.

Since the expression of b, c depends on hj, on nonuniform mesh, Ds

j , s = 0, 1, 2 have

nodes with the diﬀerent relative locations on each cell. For simplicity, below we discuss the

locations of D0

j , D1

j , D2

j for special ﬂux choices on uniform mesh.

• Alternating ﬂuxes: b = ± 2k+1

k , c = − (k+1)2

k

.

• Central ﬂux: if k is even, then b = 0, c = − (k+1)(k+2)
k(k−1)

; if k is odd, then b = 0, c = −1.

53

• IPDG ﬂuxes: if k is even, then b = 0, c = − (k+1)(k+2)−2 ˜β1
k(k−1)−2 ˜β1
−1.

; if k is odd, then b = 0, c =

For central and IPDG ﬂuxes, if k is odd, Rj,k+1 = Lj,k+1−Lj,k−1, d

dx Rj,k+1 = 4k+2
hj

Lj,k,

d2
dx2 Rj,k+1 = 8k+4
h2
j
points of order k and Lobatto points of order k + 1 excluding end points, respectively, on

j are Lobatto points of order k + 1, Gauss

L(cid:48)
j,k, implying that D0

j , D1

j , D2

interval Ij. Therefore, card(D0

j ) = k + 1, card(D1

j ) = k, card(D2

j ) = k − 1.

Cao et al. proved there exists k + 1 superconvergence points (Radau points) when using

upwind ﬂux for linear hyperbolic problem in [12], k + 1 superconvergence points (Lobatto

points) using special ﬂux parameter in DDG method in [10] and k + 1 or k superconvergence

points, depending on parameters, for using upwind-biased ﬂux for linear hyperbolic problem

in [9]. Analyzing the number and location of superconvergent points for our scheme is more

challenging. We shall only provide lower bound estimates for the number of superconvergence

points. For general parameters choices, when k ≥ 2, Rj,k+1 ⊥ P k−2
(Ij), by Theorem 3.3
and Corollary 3.4 in [67], we can easily show Rj,k+1 has at least k − 1 simple zeros, i.e.,
j ) ≥ k − 2, and
card(D0
when k ≥ 4, card(D2

j ) ≥ k − 1. By the same approach, we can show when k ≥ 3, card(D1

j ) ≥ k − 3. For small k values, D1

j can possibly be empty sets.

j , D2

c

3.3 Superconvergence properties

In this section, we study superconvergence of the numerical solution. We investigate the

superconvergence of UWDG ﬂuxes, cell averages, towards a particular projection and at

some special points. This analysis is done by decomposing the error into

e = u − uh = h + ζh, h = u − uI , ζh = uI − uh

(3.15)

54

for some uI ∈ V k

h . For error analysis of DG schemes, uI is usually taken as a projection

of u. While for our purpose of superconvergence analysis, uI needs to be carefully designed

as illustrated in Section 3.3.2. Before that, we prove some intermediate superconvergence

results in Section 3.3.1 without specifying uI . Then, the choice of uI is made in Section

3.3.2 and the main results are obtained.

(cid:80)N
j=1 |[ζh]|2

3.3.1 Some intermediate superconvergence results

This subsection will collect superconvergence results of (cid:107)(ζh)xx(cid:107)L2(I), ( 1
(cid:80)N
j=1 |[(ζh)x]|2
( 1
N

1
2 without specifying uI . The main idea is to choose special test func-

j+ 1
2

1
2 ,

)

N

)

j+ 1
2

tions in error equation, similar to the techniques used in [19] for hyperbolic problems. This

is an essential step to elevate the superconvergence order in Theorem 3.3.4 when k is even.

Lemma 3.3.1. For k ≥ 2, let u be the exact solution to (3.1) and uh be the DG solution
(cid:82)

h, ζh are deﬁned in (3.15). We choose sh to be a function in V k

h , such that

h . Then, when any of the assumptions A0/A1 is satisﬁed,

I shvhdx = a(h, vh), ∀vh ∈ V k

in (3.2).

(3.16)

(3.17)

(3.18)

Proof. The proof is given in Appendix.

(cid:107)(ζh)xx(cid:107) ≤ C(cid:107)sh + (ζh)t(cid:107),

|[ζh]|2

j+ 1
2

1

2 ≤ Ch2(cid:107)sh + (ζh)t(cid:107),

)

(

N(cid:88)
N(cid:88)

1
N

j=1

j=1

(

1
N

|[(ζh)x]|2

j+ 1
2

1

2 ≤ Ch(cid:107)sh + (ζh)t(cid:107).

)

55

3.3.2 Correction functions and the main results

In this section, we shall present the main superconvergence results. The proof depends on

Lemma 3.3.1 and the correction function technique introduced by Cao et al.

in [12, 10],

which is essential for superconvergence. We let uI = P (cid:63)
when k ≥ 3, where w ∈ V k

h u when k = 2, and uI = P (cid:63)

h u − w,

h is a specially designed correction function deﬁned below.

Similar to [10], we start the construction by deﬁning wq, 1 ≤ q ≤ (cid:98) k−1

2 (cid:99). For k ≥ 3, we

denote w0 = u − P (cid:63)

h u and deﬁne a series of functions wq ∈ V k

h , as follows

(cid:90)

Ij

(cid:90)

Ij

wq(vh)xxdx = −i
(cid:99)wq = 0,
(cid:94)(wq)x = 0,

(wq−1)tvhdx,

∀vh ∈ P k

c (Ij) \ P 1

c (Ij),

at x

at x

,

,

j+ 1
2

j+ 1
2

(3.19a)

(3.19b)

(3.19c)

for all j ∈ ZN . (3.19b) and (3.19c) is equivalent to

 wq

(wq)x

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


G

 wq

(wq)x

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


+

x
j+ 1
2

= 0.

(3.20)

−

x

j+ 1
2

+ H

wq exists and is unique when any of the assumptions A0/A1/A2/A3 is satisﬁed for the same

reason as the existence and uniqueness of P (cid:63)
h .

With the construction of wq, we deﬁne

wq(x, t),

(3.21)

w(x, t) =

(cid:98) k−1

2 (cid:99)(cid:88)

q=1

56

then

aj(h, vh) = aj(u − P (cid:63)

h u, vh) +

aj(wq, vh)

(cid:98) k−1

q=1

(cid:98) k−1

2 (cid:99)(cid:88)
(cid:32)(cid:90)
2 (cid:99)(cid:88)
(cid:90)
2 (cid:99)(cid:88)

q=1

(cid:98) k−1

(cid:90)
(cid:90)
(cid:90)

Ij

Ij

Ij

=

=

=

(w0)tvhdx +

(w0)tvhdx +

(wq)tvhdx − i

Ij

(wq − wq−1)tvhdx

q=1

(w(cid:98) k−1

2 (cid:99))tvhdx,

Ij
∀vh ∈ V k

h (I).

(cid:90)

Ij

(cid:33)

wq(vh)xxdx

(3.22)

The approximation property of wq and aj(h, vh) are presented in the following Lemma.

Lemma 3.3.2. For k ≥ 3, suppose u satisﬁes the condition in Theorem 3.2.2. For wq, 1 ≤
q ≤ (cid:98) k−1

2 (cid:99) + 1, we have

2 (cid:99), q + r ≤ (cid:98) k−1
k(cid:88)

t wq|Ij
∂r

=

m=k−1−2q

and then

For any vh ∈ V k
h ,

t cq
∂r

j,mLj,m,

t cq
∂r

j,k−1−2q = Ch2q

j ∂q+r

t

(cid:12)(cid:12)(cid:12)∂r

t cq

j,m

(uj,k−1 − ´uj,k−1),

(cid:12)(cid:12)(cid:12) ≤ C2r,qhk+1+2q,

(cid:107)∂r

t wq(cid:107) ≤ C2r,qhk+1+2q.

(3.23)

(3.24)

(3.25)

|a(h, vh)| ≤ C

Proof. The proof is given in Appendix.

2 (cid:99)hk+1+2(cid:98) k−1

2,(cid:98) k−1

2 (cid:99)(cid:107)vh(cid:107).

Lemma 3.3.3. For k ≥ 2, suppose u satisﬁes the condition in Theorem 3.2.2.

If the

57

parameters satisfy any of the assumptions A0/A1 and uh|t=0 = uI|t=0, we have

N(cid:88)
N(cid:88)

j=1

j=1

(

1
N

(

1
N

(cid:107)(ζh)xx(cid:107) ≤ C

|[(ζh)]|2

j+ 1
2

1

2 ≤ C

)

4+2(cid:98) k−1

4+2(cid:98) k−1

|[(ζh)x]|2

1

2 ≤ C

)

j+ 1
2

4+2(cid:98) k−1

2 (cid:99)hk+1+2(cid:98) k−1
2 (cid:99)
2 (cid:99)hk+3+2(cid:98) k−1
2 (cid:99)
2 (cid:99)hk+2+2(cid:98) k−1
2 (cid:99)

.

.

.

(3.26)

(3.27)

(3.28)

Proof. This Lemma is a direct result of Lemma 3.3.1, once the estiamtes of (cid:107)sh + (ζh)t(cid:107) is

acquired. When k = 2, w = 0. a(h, vh) =(cid:82)

h . That is,
sh = (h)t in the condition of Lemma 3.3.1. To bound (cid:107)(ζh)t(cid:107), we take the time derivative

I (h)tvhdx from the deﬁnition of P (cid:63)

of the error equation and obtain

a(et, vh) = a((h)t, vh) + a((ζh)t, vh) = 0.

Let vh = (ζh)t, since a((ζh)t, (ζh)t) + a((ζh)t, (ζh)t) = d

dt(cid:107)(ζh)t(cid:107)2, by the property of

P (cid:63)
h u, we have

(cid:107)(ζh)t(cid:107)2 = −a((h)t, (ζh)t) − a((h)t, (ζh)t) ≤ 2(cid:107)(h)tt(cid:107)(cid:107)(ζh)t(cid:107),

d
dt

dt(cid:107)(ζh)t(cid:107) ≤ (cid:107)(h)tt(cid:107). To estimate (cid:107)(ζh)t|t=0(cid:107), we let t = 0 in the error

which implies d
equation. Since ζh|t=0 = (uh − uI )|t=0 = 0, we have

(cid:90)

I

a(h, vh) +

(ζh)t|t=0vhdx = 0.

58

Let vh = (ζh)t|t=0, then

(cid:107)(ζh)t|t=0(cid:107)2 ≤ (cid:107)(h)t(cid:107)(cid:107)(ζh)t|t=0(cid:107).

Therefore,

(cid:107)(ζh)t(cid:107) ≤ (cid:107)(h)t(cid:107) + t(cid:107)(h)tt(cid:107).

By Lemma 3.3.1, estimates in (3.5) and the inequality above, we can get (3.26)-(3.28).

For k ≥ 3, by (3.22), we have a(h, vh) =(cid:82)

2 (cid:99))t in
the condition of Lemma 3.3.1. Then, following the same lines of proof as above, by replacing

2 (cid:99))tvhdx, that is, sh = (w(cid:98) k−1

I (w(cid:98) k−1

h with w(cid:98) k−1

2 (cid:99) and using Lemma 3.3.2, we are done.

Now we are ready to state the following estimates of (cid:107)ζh(cid:107).

Theorem 3.3.4. For k ≥ 2, suppose u satisﬁes the condition in Theorem 3.2.2. Assume
uh|t=0 = uI|t=0, then ∀t ∈ (0, Te],

C

2

(cid:107)ζh(cid:107) ≤



h2k

2, k−1

(Ck+2h4k +(cid:80)

h2k−1,

C

2, k−2

2

Ij⊂IN U Ckh4k−1)

if k is odd and A0/A1/A2/A3,

1
2

if k is even and A0/A1,

(3.29)

if k is even and A2/A3,

where IN U is the collection of cells in which the length of Ij is diﬀerent with at least one of

its neighbors.

59

Proof. From error equation, a(e, ζh) = a(h, ζh) + a(ζh, ζh) = 0, which gives us

(cid:107)ζh(cid:107)2 = −a(h, ζh) − a(h, ζh) ≤

d
dt

2(cid:107)(h)t(cid:107)(cid:107)ζh(cid:107),

2(cid:107)(w(cid:98) k−1

k = 2,

(3.30)

2 (cid:99))t(cid:107)(cid:107)ζh(cid:107), k ≥ 3.

By (3.5), (3.24) and Gronwall’s inequality, we have

(cid:107)ζh(cid:107) ≤ C

2 (cid:99)thk+1+2(cid:98) k−1
2 (cid:99)

2,(cid:98) k−1

,

∀t ∈ (0, Te].

Therefore, when k is odd, or k is even and parameters satisfy any of the assumptions A2/A3,

the proof is complete.

When k is even and parameters satisfy any of the assumptions A0/A1, we make use of

Lemma 3.3.3 to show the improved estimates. We let l = (cid:98) k−1

2 (cid:99) = k−2

2 , then

(wl)tζhdx =

Lj,mζhdx

(cid:90)
N(cid:88)

I

j=1

(cid:90)

Ij

∂tcl

j,1

N(cid:88)

k(cid:88)

j=1

m=1

Lj,1ζhdx +

(cid:90)
k(cid:88)

Ij

∂tcl

j,m

N(cid:88)

j=1

m=2

(cid:90)

Ij

a(h, ζh) =

=

∂tcl

j,m

Lj,mζhdx

= A1 + A2,
.

where we denote the ﬁrst term in the summation by A1, and the other term in summation
as A2. Note that D−1Lj,m ⊥ P 0, m ≥ 1 in the inner product sense, thus D−2Lj,m(±1) =

60

0, m ≥ 2. By integration by parts, we get

(cid:90)

Ij

A2 =

∂tcl

j,m

D−2Lj,m(ζh)xxdx
(cid:90)

N(cid:88)

k(cid:88)

(
m=2

Ij

j=1

|h2

j
4

∂tcl

j,m|2 + h

N(cid:88)

j=1

k(cid:88)
N(cid:88)

m=2

h2
j
4

≤ Ch−1

j=1

≤ Ck+2h4k

D−2Lj,m(ζh)xxdx)2 ≤ Ckh4k + Ch2(cid:107)(ζh)xx(cid:107)2

where we have used (3.23) in the ﬁrst inequality, and (3.26) in the third inequality.
2 ( ξ2
hj

To estimate A1, we take the ﬁrst and second antiderivative of Lj,1 = ξ as
2 )2 ξ3−ξ

and apply integration by parts twice,

hj

(

6

2 − 1
6),

(cid:33)

j=1

N(cid:88)
N(cid:88)
N(cid:88)
N(cid:88)

j=1

j=1

A1 =

=

=

+

hj
2

hj
2

∂tcl

j,1

∂tcl

j,1

j,1

∂tcl

hj
2

(cid:18)(cid:16)hj

(cid:17)

∂tcl

2

j=1

(cid:90)

(cid:12)(cid:12)x

x

(

)¯ζh

ξ2
2

− 1
6

(cid:32)
(cid:32)
(cid:32)
j,1 −(cid:16)hj+1

(¯ζh|−
j+ 1
2

−1
3

j+ 1
2

[¯ζh]

1
3

2

−

j+ 1
2
j− 1
2

ξ2
2

(

Ij

− ¯ζh|+
j− 1
2

) +

hj
2

(cid:90)

Ij

hj
2

+

(cid:17)

ξ3 − ξ

6

(cid:19) 1

∂tcl

j+1,1

¯ζh|+
j+ 1
2

,

3

(cid:33)

)(¯ζh)xdx

− 1
6
ξ3 − ξ

(cid:90)

(¯ζh)xxdx

(cid:33)

6

Ij

(¯ζh)xxdx

where we have used the periodicity in the last equality. Therefore,

|A1| ≤ 1
2

h

N(cid:88)

j=1

2

1
9

+ h−1

(cid:18)(cid:16)hj
(cid:19)2
(cid:17)|∂tcl
N(cid:88)
j,1|
(cid:12)(cid:12)(cid:12)(cid:12)(cid:18)(cid:16) hj
(cid:17)
j,1 −(cid:16) hj+1
(cid:17)
N(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:18)(cid:16)hj
N(cid:88)

∂tcl

j=1

j=1

2

2

2

∂tcl

j=1

≤ Ck+2h4k + C(cid:107)ζh(cid:107)2 + Ch−1

+

h−1

1
18

j=1

+ Ch2

N(cid:88)
(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)2
N(cid:88)
j,1 −(cid:16)hj+1

+ h

j=1

2

|[¯ζh]|2

j+ 1
2

j+1,1

(cid:17)

∂tcl

(cid:107)(ζh)xx(cid:107)2

L2(Ij )

(cid:107)ζh(cid:107)2
(cid:17)

∂tcl

j+1,1

L2(∂Ij )

(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)2

61

where we used (3.23), inverse inequality, (3.26) and (3.27) in the last inequality.

We estimate the last term in A1 by the estimation of the diﬀerence of uj,m in neighboring

cells, similar to that in Proposition 3.1 of [6]. If hj (cid:54)= hj+1, then

hj
2

∂tcl

j,1 − hj+1

2

∂tcl

j+1,1 ≤ Ckh2k.

If hj = hj+1, by (3.23) and (3.6),

(cid:16)hj
(cid:17)2l+1
≤ Ck+1h2k+1.

2

hj
2

∂tcl

j,1 − hj+1

2

∂tcl

j+1,1 = C

∂l+1
t

(uj,k−1 − ´uj,k−1 − (uj+1,k−1 − ´uj+1,k−1))

Therefore, we have

|A1| ≤ Ck+2h4k + C(cid:107)ζh(cid:107)2 +

(cid:88)

Ij⊂IN U

Ckh4k−1.

Combine with the estimates for A2, we have

(cid:107)ζh(cid:107)2 ≤ Ck+2h4k + C(cid:107)ζh(cid:107)2 +

d
dt

(cid:88)

Ij⊂IN U

Ckh4k−1.

By Gronwall’s inequality and the numerical initial condition, we obtain

(cid:107)ζh(cid:107) ≤ (Ck+2h4k +

Ckh4k−1)

1
2 .

(cid:88)

Ij⊂IN U

The proof is now complete.

Above theorem states that for k ≥ 2, when k is odd or k is even and any of the as-

62

sumptions A0/A1 is satisﬁed, (cid:107)ζh(cid:107) has the desired 2k-th order convergence rate. When k

is even and any of the assumptions A2/A3 is satisﬁed, we are unable to improve the order

because that a middle step, Lemma 3.3.3, is proved only under assumption A0/A1. However,
numerical result shows the superconvergence results of (cid:107)ζh(cid:107) and the quantities in following

two theorems still hold when k is even and any of the assumptions A2/A3 is satisﬁed. There

is room for improving the proof under such assumption.

Our main superconvergence results are listed in the following two theorems.

Theorem 3.3.5 (Superconvergence of numerical ﬂuxes and cell averages). Let

(cid:17) 1

(cid:16) 1
(cid:16) 1

N

N(cid:88)
N(cid:88)

j=1

N

j=1

(u −(cid:99)uh)|2
(cid:90)
(cid:12)(cid:12)(cid:12) 1

hj

Ij

Ef =

Ec =

2 , Efx =

j+ 1
2

(cid:12)(cid:12)(cid:12)2(cid:17) 1

2

u − uhdx

(cid:16) 1

N(cid:88)

N

j=1

(ux − (cid:94)(uh)x)|2

j+ 1
2

(cid:17) 1

2 ,

(3.31)

be the errors in the two numerical ﬂuxes and the cell averages, respectively. For k ≥ 2,
suppose u satisﬁes the condition in Theorem 3.2.2. Assume uh|t=0 = uI|t=0, then ∀t ∈

(0, Te]

• if k is odd, parameters satisfy any of the assumptions A0/A1/A2/A3, we have

Ef ≤ C

2, k−1

2

h2k, Efx ≤ C

2, k−1

2

h2k−1, Ec ≤ C

h2k,

2, k−1

2

(3.32)

63

• if k is even, parameters satisfy any of the assumptions A0/A1, we have

Ef ≤ (Ck+2h4k +

Ec ≤ (Ck+2h4k +

(cid:88)

Ij⊂IN U

(cid:88)

Ij⊂IN U

(cid:88)

Ij⊂IN U

Ckh4k−1)

1

2 , Efx ≤ (Ck+2h4k +

Ckh4k−1)

1
2 ,

Ckh4k−1)

1

2 h−1,

(3.33)

(3.34)

where IN U is the collection of cells in which the length of Ij is diﬀerent with at least

one of its neighbors.

• if k is even and parameters satisfy assumption A2/A3, we have

Ef ≤ C

2, k−2

2

h2k−1, Efx ≤ C

2, k−2

2

h2k−2, Ec ≤ C

h2k−1.

2, k−2

2

Proof. We ﬁrst prove the estimates for Ef . By (3.19b) and the deﬁnition of P (cid:63)
(cid:92)u − uI (x

) = 0, then

j+ 1
2

(u −(cid:99)uh)|

j+ 1
2

= ((cid:98)ζh)|

j+ 1
2

=(cid:0){ζh} − α1[ζh] + β2[(ζh)x](cid:1)|

.

j+ 1
2

(3.35)

) =

j+ 1
2

h ,(cid:98)h(x

Therefore, by inverse inequality and the fact β2 = ˜β2h,

(cid:16) 1

N

Ef ≤ C

(cid:107)ζh(cid:107)2

L2(∂IN )

(cid:17) 1
2 ≤ C(cid:107)ζh(cid:107),

and the desired estimates for Ef is obtained by (3.29). The estimates for Efx can be obtained

following same lines.

64

Next, we prove the estimates for Ec. If k is odd, then (cid:82)

Ij

(3.23) and orthogonality of Legendre polynomials. Thus,

wqdx = 0, 1 ≤ q ≤ k−3

2 , by

(cid:90)

Ij

Thus,

u − uhdx =
(cid:90)
(cid:12)(cid:12)(cid:12) 1
If k is even, then (cid:82)

hj

(cid:90)

Ij

(cid:98) k−1

2 (cid:99)(cid:88)

q=1

u − P (cid:63)

h u +

wq + ζhdx =

(cid:90)

Ij

(cid:90)

Ij

ζhdx.

w(cid:98) k−1

2 (cid:99)dx +

(cid:12)(cid:12)(cid:12)2 ≤ 2

hj

u − uhdx

((cid:107)ζh(cid:107)2

L2(Ij )

+ (cid:107)w k−1

2

(cid:107)2
L2(Ij )

).

polynomials. Thus, by similar step, we have

Ij

Ij
wqdx = 0, 1 ≤ q ≤ k−2
(cid:90)

(cid:90)

ζhdx,

Ij

Ij

u − uhdx =

(cid:90)

Ij

Therefore,

2 , by (3.23) and orthogonality of Legendre

(cid:12)(cid:12)(cid:12) 1

hj

(cid:12)(cid:12)(cid:12)2 ≤ 2

hj

u − uhdx

(cid:107)ζh(cid:107)2

L2(Ij )

.

Ec ≤ C((cid:107)ζh(cid:107)2 + (cid:107)w k−1

2

(cid:107)2)1/2 if k is odd, Ec ≤ C(cid:107)ζh(cid:107) if k is even,

and the desired estimate for Ec is obtained by (3.29) and (3.24).

Theorem 3.3.6 (Superconvergence towards projections and at special points). Suppose u
satisﬁes the condition in Theorem 3.2.2. Assume uh|t=0 = P (cid:63)

h u0, then ∀t ∈ (0, Te],

(C4h4k +(cid:80)

C2,1(1 + t)hk+3

(cid:107)uh − P (cid:63)

h u(cid:107) ≤

Ij⊂IN U C2h4k−1)

1
2

k = 2, if A0 or A1

k ≥ 3,

(3.36)

where IN U is the collection of cells in which the length of Ij is diﬀerent with at least one of

65

its neighbors.

Assume Ds, s = 0, 1, 2 deﬁned in (3.14) are not empty sets. Let

Eu =

(cid:16) 1
(cid:16) 1

|D0|

|(u − uh)(x)|2(cid:17) 1
(cid:88)
|(u − uh)xx(x)|2(cid:17) 1
(cid:88)

2

x∈D0

2 , Eux =

Euxx =

|D2|

x∈D2

(cid:16) 1

|D1|

(cid:88)

|(u − uh)x(x)|2(cid:17) 1

2 ,

x∈D1

(3.37)

be the average point value error for the numerical solution, the derivative of solution and the

second order derivative of solution at corresponding sets of points. Then

• if k = 2 and any of the assumptions A0/A1 is satisﬁed, we have

(cid:88)

Eu ≤ (C4h4k +

Ij⊂IN U
Euxx ≤ h−2(C4h4k +

C2h4k−1)
(cid:88)

C2h4k−1)

1
2 .

1

2 , Eux ≤ h−1(C4h4k +

Ij⊂IN U

(cid:88)

Ij⊂IN U

C2h4k−1)

1
2 ,

(3.38)

• if k ≥ 3 and any of the assumptions A0/A1/A2/A3 is satisﬁed, we have

Eu ≤ C2,1hk+2, Eux ≤ C2,1hk+1, Euxx ≤ C2,1hk.

(3.39)

Proof. When k = 2, we have uh − P (cid:63)

h u = −ζh. If any of the assumptions A0/A1 is satisﬁed,

by (3.29), we have

(cid:107)uh − P (cid:63)

h u(cid:107) ≤ (C4h4k +

C2h4k−1)

1
2 .

(cid:88)

Ij⊂IN U

When k ≥ 3, to relax the regularity requirement, we follow the same steps in Lemma

66

3.3.2, and change the deﬁnition of uI to uI = P (cid:63)

h u − w1. Then h = u − uI , ζh = uI − uh

and we obtain

|a(h, vh)| ≤ C2,1hk+3(cid:107)vh(cid:107),

∀vh ∈ V k
h .

By the estimates above, (3.24) and the error equation, we obtain

(cid:107)ζh(cid:107)2 ≤ 2(cid:107)(w1)t(cid:107)(cid:107)ζh(cid:107) ≤ C2,1hk+3.

d
dt

By Gronwall’s inequality,

(cid:107)ζh(cid:107) ≤ C2,1thk+3 + (cid:107)(ζh)|t=0(cid:107) = C2,1thk+3 + (cid:107)w1|t=0(cid:107) ≤ C2,1(1 + t)hk+3,

∀t ∈ (0, Te],

where uh|t=0 = P (cid:63)
that ∀t ∈ (0, Te],

h u0 is used in the ﬁrst equality. Since uh − P (cid:63)

h u = −ζh − w1, it follows

(cid:107)uh − P (cid:63)

h u(cid:107) ≤ 2((cid:107)ζh(cid:107) + (cid:107)w1(cid:107)) ≤ C2,1(1 + t)hk+3.

Then the proof for (3.36) is complete.

If any of the assumptions A0/A1/A2/A3 is satisﬁed, then

|(u − P

†
hu)(x)|2 + |(P (cid:63)

h u − uh)(x)|2 + |(P (cid:63)

h u − P

hu)(x)|2(cid:17) 1

†

2

Eu ≤(cid:16) 1

|D0|

(cid:88)

x∈D0

≤ Chk+2|u|

W k+2,2(I)

+ C(cid:107)P (cid:63)

h u − uh(cid:107) + C(cid:107)P (cid:63)

h u − P

†
hu(cid:107),

where (3.14), inverse inequality, and (3.10) are used in the last inequality. Then the estimates

for Eu is proven by Lemma 3.2.3 and (3.36). The estimates for Eux and Euxx can be proven

following the same lines.

67

Remark 3.3.1. If the initial discretization is taken as uh|t=0 = uI|t=0, the theorem above

still holds. However, the regularity requirement will be higher.

3.4 Superconvergence of postprocessed solution

In this section, we analyze the superconvergence property of the postprocessed DG so-

lutions for linear Schr¨odinger equation (3.1) on uniform mesh by using negative Sobolev

norm estimates. The postprocessor was originally introduced in [7, 55] for ﬁnite diﬀerence

and ﬁnite element methods, and later applied to DG methods in [23]. The postprocessed

solution is computed by the convolution of numerical solution uh and a kernel function
h), where d is the number of spatial dimensions, and l is the index of H−l

h (x) = 1

Kν,l

hd Kν,l( x

norm we’re trying to estimate later. The convolution kernel has three main properties. First,

it has compact support, making post processing computationally advantageous. Second, it
preserves polynomials of degree up to ν − 1 by convolution, thus the convergence rate is not

deteriorated. Third, the kernel Kν,l is a linear combination of B-splines, which allows us to

express the derivatives of kernel by diﬀerence quotients (see section 4.1 in [23]).

We give the formula for the convolution kernel when the DG scheme uses approximation

space V k
h :

K2(k+1),k+1(x) =

k(cid:88)

γ=−k

2(k+1),k+1
k
γ

ψ(k+1)(x − γ),

2(k+1),k+1
where ψ(k+1) are the B-spline bases and the computation of coeﬃcients k
γ

can be

68

found in [65]. Then we can deﬁne the postprocessed DG solution as

(cid:90) ∞
−∞ K

u∗ =

2(k+1),k+1
h

(y − x)uh(y)dy.

(3.40)

u∗ is an “averaged” version of uh such that it is closer as an approximation to the exact

solution u. Lastly, we deﬁne divided diﬀerence as

dhv(x) =

1
h

(v(x +

1
2

h) − v(x − 1
2

h)).

Now we are ready to state an approximation result showing the smoothness of u and

negative Sobolev norm of divided diﬀerence lead to a bound on u − u∗.

Theorem 3.4.1 (Bramble and Schatz [7]). Suppose u∗ is deﬁned in (3.40) and K

2(k+1),k+1
h

=

1
hK2(k+1),k+1( x

h), where K2(k+1),k+1 is a kernel function as deﬁned above. Let u be the

exact solution of linear Schr¨odinger equation (3.1) satisfying periodic boundary condition,
u ∈ H2k+2(I). Then for arbitrary time t ∈ (0, Te], h suﬃciently small, we have

(cid:88)

α≤k+1

(cid:107)dα
h(u − uh)(cid:107)

H−(k+1)(IN )

,

(3.41)

(cid:107)u − u∗(cid:107) ≤ Ch2k+2|u|

H2k+2(I)

+

where C is independent of u and h.

The right hand side of (3.41) indicates that if (cid:107)dα

h(u − uh)(cid:107)

H−(k+1)(IN )

converges at

a rate higher than k + 1, then we have superconvergence property for the postprocessed

solution. In what follows, we estimate the negative-norm term following the steps in [23].
First, we introduce a dual problem: ﬁnd a function v such that v(·, t) is periodic function

69

with period equal to the length of I, i.e., b − a for all t ∈ (0, Te] and

ivt − vxx = 0,

in R × (0, Te),

where Φ is an arbitrary function in C∞

v(x, Te) = Φ(x), x ∈ R,

0 (I). We use the notation (φ, ψ) :=(cid:82)

(3.42)

I φ ψdx in this

section. At ﬁnal time Te,

(u(Te) − uh(Te), Φ) = (u, v)(Te) − (uh, v)(Te)

= (u, v)(0) +

{(u, vt) + (ut, v)}dt − (uh, v)(Te)
(cid:90) Te
= (u, v)(0) − (uh, v)(0) −
= (u − uh, v)(0) −

{((uh)t, v) + (uh, vt)}dt,

{((uh)t, v) + (uh, vt)}dt

(cid:90) Te

0

(cid:90) Te

0

0

where the property uvt + utv = 0 is used to obtain the third equality.
The DG solution uh satisﬁes (3.2). Therefore, we have ∀vh ∈ V k

h

((uh)t, v) = ((uh)t, v − vh) + ((uh)t, vh)
= ((uh)t, v − vh) + iA(uh, vh)
= ((uh)t, v − vh) − iA(uh, v − vh) + iA(uh, v).

Then we obtain

(u(Te) − uh(Te), Φ) = ΘM + ΘN + ΘC ,

70

where

(cid:90) Te
(cid:90) Te

ΘM = (u − uh, v)(0),
ΘN = −
ΘC = −

0

0

{((uh)t, v − vh) − iA(uh, v − vh)}dt,
{(uh, vt) + iA(uh, v)}dt.

∀vh ∈ V k
h ,

By choosing the initial numerical discretization uh(0) = P 0

h u0 and vh = P 0

h v, we have

ΘM = (u − uh, v)(0) = (u − uh, v − vh)(0) and

|ΘM| ≤ (cid:107)(u − uh)(0)(cid:107) · (cid:107)(v − vh)(0)(cid:107) ≤ Ch2k+2(cid:107)u(cid:107)

Hk+1(I)

(cid:107)v(cid:107)

Hk+1(I)

.

Since v is a smooth function, we have

(cid:90) Te

0

ΘC = −

{(uh, vt) + i(uh, vxx)}dt = 0.

Choose vh = P 0

|ΘN| =

0

0

j=1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

0

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:90) Te

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

A(v − vh, uh)dt

h v and from the symmetry of the operator A(·,·), we get
(cid:90) Te
(cid:90) Te
(cid:90) Te

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:0)(cid:92)v − vh[(uh)x] − (cid:94)(v − vh)x[uh](cid:1)(cid:12)(cid:12)j+ 1
(cid:0)(cid:92)v − vh[ux − (uh)x] − (cid:94)(v − vh)x[u − uh](cid:1)(cid:12)(cid:12)j+ 1
(cid:0)(cid:107)u − uh(cid:107)L2(∂IN )(cid:107) (cid:94)(v − vh)x(cid:107)L2(∂IN )

A(uh, v − vh)dt
N(cid:88)
N(cid:88)

≤ CTe max
t∈(0,Te]
+ (cid:107)(u − uh)x(cid:107)L2(∂IN )(cid:107)(cid:92)v − vh(cid:107)L2(∂IN )

=

=

0

j=1

(cid:1).

dt

2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

dt

2

71

By (3.7),

(cid:107)u − uh(cid:107)L2(∂IN ) = (cid:107)u − P (cid:63)
≤ C0hk+ 1

2 + Ch

− 1
2(cid:107)P (cid:63)

h u(cid:107)L2(∂IN ) + (cid:107)P (cid:63)

h u − uh(cid:107)L2(∂IN )
h u − uh(cid:107) ≤ C2hk+ 1

2 ,

where we have used Lemma 2.2.3 and Theorem 3.2.2. Similarly, we have (cid:107)ux−(uh)x(cid:107)L2(∂IN )
≤ C2hk− 1
2(cid:107)vx − (vh)x(cid:107)L2(∂IN ) + h(cid:107)vx − (vh)x(cid:107) +
2(cid:107)v − vh(cid:107)L2(∂IN ) + (cid:107)v − vh(cid:107) ≤ Chk+1(cid:107)v(cid:107)

2 . By the property of L2 projection h

. Then it is straightforward that for

Hk+1(I)

h

1

3

scale invariant ﬂuxes,

(cid:107)(cid:92)v − vh(cid:107)L2(∂IN ) ≤ Chk+ 1

2(cid:107)v(cid:107)

Hk+1(I)

,

(cid:107) (cid:94)(v − vh)x(cid:107)L2(∂IN ) ≤ Chk− 1

2(cid:107)v(cid:107)

Hk+1(I)

.

Therefore, we have

|ΘN| ≤ C2h2k(cid:107)v(cid:107)

Hk+1(I)

.

(3.43)

Combine the above three estimate and the fact (cid:107)v(cid:107)

= (cid:107)Φ(cid:107)

Hk+1(I)

Hk+1(I)

, we have

(cid:107)u(Te) − uh(Te)(cid:107)

H−(k+1)(I)

≤ C2h2k.

the divided diﬀerence dα

Since we consider uh with optimal error estimates on uniform mesh with mesh size h, then
hu0, α ≤ k+1
huh also satisﬁes the DG scheme (3.2) but with shifted mesh
hu0. Then by the same proof for u − uh

hu satisﬁes the linear Schr¨odinger but with initial data dα

and initial numerical discretization dα

on shifted mesh. Similarly, dα

huh = P 0

h dα

above,

(cid:107)dα
h(u − uh)(Te)(cid:107)

H−(k+1)(I)

≤ C2+αh2k,

(3.44)

72

where we used Taylor expansion to estimate dα

hu to obtain the last inequality.

The following theorem is a result of (3.44) and Theorem 3.4.1.

Theorem 3.4.2. Let uh be the UWDG solution of (3.2), suppose the conditions in Theorem

3.4.1 and any of the assumptions A0/A1/A2/A3 is satisﬁed, then on a uniform mesh

(cid:107)u(Te) − u∗(Te)(cid:107) ≤ Ck+3h2k.

(3.45)

3.5 Numerical results

In this section, we provide numerical tests demonstrating superconvergence properties. In
the proof, we see that the initial value of uh matters in estimating (cid:107)uh − uI(cid:107), thus will

impact the superconvergence estimation for Ef and Efx. Therefore, in our numerical tests,

we apply two types of initial discretization for uh. For computing the postprocessed solution
u∗, we use the standard L2 projection P 0

h u as numerical initialization to demonstrate the

convergence enhancement ability of postprocessor. For verifying other superconvergence
quantities, we apply the initial condition uh|t=0 = uI|t=0. In order not to deteriorate the

high order convergence rates, for temporal discretization, we use explicit Runge-Kutta fourth
order method with dt = c · h2.5, c = 0.05 when k = 2 and c = 0.01 when k = 3, 4.

Example 3.5.1. We compute (3.1) on [0, 2π] with exact solution u(x, t) = exp(i3(x − 3t))

using UWDG scheme (3.2). We verify the results with several ﬂux parameters.

In the following tables, we show the convergence rate for quantities Ef , Efx, Ec, Eu, Eux,

Euxx as deﬁned in (3.31) and (3.37) as well as

E∗ = (cid:107)u − u∗(cid:107), EP = (cid:107)uh − P (cid:63)

h u(cid:107),

(3.46)

73

which represent the error after postprocessing, and the error between numerical solution

and the projected exact solution P (cid:63)

h u.

In addition, we test the superconvergence of the

intermediate quantities ζh = P (cid:63)

h u − w − uh as in Lemma 3.3.3, and introduce the following

notations:

N(cid:88)

j=1

E[ζh] = (

1
N

|[ζh]|2

j+ 1
2

)

1
2 , E[(ζh)x] = (

1
N

N(cid:88)

j=1

|[(ζh)x]|2

j+ 1
2

1
2 .

)

The numerical ﬂuxes we tested include

1. Tables 3.1, 3.5: A0 parameters, alternating ﬂux, α1 = 0.5, β1 = β2 = 0, with nonuni-

form mesh;

2. Tables 3.2, 3.6: A0 parameters, a scale invariant ﬂux, α1 = 0.3, β1 = 0.4

h , β2 = 0.4h,

with nonuniform mesh;

3. Tables 3.3, 3.7: A1 parameters, central ﬂux, α1 = β1 = β2 = 0, with uniform mesh;

4. Tables 3.4: A3 parameters, α1 = 0.25, β2 = 0, β1 = 2

h for k = 2, 3, 4, respectively,

h , 5

h, 9

with uniform mesh;

5. Table 3.8: all parameters mentioned above, with uniform mesh,

where the nonuniform mesh is generated by perturbing the location of the nodes of a uniform

mesh by 10% of mesh size.

We ﬁrst verify the results in Theorems 3.3.5, 3.3.6 by examining Tables 3.1, 3.2, 3.3, 3.4,

where the parameters satisfy assumption A0, A0, A1, A3, respectively. We observe that

the scheme can achieve at least the theoretical order of convergence for the quantities in

these two theorems. To be more speciﬁc, EP shows (k + min(3, k))-th order of convergence.

74

Table 3.1: Example 3.5.1. Error table when using alternating ﬂux on nonuniform mesh.
Ending time Te = 1, x ∈ [0, 2π].

N
10
20
40
80
160
10
20
40
80
160
10
20
40
80
160
N
10
20
40
80
160
10
20
40
80
160
10
20
40
80
160

L2 error
2.68E-01
2.68E-02
2.00E-03
1.91E-04
2.19E-05
1.02E-02
5.65E-04
2.94E-05
1.87E-06
1.18E-07
6.45E-04
2.06E-05
6.83E-07
2.04E-08
6.10E-10

Eu

2.83E-01
2.30E-02
1.47E-03
9.12E-05
5.78E-06
6.81E-03
1.40E-04
2.16E-06
3.73E-08
8.04E-10
1.09E-04
1.04E-06
1.28E-08
1.94E-10
2.85E-12

P 2

P 3

P 4

P 2

P 3

P 4

order

EP

order

Euxx

order

Eux

order

-

3.32
3.75
3.39
3.12

-

4.18
4.26
3.98
3.98

-

4.97
4.91
5.07
5.06
order

-

3.62
3.97
4.01
3.98

-

5.61
6.01
5.86
5.53

-

6.72
6.34
6.04
6.09

2.53E-01
2.47E-02
1.42E-03
9.22E-05
5.83E-06
7.51E-03
1.24E-04
2.13E-06
3.02E-08
4.57E-10
8.76E-05
6.12E-07
2.52E-09
1.55E-11
1.02E-13

Ef

2.79E-01
2.31E-02
1.46E-03
9.11E-05
5.77E-06
6.78E-03
1.36E-04
2.02E-06
3.05E-08
4.59E-10
9.69E-05
5.02E-07
1.76E-09
6.99E-12
2.79E-14

-

3.36
4.12
3.95
3.98

-

5.93
5.86
6.14
6.05

-

7.16
7.92
7.35
7.24
order

-

3.60
3.98
4.01
3.98

-

5.64
6.07
6.05
6.05

-

7.59
8.16
7.97
7.97

2.36E+00
2.78E-01
6.02E-02
1.50E-02
3.78E-03
1.60E-01
2.00E-02
2.43E-03
2.99E-04
3.76E-05
1.79E-02
1.25E-03
7.45E-05
4.70E-06
2.85E-07

Efx

9.10E-01
6.94E-02
4.45E-03
2.75E-04
1.74E-05
1.98E-02
4.08E-04
6.06E-06
9.13E-08
1.38E-09
2.83E-04
1.57E-06
5.26E-09
2.09E-11
5.55E-13

-

3.09
2.21
2.00
1.99

-

3.00
3.04
3.02
2.99

-

3.84
4.06
3.99
4.04
order

-

3.71
3.96
4.02
3.98

-

5.60
6.07
6.05
6.05

-

7.49
8.22
7.97
5.23

8.92E-01
7.43E-02
5.29E-03
3.83E-04
3.24E-05
2.50E-02
8.23E-04
3.83E-05
2.29E-06
1.45E-07
8.67E-04
3.06E-05
9.00E-07
2.78E-08
8.34E-10

Ec

5.00E-01
4.48E-02
2.91E-03
1.82E-04
1.15E-05
1.16E-02
2.64E-04
4.03E-06
6.11E-08
9.25E-10
1.62E-04
9.69E-07
3.50E-09
1.40E-11
5.07E-14

-

3.59
3.81
3.79
3.56

-

4.93
4.42
4.06
3.98

-

4.82
5.09
5.02
5.06
order

-

3.48
3.94
4.00
3.98

-

5.45
6.03
6.04
6.05

-

7.39
8.11
7.97
8.10

Eu, Eux, Euxx are shown to have (k + 2)-th, (k + 1)-th and k-th order of convergence ,

respectively. Note that when k = 2, in Tables 3.2 and 3.4, there are situations when no

superconvergence point exists. This ﬁnding shows an evidence to the assertion that Ds

deﬁned in (3.14) could be empty sets. The order of convergence for Ef , (Ef )x, Ec in all

tables are 2k.

In addition, Table 3.4 shows that when k is even and assumption A3 is

satisﬁed, the convergence order for all quantities are the same as when any of assumption

75

A0/A1 is satisﬁed, which is one order higher than the estimates in Theorems 3.3.5, 3.3.6.

Table 3.2: Example 3.5.1. Error table when using ﬂux parameters: α1 = 0.3, β1 = 0.4
0.4h on nonuniform mesh. Ending time Te = 1, x ∈ [0, 2π].

h , β2 =

N
40
80
160
320
640
10
20
40
80
160
10
20
40
80
160
N
40
80
160
320
640
10
20
40
80
160
10
20
40
80
160

L2 error
1.66E-02
1.50E-03
1.70E-04
2.16E-05
2.64E-06
2.08E-02
1.14E-03
5.91E-05
3.75E-06
2.39E-07
9.72E-04
3.17E-05
1.05E-06
3.14E-08
9.39E-10

Eu

1.26E-02
7.43E-04
4.82E-05
3.09E-06
1.94E-07
1.44E-02
3.08E-04
4.70E-06
7.56E-08
1.03E-09
1.77E-04
1.56E-06
1.83E-08
2.82E-10
4.12E-12

P 2

P 3

P 4

P 2

P 3

P 4

order

EP

order

-

3.47
3.14
2.98
3.03

-

4.19
4.27
3.98
3.97

-

4.94
4.91
5.07
5.06
order

-

4.08
3.95
3.96
3.99

-

5.54
6.04
5.96
6.20

-

6.83
6.41
6.02
6.10

1.28E-02
7.46E-04
4.91E-05
3.12E-06
1.96E-07
1.57E-02
2.75E-04
4.83E-06
6.81E-08
1.09E-09
1.48E-04
1.04E-06
4.15E-09
2.51E-11
1.63E-13

Ef

1.26E-02
7.42E-04
4.82E-05
3.09E-06
1.94E-07
1.41E-02
3.05E-04
4.56E-06
6.88E-08
1.10E-09
1.65E-04
8.66E-07
3.04E-09
1.21E-11
4.41E-14

-

4.10
3.92
3.98
3.99

-

5.84
5.83
6.15
5.96

-

7.16
7.97
7.37
7.27
order

-

4.08
3.95
3.96
3.99

-

5.53
6.06
6.05
5.97

-

7.57
8.15
7.98
8.10

Euxx

2.38E-01
2.52E-02
4.51E-03
8.89E-04
2.00E-04
2.00E-01
2.23E-02
2.72E-03
3.36E-04
4.23E-05
1.93E-02
1.37E-03
8.18E-05
5.16E-06
3.13E-07

Efx

3.83E-02
2.28E-03
1.47E-04
9.39E-06
5.86E-07
4.38E-02
9.17E-04
1.37E-05
2.06E-07
3.29E-09
4.69E-04
2.52E-06
8.85E-09
3.76E-11
1.30E-13

order

-

3.24
2.48
2.34
2.15

-

3.17
3.03
3.02
2.99

-

3.82
4.06
3.99
4.04
order

-

4.07
3.95
3.97
4.00

-

5.58
6.06
6.05
5.97

-

7.54
8.15
7.88
8.18

Eux
DNE
DNE
DNE
DNE
DNE

4.47E-02
1.42E-03
6.57E-05
3.91E-06
2.47E-07
1.32E-03
4.63E-05
1.35E-06
4.27E-08
1.27E-09

Ec

2.49E-02
1.48E-03
9.63E-05
6.18E-06
3.88E-07
2.49E-02
5.92E-04
9.12E-06
1.38E-07
2.21E-09
2.73E-04
1.66E-06
6.02E-09
2.41E-11
8.73E-14

order

-
-
-
-
-
-

4.98
4.43
4.07
3.98

-

4.83
5.10
4.99
5.07
order

-

4.07
3.94
3.96
3.99

-

5.39
6.02
6.04
5.97

-

7.36
8.11
7.97
8.11

In Tables 3.1 and 3.2, we used nonuniform mesh in numerical test. The quantities tested

have similar order of convergence compared to the order of convergence on uniform mesh.

Another interesting observation is the order of convergence of Efx. Our numerical tests show

that Efx converges at an order of 2k for all four sets of parameters, which is at least one

order higher than the estimates in Theorem 3.3.5.

76

order

order

Eux

order

order

EP

3.29
3.13
3.04
3.01

Table 3.3: Example 3.5.1. Error table when using central ﬂux on uniform mesh. Ending
time Te = 1, x ∈ [0, 2π].
L2 error
4.20E-03
4.31E-04
4.92E-05
5.99E-06
7.44E-07
3.18E-04
1.71E-05
1.03E-06
6.41E-08
4.00E-09
5.04E-04
2.10E-05
7.32E-07
2.36E-08
7.42E-10

3.21E-03
2.23E-04
1.43E-05
9.01E-07
5.60E-08
7.32E-05
1.02E-06
1.55E-08
2.41E-10
3.76E-12
7.75E-05
4.91E-07
2.65E-09
1.60E-11
1.13E-13

3.39E-02
4.49E-03
5.69E-04
7.14E-05
8.94E-06
3.34E-03
2.04E-04
1.27E-05
7.91E-07
4.94E-08
1.32E-02
6.17E-04
2.17E-05
6.76E-07
2.19E-08

2.97
2.99
3.00
3.00

1.87
1.97
1.99
2.00

2.92
2.98
2.99
3.00

3.85
3.96
3.99
4.01

6.16
6.04
6.01
6.00

4.03
4.01
4.00
4.00

4.21
4.05
4.01
4.00

-

-

-

-

-

-

-

-

-

-

-

-

Euxx

4.86E-01
1.33E-01
3.41E-02
8.57E-03
2.15E-03
4.31E-02
5.49E-03
6.89E-04
8.62E-05
1.08E-05
1.14E-01
1.11E-02
8.01E-04
5.21E-05
3.29E-06

Efx

9.58E-03
6.86E-04
3.90E-05
3.00E-06
1.51E-07
2.16E-04
3.07E-06
4.63E-08
7.18E-10
1.12E-11
2.15E-04
1.26E-06
6.17E-09
2.57E-11
9.96E-14

3.36
3.79
3.94
3.99
order

-

3.80
4.14
3.70
4.31

-

6.14
6.05
6.01
6.00

-

7.42
7.67
7.91
8.01

4.42
4.83
5.01
4.95
order

-

3.84
3.96
3.99
4.01

-

6.12
6.03
6.01
6.00

-

7.20
7.74
7.93
7.98

Ec

6.36E-03
4.44E-04
2.86E-05
1.80E-06
1.12E-07
1.41E-04
2.03E-06
3.10E-08
4.81E-10
7.51E-12
1.27E-04
8.66E-07
4.04E-09
1.66E-11
6.55E-14

N
40
80
160
320
640
20
40
80
160
320
10
20
40
80
160
N
40
80
160
320
640
20
40
80
160
320
10
20
40
80
160

P 2

P 3

P 4

P 2

P 3

P 4

4.58
4.84
4.96
4.99
order

-

3.85
3.96
3.99
4.01

-

5.19
4.99
5.01
5.01

-

5.28
5.76
5.93
5.98

7.30
7.53
7.37
7.15
order

-

3.85
3.96
3.99
4.01

-

6.16
6.04
6.01
6.00

-

7.40
7.78
7.94
7.99

Ef

3.21E-03
2.23E-04
1.43E-05
9.01E-07
5.60E-08
7.28E-05
1.02E-06
1.54E-08
2.39E-10
3.73E-12
7.63E-05
4.52E-07
2.05E-09
8.31E-12
3.27E-14

Eu

3.24E-03
2.25E-04
1.45E-05
9.10E-07
5.66E-08
1.88E-04
5.17E-06
1.63E-07
5.04E-09
1.57E-10
2.21E-04
5.70E-06
1.05E-07
1.72E-09
2.72E-11

Next, we test the order of convergence for quantities in Lemma 3.3.3. In Tables 3.5 and

3.6, we observe clean convergence order of 2k − 1, 2k + 1, 2k for (cid:107)(ζh)xx(cid:107), E[ζh], E[(ζh)x]

when k is even and 2k, 2k + 2, 2k + 1 for these three quantities when k is odd. In Table 3.7,

the order of convergence has some ﬂuctuation, but the quantities are shown to have the same

order of convergence as those in Tables 3.5 and 3.6. These convergence rates are consistent

with the results in Lemma 3.3.3.

77

Table 3.4: Example 3.5.1. Error table when using ﬂux parameters: α1 = 0.25, β1 =
2
h, 5

h , β2 = 0 on uniform mesh. Ending time Te = 1, x ∈ [0, 2π].

h, 9

N
80
160
320
640
1280
20
40
80
160
320
20
40
80
160
N
40
80
160
320
640
20
40
80
160
320
10
20
40
80
160

L2 error
1.41E-03
1.65E-04
2.03E-05
2.53E-06
3.16E-07
8.27E-04
3.92E-05
2.29E-06
1.40E-07
8.74E-09
5.10E-04
8.28E-06
1.87E-07
5.44E-09

Eu

3.24E-03
2.25E-04
1.45E-05
9.10E-07
5.66E-08
1.88E-04
5.17E-06
1.63E-07
5.04E-09
1.57E-10
2.21E-04
5.70E-06
1.05E-07
1.72E-09
2.72E-11

P 2

P 3

P 4

P 2

P 3

P 4

order

EP

order

-

3.09
3.02
3.01
3.00

-

4.40
4.10
4.03
4.01

-

5.95
5.47
5.10
order

-

3.85
3.96
3.99
4.01

-

5.19
4.99
5.01
5.01

-

5.28
5.76
5.93
5.98

8.17E-05
4.74E-06
2.92E-07
1.80E-08
1.22E-09
4.58E-05
5.20E-07
7.54E-09
1.16E-10
1.80E-12
2.08E-04
2.38E-07
1.04E-09
7.11E-12

Ef

3.21E-03
2.23E-04
1.43E-05
9.01E-07
5.60E-08
7.28E-05
1.02E-06
1.54E-08
2.39E-10
3.73E-12
7.63E-05
4.52E-07
2.05E-09
8.31E-12
3.27E-14

-

4.11
4.02
4.03
3.88

-

6.46
6.11
6.03
6.01

-

9.77
7.84
7.19
order

-

3.85
3.96
3.99
4.01

-

6.16
6.04
6.01
6.00

-

7.40
7.78
7.94
7.99

Euxx
DNE
DNE
DNE
DNE
DNE

2.98E-01
3.11E-02
3.72E-03
4.60E-04
5.74E-05
3.76E-01
1.24E-02
5.64E-04
3.29E-05

Efx

9.58E-03
6.86E-04
3.90E-05
3.00E-06
1.51E-07
2.16E-04
3.07E-06
4.63E-08
7.18E-10
1.12E-11
2.15E-04
1.26E-06
6.17E-09
2.57E-11
9.96E-14

order

Eux

order

-
-
-
-
-
-

3.26
3.06
3.02
3.00

-

4.92
4.47
4.10
order

-

3.80
4.14
3.70
4.31

-

6.14
6.05
6.01
6.00

-

7.42
7.67
7.91
8.01

1.15E-02
1.34E-03
1.65E-04
2.05E-05
2.55E-06
6.63E-03
3.60E-04
2.18E-05
1.35E-06
8.43E-08
3.96E-03
6.76E-05
1.55E-06
4.53E-08

Ec

6.36E-03
4.44E-04
2.86E-05
1.80E-06
1.12E-07
1.41E-04
2.03E-06
3.10E-08
4.81E-10
7.51E-12
1.27E-04
8.66E-07
4.04E-09
1.66E-11
6.55E-14

-

3.11
3.02
3.01
3.01

-

4.20
4.05
4.01
4.00

-

5.87
5.45
5.09
order

-

3.84
3.96
3.99
4.01

-

6.12
6.03
6.01
6.00

-

7.20
7.74
7.93
7.98

Lastly, we test the order of convergence for E∗ on uniform mesh for the four sets of
parameters. Table 3.8 shows that E∗ has a convergence rate of at least 2k, and can go up

to 2k + 2. Similar higher order of convergence behaviors exists in the literature [23, 65].

78

Table 3.5: Example 3.5.1. Error table for intermediate quantities when using alternating
ﬂux on nonuniform mesh. Ending time Te = 1, x ∈ [0, 2π].

N
10
20
40
80
160
10
20
40
80
160
10
20
40
80
160

(cid:107)ζh(cid:107) error
3.96E-01
3.28E-02
2.08E-03
1.29E-04
8.17E-06
9.54E-03
1.93E-04
2.86E-06
4.31E-08
6.87E-10
1.35E-04
7.10E-07
2.50E-09
9.90E-12
3.58E-14

order

-

3.60
3.98
4.01
3.98

-

5.63
6.08
6.05
5.97

-

7.57
8.15
7.98
8.11

(cid:107)(ζh)xx(cid:107)
3.02E+00
2.23E-01
1.42E-02
7.95E-04
9.54E-05
8.83E-02
1.76E-03
2.59E-05
3.91E-07
6.19E-09
1.41E-03
8.60E-06
4.24E-08
2.55E-10
2.22E-12

P 2

P 3

P 4

order

-

3.76
3.98
4.15
3.06

-

5.65
6.09
6.05
5.98

-

7.36
7.66
7.38
6.85

E[ζh]

4.51E-02
1.57E-03
4.21E-05
1.17E-06
3.83E-08
7.52E-05
1.75E-07
3.42E-10
1.45E-12
6.76E-15
4.97E-07
1.20E-09
2.56E-12
2.98E-15
8.90E-18

order

-

4.84
5.22
5.16
4.94

-

8.75
9.00
7.88
7.74

-

8.69
8.88
9.74
8.39

E[(ζh)x]
3.37E-01
1.82E-02
9.42E-04
5.50E-05
3.55E-06
1.31E-03
6.16E-06
2.45E-08
2.71E-10
2.59E-12
2.81E-05
9.88E-08
2.90E-10
7.48E-13
5.46E-15

order

-

4.21
4.28
4.10
3.96

-

7.74
7.97
6.50
6.71

-

8.15
8.41
8.60
7.10

Table 3.6: Example 3.5.1. Error table for intermediate quantities when using ﬂux parameters:
α1 = 0.3, β1 = 0.4

order

order

E[ζh]

order

order

3.97
3.97
3.99
3.98

3.38
3.17
3.11
2.96

h , β2 = 0.4h on nonuniform mesh. Ending time Te = 1, x ∈ [0, 2π].
(cid:107)ζh(cid:107) error
1.46E-02
9.35E-04
5.96E-05
3.76E-06
2.38E-07
2.02E-02
4.31E-04
6.46E-06
9.72E-08
1.55E-09
2.27E-04
1.22E-06
4.30E-09
1.71E-11
6.17E-14

(cid:107)(ζh)xx(cid:107)
2.67E-01
2.57E-02
2.86E-03
3.30E-04
4.25E-05
1.78E-01
3.90E-03
5.83E-05
8.76E-07
1.40E-08
2.23E-03
1.30E-05
5.93E-08
4.66E-10
3.19E-12

E[(ζh)x]
2.13E-02
1.25E-03
7.56E-05
4.76E-06
3.19E-07
1.15E-03
5.48E-06
2.99E-08
1.84E-10
1.56E-12
2.06E-06
1.66E-08
6.95E-11
2.33E-13
8.34E-16

2.55E-03
7.74E-05
2.52E-06
7.74E-08
2.57E-09
5.73E-04
1.09E-06
3.27E-09
9.55E-12
4.28E-14
1.03E-06
4.13E-09
1.01E-11
1.82E-14
3.61E-17

5.04
4.94
5.02
4.91

9.04
8.38
8.42
7.80

5.52
6.06
6.06
5.97

5.55
6.06
6.05
5.97

7.96
8.67
9.12
8.98

7.42
7.78
6.99
7.19

7.54
8.15
7.98
8.11

4.09
4.05
3.99
3.90

7.71
7.52
7.34
6.89

-

-

-

6.96
7.90
8.22
8.13

-

-

-

-

-

-

-

-

-

N
40
80
160
320
640
10
20
40
80
160
10
20
40
80
160

P 2

P 3

P 4

79

Table 3.7: Example 3.5.1. Error table for intermediate quantities when using central ﬂux on
uniform mesh. Ending time Te = 1, x ∈ [0, 2π].

N
40
80
160
320
640
20
40
80
160
320
10
20
40
80
160

(cid:107)ζh(cid:107) error
4.53E-03
3.15E-04
2.02E-05
1.27E-06
7.92E-08
1.03E-04
1.44E-06
2.18E-08
3.38E-10
5.28E-12
1.06E-04
6.37E-07
2.88E-09
1.17E-11
4.63E-14

order

-

3.85
3.96
3.99
4.01

-

6.16
6.04
6.01
6.00

-

7.37
7.79
7.94
7.98

(cid:107)(ζh)xx(cid:107)
3.84E-02
3.03E-03
1.29E-04
1.59E-05
5.73E-07
9.27E-04
1.29E-05
1.99E-07
3.05E-09
4.76E-11
1.05E-03
7.12E-06
3.20E-08
1.84E-10
2.45E-12

P 2

P 3

P 4

order

-

3.66
4.55
3.03
4.79

-

6.17
6.02
6.03
6.00

-

7.21
7.80
7.44
6.23

E[ζh]

3.04E-05
1.16E-06
1.79E-07
7.14E-09
2.90E-10
4.27E-08
1.75E-10
4.23E-13
2.91E-16
1.28E-18
7.67E-07
3.03E-09
5.78E-12
9.33E-15
3.04E-17

order

-

4.71
2.70
4.65
4.62

-

7.93
8.69
10.50
7.83

-

7.99
9.03
9.28
8.26

E[(ζh)x]
2.79E-04
8.31E-06
1.31E-06
5.08E-08
2.11E-09
5.25E-06
4.32E-08
1.64E-10
3.24E-14
1.41E-15
2.26E-05
7.70E-08
6.36E-11
5.24E-13
8.82E-16

order

-

5.07
2.66
4.69
4.59

-

6.93
8.04
12.31
4.52

-

8.20
10.24
6.92
9.21

Table 3.8: Example 3.5.1. Postprocessing error table for the four sets of parameters. Ending
time Te = 1, uniform mesh on x ∈ [0, 2π]. The ﬁrst row below labels the parameters by
( ˜α1, ˜β1, ˜β2).

P 2

Fluxes
N
10
20
40
80
160
10
20
40
80
160
10
20
40
80
160

P 3

P 4

(0.5,0,0)
E∗

order

(0, 0, 0)
E∗

order

(0.3, 0.4, 0.4)
E∗

order

(0.25, {2, 5, 9}, 0)
order

E∗

1.00E+00
2.84E-01
2.11E-02
1.37E-03
8.69E-05
1.00E+00
6.04E-02
5.39E-04
3.28E-06
3.14E-08
1.00E+00
4.54E-02
1.32E-04
1.70E-07
1.79E-10

-

1.81
3.75
3.94
3.98

-

4.05
6.81
7.36
6.70

-

4.46
8.42
9.60
9.89

2.81E-01
3.71E-02
3.23E-03
2.24E-04
1.44E-05
1.00E+00
6.29E-02
6.05E-04
5.04E-06
6.49E-08
1.00E+00
4.54E-02
1.32E-04
1.70E-07
1.80E-10

-

2.92
3.52
3.85
3.96

-

3.99
6.70
6.91
6.28

-

4.46
8.42
9.60
9.89

80

1.00E+00
1.20E-01
9.63E-03
7.55E-04
5.13E-05
1.00E+00
6.05E-02
5.26E-04
2.82E-06
2.04E-08
1.00E+00
4.54E-02
1.32E-04
1.70E-07
1.79E-10

-

3.06
3.64
3.67
3.88

-

4.05
6.85
7.54
7.11

-

4.46
8.42
9.60
9.89

1.53E-01
8.05E-02
2.68E-03
1.20E-04
6.99E-06
1.00E+00
7.02E-02
5.46E-04
2.91E-06
1.79E-08
1.00E+00
4.54E-02
1.36E-04
1.66E-07
1.75E-10

-

0.93
4.91
4.49
4.10

-

3.83
7.01
7.55
7.34

-

4.46
8.39
9.67
9.89

Chapter 4

Sparse grid central DG methods for

linear hyperbolic systems

In this chapter, we develop sparse grid central discontinuous Galerkin (CDG) method for

the following time-dependent linear hyperbolic system with variable coeﬃcients

∂(Ai(t, x)u)

= 0, x ∈ Ω,

(4.1)

d(cid:88)

∂u
∂t

+

i=1

∂xi

subject to appropriate initial and boundary conditions. In the expression above, d ≥ 2 is
the spatial dimension of the problem, u(t, x) = (u1(t, x),··· , um(t, x))T is the unknown
function, Ai(t, x) ∈ Rm×m, i = 1, . . . , d are the given smooth variable coeﬃcients. We
assume Ω = [0, 1]d in this chapter, but the discussion can be easily generalized to arbitrary

box-shaped domains. The model (4.1) arises in many contexts [46], such as simulations of

acoustic, elastic waves, and Maxwell’s equations in free space. The scheme we develop in this

chapter can also apply to the case when Ai(t, x) is deﬁned through another set of equations

that can be nonlinearly coupled with u, such as the models in kinetic plasma waves and

incompressible ﬂows.

Similar to [38], in this chapter, we restrict our attention to smooth solutions of (4.1). It is

known that for non-smooth solutions, adaptivity should be invoked to capture discontinuity

81

like structures. This can be achieved using the idea in [39] and is left for our future work.

Based on the sparse grid DG scheme constructed in [38], the goal of the present chapter is

to design and analyze the sparse grid CDG method. Motivated by the Riemann-solver-free

property and large CFL number allowance of CDG methods, we develop sparse grid CDG

method to compute hyperbolic systems eﬃciently. We investigate stability, convergence rate

and CFL condition of the resulting scheme. A novelty of this work is the design of the

scheme for non-periodic problems, where a new hierarchical representation of the solution

is presented, which results in a sparse ﬁnite element space that can be deﬁned on the dual

mesh. L2 projection results are studied for this space, which helps the convergence proof of

the schemes for initial-boundary value problems.

The rest of this chapter is organized as follows: in Section 4.1, we construct the sparse

grid CDG formulations for periodic and non-periodic problems, and perform numerical study

of the CFL conditions. In Section 4.2, we prove L2 stability and error estimates for scalar

equations. The numerical performance is validated in Section 4.3 by several benchmark tests,

including scalar transport equations, acoustic and elastic waves.

The contents of this chapter has been published in [75].

4.1 Numerical Scheme

In this section, we deﬁne and discuss the properties of the proposed sparse grid CDG meth-

ods. For convenience of notations, we rewrite (4.1) in a component-wise form as

∂ul
∂t

+ ∇ · (Al(t, x)u) = 0,

l = 1,··· , m, x ∈ Ω,

(4.2)

82

where Al(t, x) = (Al

d(t, x))T ∈ Rd×m denotes a collection of the l-th row of
each matrix Ai. The problem is solved with given initial value u(0, x) = u0(x), and periodic

1(t, x),··· , Al

or Dirichlet type boundary conditions.

We proceed as follows. First, we introduce the scheme for periodic problems. In this

setting, the ﬁnite element space on the primal and dual mesh can be deﬁned in similar

ways. Then, we discuss the implementation details and perform numerical study of the CFL

conditions. Finally, we consider the more complicated non-periodic problems, for which a

new sparse ﬁnite element space will be introduced on the dual mesh.

4.1.1 Periodic problems

To deﬁne the sparse ﬁnite element space, we ﬁrst review the hierarchical decomposition of

piecewise polynomial space in one dimension [76]. Consider a general interval [a, b], we deﬁne
the n-th level mesh Ωn([a, b]) to be a uniform partition of 2n cells with length hn = 2−n(b−a)
and Ij

n = [a + jhn, a + (j + 1)hn], j = 0, . . . , 2n − 1, for any n ≥ 0. Let

n ([a, b]) := {v : v ∈ P k(Ij
V k

n), ∀ j = 0, . . . , 2n − 1}

be the usual piecewise polynomials of degree at most k on Ωn. Then, we have the nested

structure

0 ([a, b]) ⊂ V k
V k

1 ([a, b]) ⊂ V k

2 ([a, b]) ⊂ V k

3 ([a, b]) ⊂ ···

Similar to [76], we can now deﬁne the multiwavelet subspace W k

n ([a, b]), n = 1, 2, . . . as

the orthogonal complement of V k

n−1([a, b]) in V k

n ([a, b]) with respect to the L2 inner product

83

on [a, b], i.e.,

n−1([a, b]) ⊕ W k
V k

n ([a, b]) = V k

n ([a, b]), W k

n ([a, b]) ⊥ V k

n−1([a, b]).

For notational convenience, we let W k

0 ([a, b]) := V k

0 ([a, b]), which is the standard piecewise

polynomial space of degree k on [a, b]. This gives the hierarchical decomposition V k

n ([a, b])

n ([a, b]) =(cid:76)

on Ωn as V k

0≤l≤n W k

l ([a, b]).

For a d dimensional domain [a, b]d, we recall some basic notations about multi-indices.
0, where N0 denotes the set of nonnegative integers,

For a multi-index α = (α1,··· , αd) ∈ Nd
the l1 and l∞ norms are deﬁned as

(cid:88)d

|α|1 :=

αi,

|α|∞ := max
1≤i≤d

αi.

i=1

The component-wise arithmetic operations and relational operations are deﬁned as

α · β := (α1β1, . . . , αdβd),

c · α := (cα1, . . . , cαd),

2α := (2α1, . . . , 2αd),

α ≤ β ⇔ αi ≤ βi, ∀i, α < β ⇔ α ≤ β and α (cid:54)= β.

By making use of the multi-index notation, we denote by l = (l1,··· , ld) ∈ Nd

0 the

mesh level in a multivariate sense. We deﬁne the tensor-product mesh grid Ωl([a, b]d) =

([a, b]) ⊗ ··· ⊗ Ωld

Ωl1
the grid Ωl, we denote by Ij

([a, b]) and the corresponding mesh size hl = (hl1

,··· , hld

). Based on

l = {x : xi ∈ I
ji
li

, i = 1,··· , d} as an elementary cell, and

l ([a, b]d) := {v : v(x) ∈ Qk(Ij
Vk

l ), 0 ≤ j ≤ 2l − 1} = V k

l1,x1

([a, b]) × ··· × V k

ld,xd

([a, b])

84

as the standard tensor-product piecewise polynomial space on this mesh, where Qk(Ij
l )

denotes the collection of polynomials of degree up to k in each dimension on cell Ij
l .
l = (N,··· , N ), the grid and space will be further denoted by ΩN ([a, b]d) and Vk

N ([a, b]d),

If

respectively.

Based on a tensor-product construction, the multi-dimensional increment space can be

deﬁned as

Therefore, we have Vk

mation space we consider, is deﬁned by

Wk

l1,x1

l ([a, b]d) = W k

N ([a, b]d) =(cid:76)|l|∞≤N

l∈Nd
0

ˆVk

N ([a, b]d) :=

([a, b]) × ··· × W k

([a, b]).

ld,xd

Wk

l ([a, b]d). The sparse ﬁnite element approxi-

(cid:77)

|l|1≤N
l∈Nd
0

Wk

l ([a, b]d).

This is a subset of Vk
1)d2N N d−1) [76], which is signiﬁcantly less than that of Vk

N ([a, b]d), and its number of degrees of freedom scales as O((k +
N ([a, b]d) with exponential de-

pendence on N d. This is the key to computational savings in high dimensions.

The standard CDG schemes [48, 50] is characterized by numerical approximations on

two sets of overlapping grids: primal and dual meshes. Now, we are ready to incorporate

the sparse ﬁnite element space deﬁned above into the CDG framework. For the domain

under consideration Ω = [0, 1]d, we let ΩN,P := ΩN ([0, 1]d) be the primal mesh and ΩN,D,
which is the periodic extension of ΩN ([−hN /2, 1 − hN /2]d) restricted to [0, 1]d, be the dual
mesh. Similarly, we let ˆVk
N,D to be the periodic extension of
N ([−hN /2, 1 − hN /2]d) restricted to [0, 1]d. Here and below, the subscripts P and D
ˆVk

N ([0, 1]d) and ˆVk

N,P := ˆVk

represent the quantities deﬁned on the primal and dual mesh, respectively.

85

The approximation properties for the sparse ﬁnite element space have been established

in previous work [76, 38]. By using a lemma in [38], we can have estimates for L2 projection

operator onto the spaces ˆVk

N,P , ˆVk

N,D.

To facilitate the discussion, below we introduce some notations about norms and semi-

Hs(I

j
N,G)

(cid:107)v(cid:107)2

0≤j≤2N−1

is the standard Sobolev norm on Ij

Hs(ΩN,G) = (cid:80)

norms. Let G = P, D, on primal or dual mesh ΩN,G, we use (cid:107) · (cid:107)Hs(ΩN,G) to denote
the standard broken Sobolev norm, i.e. (cid:107)v(cid:107)2
(cid:107)v(cid:107)
N,G, (and s = 0 is used to denote the
L2 norm). Similarly, we use | · |Hs(ΩN,G) to denote the broken Sobolev semi-norm, and (cid:107) ·
(cid:107)Hs(Ωl,G),|·|Hs(Ωl,G) to denote the broken Sobolev norm and semi-norm that are supported
on a general grid Ωl,G. For any set L = {i1, . . . ir} ⊂ {1, . . . d}, we deﬁne Lc to be the
complement set of L in {1, . . . d}. For a non-negative integer α and set L, we deﬁne the
semi-norm on any domain denoted by Ω(cid:48)

j
N,G)

, where

Hs(I

|v|

Hα,L(Ω(cid:48))

:=

and

(cid:33)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(Ω(cid:48))

v

,

··· ∂α
∂xα
ir

∂α
∂xα
i1

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:32)
 max

|v|Hq+1(Ω(cid:48)) := max
1≤r≤d

|v|

Hq+1,L(Ω(cid:48))

L⊂{1,2,··· ,d}

|L|=r

 ,

which is the norm for the mixed derivative of v of at most degree q + 1 in each direction. In
this chapter, we use the notation A (cid:46) B to represent A ≤ constant × B, where the constant

is independent of N and the mesh level considered. The following results are obtained from

Lemma 3.2 in [38].

Lemma 4.1.1 (L2 projection estimate). Let PP , PD be L2 projections onto the spaces
N,D, respectively, then for k ≥ 1, 1 ≤ q ≤ min{p, k}, and v ∈ Hp+1(Ω), which is

N,P , ˆVk

ˆVk

86

periodic on Ω, N ≥ 1, d ≥ 2, we have for G = P, D,

|PGv − v|Hs(ΩN,G)

(cid:46)

N d2−N (q+1)|v|Hq+1(Ω)

2−N q|v|Hq+1(Ω)

s = 0,

s = 1.

(4.3)

N,G, ΓN,G :=(cid:83)

This lemma shows that the L2 norm and H1 semi-norm of the projection error scale like
O(N d2−N (k+1)) and O(2−N k) with respect to N when the function v has bounded mixed

derivatives up to enough degrees. This lemma will be used in Theorem 4.2.2 to establish

convergence of the scheme.

Now, we are ready to formulate the sparse grid CDG scheme. Below we review some

standard notations about jumps and averages of piecewise functions. With G = P or D,

let Th,G be the collection of all elementary cell Ij

T∈ΩN,G

∂T be the union

of the interfaces for all the elements in ΩN,G (here we have taken into account the periodic
boundary condition when deﬁning ΓN,G) and S(ΓG) := ΠT∈ΩN,G
L2(∂T ) be the set of L2
functions deﬁned on ΓN,G. For any q ∈ S(ΓN,G) and q ∈ [S(ΓN,G)]d, we deﬁne their
averages {q},{q} and jumps [q], [q] on the interior edges as follows. Suppose e is an interior

edge shared by elements T+ and T−, either on primal or dual mesh, we deﬁne the unit normal
vectors n+ and n− on e pointing exterior of T+ and T−, respectively, then

[q] = q−n− + q+n+, {q} =
[q] = q− · n− + q+ · n+, {q} =

(q− + q+),
1
2
(q− + q+).
1
2

The semi-discrete sparse grid CDG scheme for (4.2), based on the weak formulation

introduced in [48, 50], is deﬁned as follows: we ﬁnd ul

h ∈ ˆVk

N,P and vl

h ∈ ˆVk

N,D, such that

87

∀ l = 1,··· , m

(cid:90)

Ω

(cid:90)

(ul

h)t ϕh dx =

(vl

h)t ψh dx =

Ω

τmax

1

Ω

(cid:90)
− (cid:88)
(cid:90)
− (cid:88)

e∈ΓN,P
1

Ω

τmax

e∈ΓN,D

e

(cid:90)
h − ul
(vl

h) ϕh dx +

e

Al(t, x)vh · [ϕh] ds,
(cid:90)

h) ψh dx +

Ω

Al(t, x)uh · [ψh] ds,

(cid:90)
h − vl
(ul

(cid:90)

Ω

Al(t, x)vh · ∇ϕh dx

(4.4)

Al(t, x)uh · ∇ψh dx

(4.5)

for any ϕh ∈ ˆVk

N,P and ψh ∈ ˆVk

N,D, where uh = (u1

h,··· , um

h ), vh = (v1

h,··· , vm

h ) and τmax

is an upper bound for the time step due to the CFL restriction (see Section 4.1.3 for detailed

discussions).

4.1.2 Discussions on implementations

Here, we brieﬂy discuss some details about the implementation of the scheme. We perform

the computation by using orthonormal multiwavelet bases constructed by Alpert [4]. In 1D,

the bases of W k

l ([0, 1]) are denoted by

vj
p,l(x),

p = 1,··· , k + 1,

j = 0,··· , 2l−1 − 1

and they satisfy(cid:82) b

p,l(x)vj(cid:48)
a vj

of the basis functions for k = 0, 1 and l = 0, 1, 2. The bases in W k
l

p(cid:48),l(cid:48)(x)dx = δpp(cid:48)δjj(cid:48)δll(cid:48). Figures 4.1a and 4.2a provide illustrations
in multi-dimensions are

deﬁned by tensor products

d(cid:89)

i=1

vs = vj

p,l :=

ji
v
pi,li

(xi),

pi = 1,··· , k + 1,

ji = 0,··· , max(0, 2li−1 − 1),

88

where we have used the notation s = (l, j, p) and si = (li, ji, pi) to denote the multi-index

for the bases.

As for temporal schemes, we can use the total variation diminishing Runge-Kutta (TVD-

RK) methods [73] to solve the ordinary diﬀerential equations for the coeﬃcients resulting

from the discretization. To calculate the right-hand-side of (4.4)-(4.5), the fast matrix-vector

product by LU split or LU decomposition algorithms [69, 70, 64] can be applied, by which one

can decompose all calculations into one dimensional operations. Below, we brieﬂy describe

the LU decomposition algorithm for the calculation of the following matrix-vector product

which appears at the right-hand-side of (4.4)-(4.5)

(cid:88)

bj =

s:|l|1≤N

fst1

s1,j1

··· td

sd,jd

,

where fs can be the coeﬃcient of the basis in sparse grid space and ti

si,ji

, i = 1,··· , d, are

the corresponding one-dimensional transform of coeﬃcients from basis vsi to basis vji
i-th dimension in our scheme. Note that we have n = 2N (k + 1) one-dimensional bases in

in the

each dimension, and we use vsi to denote the si-th basis. The bases are ordered according

to grid increment. Using Algorithm 1 in [70], we should calculate all the one-dimensional

transform along each direction associated with a block lower triangular matrix, and then

calculate all the one-dimensional transforms having a block upper triangular structure. The
fast matrix-vector product fs → bj on sparse grid with LU decomposition can be proceeded

as follows.

1. Calculate (block) LU decomposition ti

m,j, s, j = 1,··· , n, for
i = 1,··· , d, where P i, Qi are the permutation matrices, li, ui are lower and upper

m=1(P l)i

s,m(uQ)i

s,j =(cid:80)n

triangular matrices.

89

2. Compute the transform with a (block) lower triangular matrix for i = 1,··· , d,

b

(cid:48)
s1,··· ,si−1,s
i,si+1,··· ,sd

si:l1+···+ld≤N fs(P l)i

.

(cid:48)
si,s
i

3. Compute the transform with a (block) upper triangular matrix for i = 1,··· , d,

←(cid:80)

bs ←(cid:80)

(cid:48)
i:l1+···+li−1+l
s

(cid:48)
i+li+1+···+ld≤N

b

(cid:48)
s1,··· ,si−1,s
i,si+1,··· ,sd

(uQ)i
(cid:48)
s
i,si

.

Note that in step 1, the LU decomposition pivots only from rows or columns in the same

mesh level to maintain the hierarchical structure. This pivoting can be successfully done in

the sparse grid CDG scheme, but not in the sparse grid DG scheme, for which additional

splitting of the ﬂux terms are deemed necessary for variable coeﬃcient case.

For the integrals involving variable-coeﬃcient, we use Gaussian quadrature to compute

these terms. Since these integrals are multi-dimensional integrations, we use the so-called

unidirectional principle to separate the integration into multiplication of one-dimensional
integrals. For example, if φ(x) = φ1(x1)··· φd(xd) is separable,

(cid:90)

(cid:90)

(cid:90)

φ(x) =

Ω

[a,b]

φ1(x1)···

φd(xd).

[a,b]

When the variable coeﬃcient Ai(t, x) is separable, we can use unidirectional principle directly.

If it is not separable, we can ﬁnd Ah

i (t, x) as the L2 projection of Ai(t, x) onto the sparse

grid ﬁnite element space, and then use Ah

i (t, x) to compute the integrals.

4.1.3 Discussions on CFL conditions

It is well known that the CDG schemes allow larger CFL numbers than the standard DG

methods except for piecewise constant approximations [50, 60]. Here, we perform a numerical

study of the CFL conditions of DG [27], CDG [51], sparse grid DG [38], and the sparse

90

Table 4.1: CFL numbers of the DG method, CDG method, sparse grid DG method and
sparse grid CDG method with piecewise degree k polynomials, Runge-Kutta method of order
ν for Example 4.3.1 with d=2. The CFL numbers of the sparse grid DG/CDG methods are
measured with regard to the most reﬁned mesh hN .

k

ν = 2
ν = 3
ν = 4

1

0.33
0.40
0.46

DG

2
–

3
–

0.20
0.23

0.13
0.14

CDG

2
–

3
–

0.36
0.52

0.24
0.35

1

0.48
0.66
0.90

2
–

3
–

0.41
0.46

0.25
0.28

0.66
0.81
0.92

sparse grid DG sparse grid CDG
1

1

2
–

3
–

0.65
0.94

0.44
0.62

0.87
1.17
1.58

grid CDG schemes. We only consider the two-dimensional case solving constant coeﬃcient

equation ut + ux1 + ux2 = 0 for now. The results are listed in Table 4.1. The CFL number

of DG method is obtained from Table 2.2 in [27]. The rest of the table is computed by

eigenvalue analysis of the discretization matrix, and by requiring the ampliﬁcation of the

eigenvalues to be bounded by 1 in magnitude. We observe that the sparse grid DG method

has CFL number that is about two times the CFL number of the standard DG method.

The sparse grid CDG method oﬀers the largest CFL conditions among all four methods.

Here, as a side note, we ﬁnd that the CFL number for two-dimensional CDG method is

larger than the CFL number for one-dimensional CDG method in [51]. This table shows

that one advantage of the sparse grid CDG method is the ability to take large time steps

for time evolution problems. In general, further numerical results suggest that for equation

ut + c1ux1 + c2ux2 = 0, the CFL number for sparse grid DG and sparse grid CDG method

will change with the value of the coeﬃcients c1, c2. Results in higher dimensions are yet to

be studied. A preliminary calculation shows that for equation ut + ux1 + ux2 + ux3 = 0 the

CFL conditions for CDG, sparse grid DG and sparse grid CDG methods in 3D are all higher

than those for the 2D case in Table 4.1. The sparse grid CDG method still possesses the

largest CFL number among all four methods. Those interesting issues will be investigated

in our future work.

91

4.1.4 Non-periodic problems

Here, we consider non-periodic problems, where equation (4.1) or (4.2) is supplemented

by Dirichlet boundary condition on the inﬂow edges.

In this case, we can no longer use

periodicity to deﬁne the ﬁnite element space on the dual mesh, and a new grid hierarchy

needs to be introduced.

Recall that for standard CDG methods with non-periodic boundary condition on the

domain [0, 1], the ﬁnite element space on dual mesh with cell size hn = 1/2n is represented

by

n,D = {v : v ∈ P k(Ij
V k

n,D), ∀ j = 0, . . . , 2n},

(4.6)

where the mesh is partitioned as

I0
n,D = [0,

hn
2

],

n,D = [(j − 1
Ij

2

)hn, (j +

1
2

)hn], j = 1, . . . , 2n − 1,

n,D = [1 − hn
I2n

2

, 1],

which consists of 2n−1 cells of size hn, and two cells at the left and right ends of size hn/2. It

is easy to see that this space does not have nested structures, i.e. V k

n−1,D (cid:54)⊂ V k

n,D. Therefore,

we need a new hierarchy to deﬁne the increment polynomial spaces.

For a ﬁxed reﬁned mesh level N, we deﬁne the following grid Ωl,N,D on level l, l = 0 . . . N,

by a collection of cells as

l,N,D = [0, hl − hN
I0

2

],

l,N,D = [1 − hN
I2l
, 1],
, (j + 1)hl − hN
], j = 1, . . . , 2l − 1,
2

2

l,N,D = [jhl − hN
Ij

2

which consists of 2l − 1 cells of size hl, and a cell at the left end of size hl − hN

2 , and a cell

at the right end of size

hN
2 . This grid structure is naturally nested, and therefore V k

l,N,D

92

which consists of piecewise polynomials of degree k deﬁned on Ωl,N,D are also nested, and

V k
N,N,D = V k

N,D as deﬁned in (4.6).

Then the deﬁnitions of sparse ﬁnite element space in Section 4.1.1 can be naturally

extended here. We let W k

l,N,D, l = 1, . . . N be a complement set of V k

l−1,N,D in V k

l,N,D, i.e.

l−1,N,D ⊕ W k
V k

l,N,D = V k

l,N,D.

However, we no longer require W k

l,N,D to be L2 orthogonal to V k

l−1,N,D, because such deﬁ-

nition will be diﬃcult to implement in practice. Instead, we deﬁne W k

l,N,D to be a span of

basis functions that are shifted basis functions of W k

l space deﬁned in Section 4.1.1, namely,

W k

l,N,D = W k

l ([−hN

2

, 1 − hN
2

l ≥ 1.

])(cid:12)(cid:12)[0,1],
N,D =(cid:76)

By denoting W k

0,N,D = V k

0,N,D, we have decomposed V k

0≤l≤N W k

l,N,D. Illustration

of basis functions by such deﬁnitions for k = 0, 1 and l = 0, 1, 2 can be found in Figures 4.1b

and 4.2b. The dimension of W k
are 2l−1(k + 1).

0,N,D is 2(k + 1), while the dimensions of W k

l,N,D, l = 1, . . . N

Finally, the sparse ﬁnite element space on the dual mesh of domain [0, 1]d is deﬁned as

ˆ˜Vk

N,D :=

Wk

l,N,D,

(cid:77)

|l|1≤N
l∈Nd
0

93

×···×W k

l,N,D = W k
Wk

N,D =
l,N,D, and its number of degrees of freedom scales as O(2d−1(k + 1)d2N N d−1)

. This is a subset of the full grid space Vk

ld,N,D,xd

l1,N,D,x1

where Wk

(cid:76)|l|∞≤N

l∈Nd
0

(the proof is similar to Lemma 2.3 in [76]), which is larger than that of ˆVk

N,P , but still

signiﬁcantly less than that of Vk

N,D with exponential dependence on N d.

We will now investigate the approximation property of the space ˆ˜Vk

N,D. We can obtain the

following result, which essentially states that the L2 projection onto this newly constructed

space has the same order of accuracy as PP , PD in Lemma 4.1.1.

Lemma 4.1.2 (L2 projection estimate onto ˆ˜Vk

N,D ). Let ˜PD be the L2 projection onto the
N,D, then for k ≥ 1, 1 ≤ q ≤ min{p, k}, and v ∈ Hp+1(Ω), N ≥ 1, d ≥ 2, we have

space ˆ˜Vk

| ˜PDv − v|Hs(ΩN,D)

(cid:46)

N d2−N (q+1)|v|Hq+1(Ω)

2−N q|v|Hq+1(Ω)

s = 0,

s = 1.

(4.7)

Proof. The proof follows same procedure as Appendix A in [38]. We will mainly highlight

the diﬀerence in the proof (see Steps 1 and 2 below). The main diﬀerence lies in the fact

that all the hierarchical spaces (and associated projections) have dependence not only on l,

but also on the ﬁnest mesh level N.

Step 1: Decomposition of ˜PD into tensor products of one-dimensional increment pro-

jections. We denote P k

l,N,D as the standard L2 projection operator from L2([0, 1]) to V k

l,N,D,

and the induced increment projection

P k

P k
0,N,D,

l,N,D − P k

l−1,N,D,

Qk

l,N,D :=

if l = 1, . . . N,

if l = 0,

94

and further denote

˜Pk

N,D :=

(cid:88)

|l|1≤N
l∈Nd
0

Qk

l1,N,D,x1

⊗ ··· ⊗ Qk

ld,N,D,xd

,

where the last subindex of Qk

li,N,D,xi

indicates that the increment operator is deﬁned in xi-

direction. We can verify that ˜PD = ˜Pk

N,D. In fact, for any v, it’s clear that ˜Pk

N,Dv ∈ ˆ˜Vk

N,D.

Therefore, we only need

(cid:90)

Ω

( ˜Pk

N,Dv − v)w dx = 0,

∀ w ∈ ˆ˜Vk

N,D.

(4.8)

It suﬃces to show (4.8) for v ∈ C∞(Ω) which is a dense subset of L2(Ω). In fact, we have

v = Pk

N,Dv + v − Pk

N,Dv,

( ˜Pk

N,Dv − v)wdx =

N,Dv)wdx +

is the L2 projection onto the full grid space

(cid:90)

Ω

(v − Pk

N,Dv)wdx

Qk

l1,N,D,x1

⊗ ··· ⊗ Qk

ld,N,D,xd

v) w dx.

where Pk

N,D = P k

N,N,D,x1

Vk

N,D. Therefore,

(cid:90)

Ω

⊗ ··· ⊗ PN,N,D,xd
(cid:90)

(cid:90)

Ω

( ˜Pk

N,Dv − Pk
(cid:88)

= −

(
|l|∞≤N,|l|1>N

Ω

l∈Nd
0

The last term in the ﬁrst row of the equality above vanishes because w ∈ ˆ˜Vk

N,D ⊂ Vk

N,D.

95

In addition, for any l ≥ 1, φ ∈ L2([0, 1]), ϕ ∈ V k

l−1,N,D

(cid:90)

Qk

l,N,Dφ ϕdx =

[0,1]

[0,1]

(I − P k

l−1,N,D)φ ϕdx −

(cid:90)

[0,1]

(I − P k

l,N,D)φ ϕdx = 0,

Therefore, by properties of the tensor product projections

( ˜Pk

N,Dv − v)wdx = 0,

∀w ∈ ˆ˜Vk

N,D,

and the proof for ˜PD = ˜Pk

N,D is complete.

Step 2: Estimation of the increment projections. For a function v ∈ Hp+1([0, 1]), we

have the convergence property of the L2 projection P k
1 ≤ q ≤ min{p, k}, s = 0, 1,

l,N,D as follows: for any integer q with

|P k
l,N,Dv − v|

Hs(I

j
l,N,D

)

≤ ck,s,q(hj

l,N )(q+1−s)|v|

Hq+1(I

j
l,N,D

j = 1,··· , 2l − 1,

,

)

where the mesh size hj

l,N =

hl − hN /2,

j = 0

hl,

j = 1,··· , 2l − 1,

hN /2,

j = 2l.

The estimation above directly applies for Qk

0,N,D = P k

0,N,D. For l ≥ 1, by simple algebra,

(cid:90)

(cid:90)

Ω



96

we have

|Qk

l,N,Dv|

Hs(I

|Qk

l,N,Dv|

Hs(I

j
l,N,D

)

j
l,N,D

)

≤ ˜ck,s,q2−l(q+1−s)|v|

Hq+1(I

(cid:98)j/2(cid:99)
l−1,N,D

≤ ck,s,q(hl)(q+1−s)|v|

Hq+1(I

j
l,N,D

)

j = 2,··· , 2l − 1,

,

)

+ ck,s,q(hl−1 − hN /2)(q+1−s)|v|
< ˜ck,s,q2−l(q+1−s)|v|

Hq+1(I0

l−1,N,D

,

)

Hq+1(I0

l−1,N,D

,

)

j = 0, 1,

|Qk

l,N,Dv|

Hs(I2l

l,N,D

= 0,

)

with ˜ck,s,q = ck,s,q(1 + 2q+1−s).

The rest of the proof is then very similar to Appendix A in [38], and is omitted.

We now provide a numerical validation of Lemma 4.1.2 by considering the error of pro-

jection ˜PD for a smooth function

u(x) = exp

 , x ∈ [0, 1]d.

xi

 d(cid:89)

i=1

(4.9)

In Table 4.2, we report the L2 errors and the associated orders of accuracy for k = 1, 2, 3, d =

2, 3. It is clear that the predicted order of accuracy is achieved.

With the aid of this space, the semi-discrete scheme can now be deﬁned similarly as

in (4.4)-(4.5) by using the space on the dual mesh as ˆ˜Vk

N,D, and replacing the numerical

values on the boundary of the domain by corresponding functions in the Dirichlet boundary

conditions.

We now comment on the implementation of this algorithm. As can be seen from Figures

97

Table 4.2: L2 errors and orders of accuracy for L2 projection operator ˜PD of (4.9) onto
ˆ˜Vk
N,D when d = 2 and d = 3. N is the number of mesh levels, k is the polynomial order, d
is the dimension. L2 order is calculated with respect to hN .

L2 error

order L2 error

order L2 error

order

N

hN

k = 1

3
4
5
6
7

3
4
5
6
7

1/8
1/16
1/32
1/64
1/128

1/8
1/16
1/32
1/64
1/128

8.93E-04
2.61E-04
7.34E-05
2.00E-05
5.35E-06

6.19E-04
1.90E-04
5.71E-05
1.67E-05
4.80E-06

–

1.77
1.83
1.88
1.90

–

1.70
1.73
1.77
1.80

k = 2
d = 2

9.14E-06
1.29E-06
1.77E-07
2.37E-08
3.11E-09

d = 3

4.93E-06
7.45E-07
1.10E-07
1.58E-08
2.24E-09

–

2.82
2.87
2.90
2.93

–

2.73
2.76
2.80
2.82

k = 3

6.40E-08
4.45E-09
3.01E-10
1.98E-11
1.29E-12

3.18E-08
2.36E-09
1.69E-10
1.18E-11
9.35E-13

–

3.85
3.89
3.93
3.94

–

3.75
3.80
3.84
3.66

4.1b and 4.2b, there are two types of basis functions in 1D for the dual space.

• Type 1 bases (for l ≥ 0), which are the shifted and truncated multiwavelet bases.

• Type 2 bases (for l = 0), which are the Legendre polynomials of degree up to k on

[1 − hN

2 , 1].

Clearly, Type 1 bases are orthogonal to Type 2 bases, because their support do not overlap.

Type 2 bases are orthogonal to each other due to the deﬁnition of Legendre polynomials.

However, Type 1 bases are no longer orthogonal to each other, due to the domain shift and

truncation. However, only the left-most element on each level are changed. For other bases

in that level, they will still retain orthogonality. The bases on left-most element in all level

are orthogonal to other bases, but not to each other, i.e., the bases deﬁned on left-most

element in diﬀerent levels are not orthogonal. This implies that although the mass matrix

is not identity here, it will have block structures and be sparse.

98

(a) Primal mesh. Number of bases for l = 0, 1, 2 are
1, 1, 2.

(b) Dual mesh. Number of bases for l = 0, 1, 2 are
2, 1, 2.

Figure 4.1: Illustration of one-dimensional bases on diﬀerent levels for k = 0: non-periodic
problems. Diﬀerent colors represent diﬀerent bases.

4.2 Stability and convergence

In this section, we prove L2 stability and error estimates for the sparse grid CDG scheme

for the scalar equation. We consider both periodic and non-periodic boundary conditions.

For periodic problems, (4.2) reduces to

+ ∇ · (Au) = 0, x ∈ Ω,

∂u
∂t

(4.10)

where A = (A1(t, x),··· , Ad(t, x)), and (cid:107)A(cid:107)L∞(Ω) < ∞,(cid:107)∇ · A(cid:107)L∞(Ω) < ∞. We assume
Ai (cid:54)= 0 to avoid the discussion of diﬀerent boundary conditions for degenerating coeﬃcients.

However, there is no diﬃculty to extend the proof below to degenerating case. For non-

99

(a) Primal mesh. Number of bases for l = 0, 1, 2 are
2, 2, 4.

(b) Dual mesh. Number of bases for l = 0, 1, 2 are
4, 2, 4.

Figure 4.2: Illustration of one-dimensional bases on diﬀerent levels for k = 1: non-periodic
problems. Diﬀerent colors represent diﬀerent bases.

periodic problems, the following inﬂow boundary conditions are prescribed,

u(t, x)|∂Ω

= gi(t,··· , xi−1, xi+1,··· , xd)

where

∂Ω

xin
i

:=

{x ∈ Ω|xi = 0},
{x ∈ Ω|xi = 1},

if Ai(t, x) > 0,

if Ai(t, x) < 0.

Correspondingly, we denote the outﬂow edges by

xin
i




∂Ω

xout
i

:=

{x ∈ Ω|xi = 1},
{x ∈ Ω|xi = 0},

if Ai(t, x) > 0,

if Ai(t, x) < 0.

100

The scheme for periodic case reduces to: to ﬁnd uh ∈ ˆVk
(cid:90)

N,P and vh ∈ ˆVk
(cid:90)

(cid:90)

(cid:90)

(uh)t ϕh dx =

(vh − uh) ϕh dx +

vhA · ∇ϕh dx − (cid:88)
uhA · ∇ψh dx − (cid:88)

e∈ΓN,P

e∈ΓN,D

e

(cid:90)

e

Ω

(cid:90)

Ω

N,D, such that

vhA · [ϕh] ds,

(4.11)

uhA · [ψh] ds,

(4.12)

Ω

(cid:90)

Ω

1

τmax

Ω

(cid:90)

1

τmax

Ω

(vh)t ψh dx =

(uh − vh) ψh dx +

for any ϕh ∈ ˆVk
and enforce uh|∂Ω

N,P and ψh ∈ ˆVk

= vh|∂Ω

xin
i

xin
i

N,D. For non-periodic problems, we require vh, ψh ∈ ˆ˜Vk
= gi on the boundary interface.

N,D,

We can prove that the schemes retain similar stability properties as the standard CDG

schemes.

Theorem 4.2.1 (L2 Stability). With periodic boundary condition, the numerical solutions

uh and vh of the sparse grid CDG scheme (4.11)-(4.12) for the equation (4.10) satisfy the

following L2 stability condition

(cid:107)uh(cid:107)2

L2(ΩN,P )

+ (cid:107)vh(cid:107)2

L2(ΩN,D)

(cid:46) (cid:107)uh(0, x)(cid:107)2

L2(ΩN,P )

+ (cid:107)vh(0, x)(cid:107)2

L2(ΩN,D)

.

(4.13)

For non-periodic boundary condition, the corresponding numerical solutions satisfy

(cid:107)uh(cid:107)2

L2(ΩN,P )

+ (cid:107)vh(cid:107)2

L2(ΩN,D)

+ (cid:107)vh(0, x)(cid:107)2

(cid:46) (cid:107)uh(0, x)(cid:107)2
(cid:90)
(cid:90) T
d(cid:88)

+

L2(ΩN,P )
|Ai|g2

0

i=1

∂Ω

xin
i

i ds dt

L2(ΩN,D)
if τmax (cid:46) hN(cid:107)A(cid:107)1

(4.14)

.

Proof. For periodic boundary condition, let ϕh = uh in (4.11) and ψh = vh in (4.12),

101

summing the two equalities up, we have

1
2

d
dt
1

=

τmax

(cid:90)

+

Ω

(cid:90)
(cid:90)
uhA · ∇vh dx − (cid:88)
(cid:90)
(cid:90)

Ω

e∈ΓN,D

Ω

= − 1
τmax

− (cid:88)

Ω

(uh − vh)2dx +
(cid:90)

uhA · [vh] ds.

Ω

e∈ΓN,D

e

((uh)2 + (vh)2)dx
vh uh − uh uh + uhvh − vhvh dx +

(cid:90)

vhA · ∇uh dx − (cid:88)

(cid:90)

e

vhA · [uh] ds

Ω

uhA · [vh] ds

(cid:90)
A · ∇(uhvh)dx − (cid:88)

e

e∈ΓN,P

e∈ΓN,P

(cid:90)

e

vhA · [uh] ds

(cid:90)

Ω

Apply divergence theorem, and by periodicity, we have

A · ∇(uhvh)dx − (cid:88)

(cid:90)

Avh · [uh] ds − (cid:88)

(cid:90)

e∈ΓN,P

e

e∈ΓN,D

e

Auh · [vh] ds = −

(cid:90)

Ω

∇ · Auhvhdx.

By the simple inequality ab ≤ 1

2(a2 + b2),

(cid:90)

1
2

d
dt

Ω

(cid:0)(uh)2 + (vh)2(cid:1)dx ≤ − 1

(cid:90)

(uh − vh)2dx +

(cid:107)∇ · A(cid:107)L∞(Ω)

1
2

τmax

Ω

(cid:90)

Ω

((uh)2 + (vh)2)dx.

and the proof for the periodic case is complete by using Gronwall’s inequality.

For non-periodic boundary condition, we follow the same lines and plug in the corre-

sponding boundary condition,

(cid:90)

d
dt

1
2
= − 1
τmax

Ω

(cid:90)

((uh)2 + (vh)2)dx
(uh − vh)2dx −

Ω

(cid:90)

∂Ω

A · nuhvhds

(cid:90)

Ω

∇ · Auhvhdx +

102



A · nuhvhds

(cid:90)

∂Ω

xout
i

(cid:90)

∂Ω

xout
i

(cid:90)

Ω

(cid:90)

1
2





(cid:90)
(cid:90)
(cid:90)
(cid:90)
(cid:90)
(cid:90)
(cid:90)
(cid:90)
(cid:90)

− d(cid:88)

i=1

d(cid:88)

i=1

d(cid:88)

i=1

d(cid:88)

i=1

d(cid:88)

i=1

+

+

+

+

≤ − 1
τmax

Ω

= − 1
τmax

Ω

A · ngi(uh + vh)ds + 2

∂Ω

xin
i
(uh − vh)2dx −

(cid:90)

Ω

∇ · Auhvhdx

= − 1
τmax

Ω

|A · n|(−uhvh + gi(uh + vh))ds −

|A · n|uhvhds

∂Ω

xin
i
(uh − vh)2dx +

≤ − 1
τmax

Ω

(cid:107)∇ · A(cid:107)L∞(Ω)

1
2

((uh)2 + (vh)2)dx

|A · n|(uh − vh)2ds

|A · n|(g2

i +

∂Ω

xin
i
(uh − vh)2dx +

|A · n|g2

i ds +

∂Ω

xin
i
(uh − vh)2dx +

1
2

(uh − vh)2)ds +
(cid:90)

1
2

(cid:107)∇ · A(cid:107)L∞(Ω)
(cid:90)

Ω

∪∂Ω

∂Ω

xin
i

xout
i
(cid:107)∇ · A(cid:107)L∞(Ω)

(cid:90)

∂Ω

xout
i

((uh)2 + (vh)2)dx

|A · n|1
2

(uh − vh)2ds

((uh)2 + (vh)2)dx

Ω



1
2

(cid:90)

|Ai|g2

i ds +

xin
i

∂Ω

∂Ω

xin
i

∪∂Ω

xout
i

|Ai|1
2

(uh − vh)2ds

 .

by noticing A · n|∂Ω

< 0 and A · n|∂Ω

xin
i

> 0.

xout
i

Let T i

N,D := {T ∈ ΩN,D|T ∩ ∂Ωxi (cid:54)= ∅} denote the cells on dual mesh adjacent to
(cid:46)

the boundary in the i-th direction. By inverse inequality, we have (cid:107)uh − vh(cid:107)2
h−1
N (cid:107)uh − vh(cid:107)2

N (cid:107)uh − vh(cid:107)2

L2(∂Ωxi )

≤ h−1

L2(Ω)

L2(T i

N,D)

(cid:90)

Ω

1
2

d
dt

((uh)2 + (vh)2)dx ≤ 1
2

(cid:107)∇ · A(cid:107)L∞(Ω)

Ω

. Therefore, if τmax (cid:46) hN(cid:107)A(cid:107)1
(cid:90)
d(cid:88)

((uh)2 + (vh)2)dx +

,

(cid:90)

|Ai|g2

i ds,

i=1

∂Ω

xin
i

and the proof for the non-periodic case is complete by using Gronwall’s inequality.

103

Now we are ready to prove L2 error estimate of the sparse grid CDG scheme.

Theorem 4.2.2 (L2 error estimate). Let u be the exact solution to (4.10) and uh, vh be

the numerical solution to the semidiscrete scheme (4.11) and (4.12) with initial discretiza-

tion uh(0, x) = PP u0, vh(0, x) = PDu0 for periodic boundary condition or uh(0, x) =
PP u0, vh(0, x) = ˜PDu0 for non-periodic boundary condition. If τmax (cid:46) hN , then for k ≥ 1,
u0 ∈ Hp+1(Ω), 1 ≤ q ≤ min{p, k}, N ≥ 1, d ≥ 2, we have for all t ≥ 0

(cid:107)u − uh(cid:107)L2(ΩN,P ) + (cid:107)u − vh(cid:107)L2(ΩN,D)

(cid:46) N d2−N q |u|Hq+1(Ω) .

(4.15)

Proof. For periodic problems, we ﬁrst introduce the standard notation of bilinear form

B(uh, vh; ϕh, ψh) =

+

−

Ω

(cid:90)
(cid:90)
(cid:90)
(cid:90)
(uh)t ϕh dx − 1
(cid:88)
τmax
vhA · [ϕh] ds +
(cid:90)
(cid:88)

uhA · ∇ψh dx +

e∈ΓP

e

Ω

Ω

e∈ΓD

e

(cid:90)

(vh − uh) ϕh dx −
(vh)t ψh dx − 1
(cid:90)
τmax

uhA · [ψh] ds.

Ω

Ω

(cid:90)
vhA · ∇ϕh dx
(uh − vh) ψh dx

Ω

By Galerkin orthogonality, we have the error equation

B(u − uh, u − vh; ϕh, ψh) = 0,

∀ϕh ∈ ˆVk

N,P , ψh ∈ ˆVk

N,D.

(4.16)

We take

ϕh = PP u − uh, ψh = PDu − uh,
ψe = PDu − u,
ϕe = PP u − u,

104

then the error equation (4.16) becomes

B(ϕh, ψh; ϕh, ψh) = B(ϕe, ψe; ϕh, ψh).

(4.17)

From Theorem 4.2.1, we get

(cid:90)

1
2

d
dt

Ω

(cid:0)ϕ2

h + ψ2
h

(cid:1)dx ≤ B(ϕe, ψe; ϕh, ψh) +

(cid:90)

Ω

(cid:107)∇ · A(cid:107)L∞(Ω)

1
2

(ϕ2

h + ψ2

h)dx.

(4.18)

We write the bilinear form on the right-hand side as a sum of three terms

B(ϕe, ψe; ϕh, ψh) = B1 + B2 + B3,

(4.19)

where

B1 =

(cid:90)

B2 = −

B3 =

Ω

(cid:90)
(ϕe)t ϕh dx − 1
τmax
(cid:90)
ψeA · ∇ϕh dx −
(cid:88)

Ω

ψeA · [ϕh] ds +

(ψe − ϕe) ϕh dx +
ϕeA · ∇ψh dx,
(cid:88)

(cid:90)

ϕeA · [ψh] ds.

e∈ΓN,P

e

e∈ΓN,D

e

(cid:90)

Ω

(ψe)t ψh dx − 1
τmax

(cid:90)

Ω

(ϕe − ψe) ψh dx,

(cid:90)
(cid:90)

Ω

Ω

(cid:90)

Ω

By Cauchy-Schwartz inequality, Lemma 4.1.1 and τmax (cid:46) hN , we have

B1 (cid:46)

(ϕ2

h + ψ2

h)dx + N 2d2−2N q |u|2Hq+1(Ω)

.

(4.20)

To estimate B2, B3, we use the following inverse inequalities ∀wh ∈ ˆVk

N,G, for G = P, D,

|wh|H1(ΩN,G)

(cid:46) h−1

N (cid:107)wh(cid:107)L2(ΩN,G),

(cid:107)wh(cid:107)ΓN,G

(cid:46) h

− 1
N (cid:107)wh(cid:107)L2(ΩN,G)
2

105

and trace inequality,

(cid:107)φ(cid:107)2

L2(∂T )

(cid:46) hN

Then we have

−1(cid:107)φ(cid:107)2
(cid:90)

+ hN|φ|H1(T ),

∀φ ∈ H1(T ), T ∈ ΩN,G.

L2(T )

B2 (cid:46)

Ω

(ϕ2

h + ψ2

h)dx + N 2d2−2N q |u|2Hq+1(Ω)

(4.21)

and

(cid:90)

Ω

B3 (cid:46)

(ϕ2

h + ψ2

h)dx + N 2d2−2N q |u|2Hq+1(Ω)

.

(4.22)

Combining (4.20), (4.21), (4.22) with (4.18), we obtain

(cid:90)

d
dt

Ω

(cid:0)ϕ2

h + ψ2
h

(cid:1)dx (cid:46)

(cid:90)

Ω

(ϕ2

h + ψ2

h)dx + N 2d2−2N q |u|2Hq+1(Ω)

.

Together with the estimates for initial discretization and by Gronwall’s inequality, the

proof is complete. For non-periodic problems, the argument is very similar as long as the

stability result holds. The proof is omitted for brevity.

This theorem proves L2 error of the scheme is O(N d2−N k) or O(|log hN|d hk

N ) when the

exact solution has enough smoothness in the mixed derivative norms.

4.3 Numerical results

In this section, we present several numerical tests to validate the performance of the proposed

sparse grid CDG schemes. Unless otherwise stated, we use the third-order TVD-RK temporal

discretization [73] and choose the time step ∆t =

, with c = 0.1 for k = 1, 2, where

d(cid:88)

c

ci
hN

i=1

106

ci is the maximum wave propagation speed in xi-direction. To guarantee that the spatial

error dominates for k = 3, we take ∆t = O(h

4/3
N ). τmax is taken as

1

2k+1 hN which is always

smaller than the maximum time step allowed based on the CFL number in Table 4.1. For

periodic problems, we only provide L2 errors on the primal mesh, because the results on the

dual mesh are similar. For non-periodic problems, the L2 errors are the L2 average of the

errors on the primal and dual meshes.

4.3.1 Scalar case

In this subsection, we consider the scalar case, i.e. m = 1.

Example 4.3.1 (Linear advection with constant coeﬃcients). We consider



d(cid:88)

ut +

i=1

u(0, x) = sin

uxi = 0, x ∈ [0, 1]d,

2π

d(cid:88)

i=1

 ,

xi

(4.23)

with periodic or Dirichlet boundary conditions on the inﬂow edges corresponding to the given

exact solution.

The exact solution is a smooth function,

2π

 .

xi − d t

 d(cid:88)

i=1

u(t, x) = sin

In the simulation, we compute the numerical solutions up to two periods in time, meaning

that we let ﬁnal time T = 1 for d = 2, T = 2/3 for d = 3, and T = 0.5 for d = 4.

We ﬁrst test the scheme with periodic boundary condition. In Table 4.3, we report the

107

L2 errors and orders of accuracy for k = 1, 2, 3 and up to dimension four. As for accuracy,

we observe about half order reduction from the optimal (k + 1)-th order for high-dimensional

computations (d = 4). The order is slightly better for lower dimensions. The convergence

order is similar to the performance of the sparse grid DG scheme in [38]. In Figure 4.3, we

plot the time evolution of the error of L2 norm of numerical solutions uh and vh, which is

given by

(cid:90)

Ω

(cid:0)(uh(t, x))2 + (vh(t, x))2(cid:1)dx −

(cid:90)

Ω

(cid:0)(uh(0, x))2 + (vh(0, x))2(cid:1)dx

for two-dimensional case for t = 0 to t = 100. From Theorem 4.2.1, such errors are propor-

tional to the diﬀerence between uh and vh. We can clearly see that the higher order accurate

scheme performs way better in conservation of L2 norm due to its higher order accuracy.

(a) k=1

(b) k=2

(c) k=3

Figure 4.3: Example 4.3.1. The time evolution of the error of L2 norm of numerical solutions
uh and vh of the sparse grid CDG method with d = 2. (a) k=1, (b) k=2, (c) k=3. N = 4, 5, 6.

Then, we test the scheme with Dirichlet boundary condition prescribed at the inﬂow edge

according to the exact solution. The results are listed in Table 4.4. The accuracy order is

similar to the periodic case.

Finally, we use this example to compare the performance of the DG, CDG, sparse grid

108

tK=1020406080100(cid:173)1.2(cid:173)1(cid:173)0.8(cid:173)0.6(cid:173)0.4(cid:173)0.20N=4N=5N=6tK=2020406080100(cid:173)0.06(cid:173)0.05(cid:173)0.04(cid:173)0.03(cid:173)0.02(cid:173)0.010N=4N=5N=6tK=3020406080100(cid:173)0.007(cid:173)0.006(cid:173)0.005(cid:173)0.004(cid:173)0.003(cid:173)0.002(cid:173)0.0010N=4N=5N=6Table 4.3: L2 errors and orders of accuracy for Example 4.3.1 at T = 1 when d = 2, T = 2/3
when d = 3, and T = 0.5 when d = 4. N denotes mesh level, hN is the size of the smallest
mesh in each direction, k is the polynomial order, d is the dimension. L2 order is calculated
with respect to hN .

L2 error

order L2 error

order L2 error

order

N

hN

k = 1

3
4
5
6
7

3
4
5
6
7

3
4
5
6
7

1/8
1/16
1/32
1/64
1/128

1/8
1/16
1/32
1/64
1/128

1/8
1/16
1/32
1/64
1/128

3.14E-01
6.99E-02
1.34E-02
3.43E-03
9.21E-04

6.77E-01
3.56E-01
1.05E-01
2.54E-02
7.45E-03

7.13E-01
6.48E-01
3.80E-01
1.37E-01
3.81E-02

–

2.17
2.38
1.97
1.90

–

0.93
1.76
2.05
1.77

–

0.14
0.77
1.47
1.85

k = 2
d = 2

1.20E-02
2.23E-03
4.87E-04
5.97E-05
9.33E-06

d = 3

5.27E-02
1.10E-02
1.82E-03
5.22E-04
6.89E-05

d = 4

1.26E-01
3.39E-02
6.91E-03
1.39E-03
3.56E-04

–

2.43
2.20
3.03
2.68

–

2.26
2.60
1.80
2.92

–

1.89
2.29
2.31
1.97

k = 3

5.84E-04
8.50E-05
3.84E-06
3.89E-07
1.80E-08

2.13E-03
2.62E-04
2.85E-05
2.01E-06
2.01E-07

4.41E-03
7.56E-04
9.82E-05
9.44E-06
8.16E-07

–

2.78
4.47
3.30
4.43

–

3.02
3.20
3.83
3.32

–

2.54
2.94
3.38
3.53

Table 4.4: L2 errors and orders of accuracy for Example 4.3.1 with Dirichlet boundary
condition on the inﬂow edges at T = 1 when d = 2 and T = 2/3 when d = 3. N denotes
mesh level, hN is the size of the smallest mesh on the primal mesh in each direction, k is
the polynomial order, d is the dimension. L2 order is calculated with respect to hN .

L2 error

order L2 error

order L2 error

order

N

hN

k = 1

3
4
5
6
7

3
4
5
6
7

1/8
1/16
1/32
1/64
1/128

1/8
1/16
1/32
1/64
1/128

2.66E-01
7.47E-02
1.94E-02
5.44E-03
1.49E-03

6.15E-01
2.86E-01
1.14E-01
3.23E-02
1.03E-02

–

1.83
1.95
1.83
1.87

–

1.10
1.33
1.82
1.65

k = 2
d = 2

1.66E-02
3.33E-03
5.97E-04
8.60E-05
1.35E-05

d = 3

5.34E-02
1.40E-02
2.57E-03
5.82E-04
9.81E-05

–

2.32
2.48
2.80
2.67

–

1.93
2.45
2.14
2.57

k = 3

8.21E-04
8.80E-05
4.79E-06
4.50E-07
2.20E-08

2.67E-03
2.87E-04
3.21E-05
2.60E-06
2.86E-07

–

3.22
4.20
3.41
4.35

–

3.22
3.16
3.63
3.18

109

DG and sparse grid CDG methods. We use the following non-separable initial condition

sin

2π

d(cid:88)

 , x ∈ [0, 1]d,

xi

u(0, x) = exp

(4.24)

i=1

where d = 2. When k = 1, 2, 3, Runge-Kutta methods of order ν = 2, 3, 4, respectively, are

used for time discretization. We take the time step according to the CFL numbers listed in

Table 4.1. We plot the comparison of the methods measuring L2 errors vs. CPU times in

Figure 4.4. The computations in this example are implemented by an OpenMP code using

computational resources from the Institute for Cyber-Enabled Research in Michigan State

University. We can see that the sparse grid CDG method outperforms the CDG method,

and the sparse grid DG method outperforms the DG method particularly when the mesh

level N is more reﬁned. When the mesh level increases from N to N + 1, the CPU cost

for sparse grid method grows with the rate of about 4 to 5, while the factor is about 8 to

10 for full grid calculations, respectively, for this 2D case. This shows the advantage of the

sparse grid approach. When comparing the sparse grid CDG method with the sparse grid

DG method, it seems that for this example, the sparse grid DG method is more eﬃcient. It

will be interesting to compare the results for fully nonlinear problems in higher dimensions,

for which the CDG method is more advantageous, and this is currently under investigation.

Example 4.3.2 (Solid body rotation). We consider solid-body-rotation problems, which are

in the form of (4.1) with periodic boundary conditions and

• d = 2, A1(t, x) = −x2 + 1
(cid:16)

• d = 3, A1(t, x) = −

√
2
2

2 , A2(t, x) = x1 − 1
2 ,
√
2
2

, A2(t, x) =

x2 − 1

(cid:17)

2

(cid:16)

(cid:17)

√
2
2

+

(cid:16)

x3 − 1

2

(cid:17)

x1 − 1

2

, A3(t, x) =

(cid:16)
x2 − 1

2

(cid:17)

√
2
2

−

.

110

(a) k=1

(b) k=2

(c) k=3

Figure 4.4: L2 errors and associated CPU times of DG, CDG, sparse grid DG and sparse
grid CDG methods for Example 4.1 with initial condition (4.24) at T = 1 for d=2. (a) k=1,
(b) k=2, (c) k=3.

Such benchmark tests are commonly used in the literature to assess performance of

transport schemes. Here, the initial proﬁle traverses along circular trajectories centered
at (1/2, 1/2) for d = 2 and about the axis {x1 = x3} ∩ {x2 = 1/2} for d = 3 without

deformation, and it goes back to the initial state after 2π in time. The initial conditions are

set to be the following smooth cosine bells (with C5 smoothness),

 bd−1 cos6(cid:0) πr

0,

u(0, x) =

(cid:1) ,

2b

r ≤ b,

if

otherwise,

(4.25)

where b = 0.23 when d = 2 and b = 0.45 when d = 3, and r = |x − xc| denotes the

distance between x and the center of the cosine bell with xc = (0.75, 0.5) for d = 2 and

xc = (0.5, 0.55, 0.5) for d = 3.

In Table 4.5, we summarize the convergence study of the numerical solutions computed

by the sparse CDG method, including the L2 errors and orders of accuracy. For this variable

coeﬃcients equation, we observe at least k-th order convergence for all cases. The order is

slightly lower than the corresponding ones in Example 4.3.1.

111

+++CPU timeL2 error10010110(cid:173)310(cid:173)210(cid:173)1Full DG Full CDG Sparse DG Sparse CDG ++++CPU timeL2 error10010110(cid:173)510(cid:173)410(cid:173)310(cid:173)2Full DGFull CDGSparse DGSparse CDG++++CPU timeL2 error10110210(cid:173)710(cid:173)610(cid:173)510(cid:173)410(cid:173)3Full DGFull CDGSparse DGSparse CDG+Table 4.5: L2 errors and orders of accuracy for Example 4.3.2 at T = 2π. N denotes mesh
level, hN is the size of the smallest mesh in each direction, k is the polynomial order, d is
the dimension. L2 order is calculated with respect to hN .

L2 error

order L2 error

order L2 error

order

N

hN

k = 1

5
6
7
8

5
6
7
8

1/32
1/64
1/128
1/256

1/32
1/64
1/128
1/256

1.53E-02
1.02E-02
4.66E-03
1.42E-03

4.83E-03
1.87E-03
7.46E-04
2.55E-04

–

0.58
1.13
1.71

–

1.37
1.33
1.55

k = 2
d = 2

5.81E-03
1.50E-03
1.46E-04
2.34E-05

d = 3

6.25E-04
1.20E-04
3.39E-05
8.11E-06

–

1.95
3.36
2.64

–

2.38
1.82
2.06

k = 3

1.34E-03
9.64E-05
1.16E-05
1.10E-06

7.35E-05
9.18E-06
1.36E-06
1.94E-07

–

3.80
3.05
3.40

–

3.00
2.75
2.81

Example 4.3.3 (Deformational ﬂow). We consider the two-dimensional deformational ﬂow

with velocity ﬁeld

A1(t, x) = sin2(πx1) sin(2πx2)g(t), A2(t, x) = − sin2(πx2) sin(2πx1)g(t),

where g(t) = cos(πt/T ) with T = 1.5, with periodic boundary condition.

We still adopt the cosine bell (4.25) as the initial condition for this test, but with xc =

(0.65, 0.5) and b = 0.35. Note that the deformational test is more challenging than the

solid body rotation due to the space and time dependent ﬂow ﬁeld. In particular, along the

direction of the ﬂow, the cosine bell deforms into a crescent shape at t = T /2 , then goes

back to its initial state at t = T as the ﬂow reverses. In the simulations, we compute the

solution up to t = T . The convergence study is summarized in Table 4.6. Similar orders

are observed compared with Example 4.3.2. In Figure 4.5, we plot the contour plots of the

numerical solutions on the primal mesh at t = T /2 when the shape of the bell is greatly

deformed, and t = T when the solution is recovered into its initial state. It is observed that

112

the sparse CDG scheme with higher degree k can better resolve the highly deformed solution

structure.

Table 4.6: L2 errors and orders of accuracy for Example 4.3.3 at T = 1.5. N denotes mesh
level, hN is the size of the smallest mesh in each direction, k is the polynomial order, d is
the dimension. L2 order is calculated with respect to hN . d = 2.

N

hN

L2 error

order L2 error

order L2 error

order

k = 1

k = 2

k = 3

5
6
7
8

1/32
1/64
1/128
1/256

1.73E-02
8.06E-03
3.29E-03
1.08E-03

–

1.10
1.29
1.61

4.37E-03
1.17E-03
2.04E-04
2.78E-05

–

1.90
2.52
2.88

1.14E-03
2.44E-04
2.05E-05
2.75E-06

–

2.22
3.57
2.90

4.3.2 System case

In this subsection, we consider system case, which means m > 1 in equation (4.1) or (4.2).

Example 4.3.4 (Acoustic wave equation with constant wave speed). We consider

ut = ∇ · v, x ∈ [0, 1]2,
vt = ∇u,

u(0, x) = u0(x), v(0, x) = v0(x).

(4.26)

with periodic boundary conditions. The initial conditions u0(x) and v0(x) are chosen ac-

cording to the following two types of exact solutions: the standing wave

u(t, x)

2πt) sin(2πx1) sin(2πx2)



v1(t, x)

v2(t, x)



−√

√
2 sin(2
√
cos(2
√
cos(2

2πt) cos(2πx1) sin(2πx2)

2πt) sin(2πx1) cos(2πx2)

 ,


 =

113

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4.5: Example 4.3.3. Deformational ﬂow test. The contour plots of the numerical
solutions on primal mesh at t = T /2 (a, c, e) and t = T (b, d, f). k = 1 (a, b), k = 2 (c, d),
and k = 3 (e, f). N = 7.

114

and the traveling wave

 =



√
√
2πt + 2πx1) cos(2πx2)
2 sin(2
√
sin(2
2πt + 2πx1) cos(2πx2)
√
cos(2

2πt + 2πx1) sin(2πx2)

 .

u(t, x)

v1(t, x)

v2(t, x)

We compute the solution until T = 1. Similar to the scalar case, we present the L2 errors

(cid:20)

(cid:21)T

and orders of accuracy for u(t, x) =

u(t, x), v1(t, x), v2(t, x)

in Table 4.7. From the

table, we still observe at least (k + 1/2)-th order for the solution.

Table 4.7: L2 errors and orders of accuracy for Example 4.3.4 at T = 1. N denotes mesh
level, hN is the size of the smallest mesh in each direction, k is the polynomial order, d is
the dimension. L2 order is calculated with respect to hN . d = 2.

L2 error

order L2 error

order L2 error

order

N

hN

k = 1

k = 2

k = 3

3
4
5
6
7

3
4
5
6
7

1/8
1/16
1/32
1/64
1/128

1/8
1/16
1/32
1/64
1/128

3.56E-01
7.93E-02
1.50E-02
3.72E-03
1.01E-03

3.97E-01
8.58E-02
1.97E-02
5.36E-03
1.50E-03

–

2.17
2.40
2.01
1.88

–

2.21
2.12
1.88
1.84

standing wave
1.05E-02
–
2.51
1.84E-03
2.53
3.18E-04
2.68
4.95E-05
7.60E-06
2.70
traveling wave
1.85E-02
–
3.36E-03
6.07E-04
9.66E-05
1.45E-05

2.46
2.47
2.65
2.74

5.37E-04
4.31E-05
3.39E-06
2.77E-07
2.03E-08

7.75E-04
6.76E-05
5.68E-06
4.44E-07
3.39E-08

–

3.64
3.67
3.61
3.77

–

3.52
3.57
3.68
3.71

Example 4.3.5 (Two-dimensional homogeneous isotropic elastic wave [44]). The 2D elastic

wave equation in homogeneous and isotropic medium in velocity-stress formulation without

external source, is a linear hyperbolic system of the form

ut + A1ux1 + A2ux2 = 0,

(4.27)

115





0 0 0 λ + 2µ 0

0 0 0

0 0 0

1
ρ 0 0

0 0 1
ρ

λ

0

0

0

0

µ

0

0

A1 = −

, A2 = −

0 0 0 µ



0 0 0 0

λ

0 0 0 0 λ + 2µ

0 0

0 1
ρ

1
ρ

0

0 0



,

0

0

0

(cid:113) λ+2µ

ρ

(cid:20)

(cid:21)T

where u =

σxx, σyy, σxy, v, w

, σxx, σyy represents the normal stress and σxy rep-

resents the shear stress and v, w are the velocity in x and y directions.

where λ and µ are the Lam´e constants and ρ is the mass density of material. Eigenvalues of
A1 and A2 are −cp,−cs, 0, cs, cp, which give us the wave speed cp =

and cs =

(cid:113) µ

ρ

for P-wave and S-wave respectively. We consider the homogeneous material parameters

λ = 2, µ = 1, ρ = 1, then cp = 2, cs = 1. On domain Ω = [0, 1]2, we take the solutions

√
2
2 ,

√
2
2 ) and a plane

consisting of a plane P-wave traveling along diagonal direction n = (

S-wave traveling in the opposite direction, i.e.,

u(t, x) = Rsesin(k·x+kcst) + Rpesin(k·x−kcpt),

where Rs = [−µ, µ, 0,−

√
kn, k = 2

√
2
2 cs,

√
2 cs]T , Rp = [λ + µ, λ + µ, µ,−

2

√
2 cp,−

2

√
2
2 cp]T and k =

2π. Periodic boundary condition is applied and the initial condition is chosen as

u(0, x).

We compute the solution until T = 1. The L2 errors and orders of accuracy for u(t, x)

are shown in Table 4.8. We observe that the convergence order is close to k + 1.

Example 4.3.6 (Three-dimensional isotropic elastic wave [31]). We extend the previous

116

Table 4.8: L2 errors and orders of accuracy for Example 4.3.5 at T = 1. N denotes mesh level,
hN is the size of the smallest mesh in each direction, k is the polynomial order, dimension
d = 2. L2 order is calculated with respect to hN .

N
4
5
6
7

hN
1/16
1/32
1/64
1/128

L2 error

order L2 error

order L2 error

order

k = 1

k = 2

k = 3

1.09E+00
7.47E-01
2.41E-01
7.14E-02

–

0.55
1.63
1.76

2.72E-01
6.48E-02
9.65E-03
1.12E-03

–

2.07
2.75
3.11

5.71E-02
6.19E-03
4.77E-04
2.55E-05

–

3.21
3.70
4.23

example to 3D and obtain the following linear hyperbolic system

(cid:20)

ut + A1ux1 + A2ux2 + A3ux3 = 0,

(cid:21)T

(4.28)

where u =

σxx, σyy, σzz, σxy, σyz, σxz, u, v, w

, σ is the stress tensor and

u, v, w are the velocities in each spatial direction.

0 0 0 0 0 0 0

λ

0

0 0 0 0 0 0 0 λ + 2µ 0



0 0 0 0 0 0 0

0 0 0 0 0 0 µ

, A2 = −

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 1
ρ

0 0 0

0 1

ρ 0 0 0 0 0

0 0 0 0

1
ρ 0 0



,

λ

0

0

0

0

0

0

0

0

µ

0

0

0

0



0 0 0 0 0 0 λ + 2µ 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1
ρ 0 0 0 0 0

0 0 0 1

ρ 0 0

0 0 0 0 0 1
ρ

λ

λ

0

0

0

0

0

0

0 0

0 0

µ 0

0 0

0 µ

0 0

0 0

0 0

A1 = −



117



0 0 0 0 0 0 0 0

A3 = −

0 0 0 0 0 0 0 µ

0 0 0 0 0 0 µ 0

0 0 0 0 0

0 0 0 0 1
ρ

1
ρ

0 0

0 0 0

0 0 1

ρ 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 λ + 2µ



,

λ

λ

0

0

0

0

0

0

where λ, µ and ρ take the same values as the previous example. Hence, we have the same
values for cp and cs. Eigenvalues of A1, A2 and A3 are −cp,−cs,−cs, 0, 0, 0, cs, cs, cp, which

describe the wave speed for P-wave and S-wave (with diﬀerent polarizations). On domain

Ω = [0, 1]3, we take the solutions consisting of a plane S-wave traveling along diagonal
direction n = (− 1√

) and a plane P-wave traveling in the opposite direction, i.e.,

,− 1√

,− 1√

3

3

3

u(t, x) = Rs sin(k · x − kcst) + Rp sin(k · x + kcpt),

where

Rs = [−2
3

µ,

2
3

µ, 0, 0,

1
3

µ,−1
3

Rp = [λ +

2
3

µ, λ +

2
3

µ, λ +

2
3

µ,− 1√
3
2
3

2
3

µ,

µ,

cs,

µ,

cs, 0]T ,

1√
3
µ,− 1√
3

2
3

cp,− 1√
3

cp,− 1√
3

cp]T

√
and k = kn, k = −2

3π. Similarly, we consider periodic boundary condition and u0(x) =

u(0, x) as initial condition.

118

We present the numerical results at T = 1. In Table 4.9, we get at least (k + 1/2)-th

order of accuracy for the solution u(t, x).

Table 4.9: L2 errors and orders of accuracy for Example 4.3.6 at T = 1. N denotes mesh
level, hN is the size of the smallest mesh in each direction, k is the polynomial order, d is
the dimension. L2 order is calculated with respect to hN . d = 3.

N
4
5
6
7

hN
1/16
1/32
1/64
1/128

L2 error

order L2 error

order L2 error

order

k = 1

k = 2

k = 3

2.49E+00
7.70E-01
1.76E-01
4.27E-02

–

1.69
2.13
2.04

4.93E-02
8.17E-03
1.59E-03
2.79E-04

–

2.59
2.36
2.51

8.91E-04
8.66E-05
7.12E-06
5.42E-07

–

3.36
3.60
3.72

119

APPENDIX

120

Detailed discussions on the choice of the T matrix as in

(51) or (52)

We discuss what parameters result in |b1 ± b2| = 0, under the assumption that α1 has no
dependence on h, β1 = ˜β1hp1, β2 = ˜β2hp2, ˜β1, ˜β2 are nonzero constants that do not depend

on h.

b1 − b2 = (−β1 +

k(k − 1)

)(1 − β2

2h
k(k − 1)

2k(k − 1)

k(k − 1)

) +

2α2
1

h

= (− ˜β1hp1 +

h−1)(1 − 2k(k − 1) ˜β2hp2−1) + k(k − 1)2α2

1h−1,

h

h

2

2

b1 + b2 = (−β1 +

k(k + 1)

)(1 − β2

2k(k + 1)

k(k + 1)

) +

2α2
1

h

= (− ˜β1hp1 +

h−1)(1 − 2k(k + 1) ˜β2hp2−1) + k(k + 1)2α2

1h−1.

2h
k(k + 1)

If b1 − b2 = 0,∀h < h0, then

• α1 (cid:54)= 0, then p1 = −1, p2 = 1 and ˜β1, ˜β2 satisﬁes

(− ˜β1 +

k(k − 1)

2

)(1 − 2k(k − 1) ˜β2) + k(k − 1)2α2

1 = 0.

(29)

Similarly, for b1 + b2 = 0,∀h < h0, then

• α1 (cid:54)= 0, p1 = −1, p2 = 1 and ˜β1, ˜β2 satisﬁes

(− ˜β1 +

k(k + 1)

2

)(1 − 2k(k + 1) ˜β2) + k(k + 1)2α2

1 = 0.

(30)

121

Detailed discussions on assumption A2

Parameter choices for |Γ| = |Λ| imply

Γ ± Λ = β1 +

k2(k2 − 1)
= (β1 − k(k ∓ 1)

h2

2h

β2 +
)(1 − 2β2

k(k ± 1)

h
k(k ± 1)

(−2α2

1 − 2β1β2) +
) − k(k ± 1)

2α2

h

h

−k2 ± k

2h

1 = 0,

which indicates

• if α1 (cid:54)= 0, then b1 ± b2 can be greatly simpliﬁed as follows.

– If Γ + Λ = 0, then k is odd, and

b1 + b2 =

(cid:18)
(cid:18)
1 − β2
(cid:18)
b1 − b2 = − 2
k + 1

k
h

2k(k + 1)

(cid:19)
β1 − k(k − 1)
,
k2(k2 − 1)

(cid:19)

2h

h

,

Λ = − 1

k + 1

β1 − k2

h

+

(cid:19)

.

β2

(cid:18)

122

– If Γ − Λ = 0, then k is even, and

h2

(cid:19)
(cid:19)

,

β1 − k(k + 1)
2k(k − 1)

2h

2

b1 + b2 =

(cid:18)
k − 1
(cid:18)
b1 − b2 = − k
1 − β2
h
β1 − k2

Λ = − 1

k − 1

h

(cid:19)

,
h
k2(k2 − 1)

+

h2

β2

,

k > 1.

• If α1 = 0, then

k(k ± 1)

2h

β1 =

, or β2 =

h

2k(k ± 1)

.

(31)

More estimates related to Legendre coeﬃcients

We provide estimates of the Legendre coeﬃcients, especially their diﬀerence in neighboring

cells of equal size.

If u ∈ W k+2+n,∞(I), then expand ˆuj(ξ) at ξ = −1 in (2.24) by Taylor series, we have

for m ≥ k + 1, ∃z ∈ [−1, 1], s.t.

(cid:90) 1

uj,m = C

d

−1

dξk+1

(cid:16) n(cid:88)

s=0

(ξ + 1)s

s!

d

dξs ˆuj(−1)
(cid:17)

d

dξn+1 ˆuj(z)

(ξ + 1)n+1
(n + 1)!

dξm−k−1

(ξ2 − 1)mdξ,

(32)

θshk+1+s

j

u(k+1+s)(x

) + O(hk+2+n

j

j− 1
2

|u|

W k+2+n,∞(Ij )

),

where θs are constants independent of u and hj.

When hj = hj+1, we use Taylor expansion again, and compute the diﬀerence of two uj,m

from neighboring cells

uj,m − uj+1,m =

µshk+1+s

j

u(k+1+s)(x

) + O(hk+2+n

j

j− 1
2

|u|

W k+2+n,∞(Ij∪Ij+1)

). (33)

+

=

d

n(cid:88)

s=0

n(cid:88)

s=1

n(cid:88)

s=1

Then we obtain the estimates

|uj,m − uj+1,m +

µshk+1+s

j

u(k+1+s)(x

j− 1
2

)| ≤ Chk+2+n|u|

W k+2+n,∞(Ij∪Ij+1)

, (34)

123

where µs are constants independent of u and hj.

Two convolution-like operators

To facilitate the analysis in Chapter 2 and 3, we deﬁne two operators on a periodic functions

u in L2(I):

(cid:1)u(x) =

(cid:2)λu(x) =

N−1(cid:88)

l=0

(−1)l−N + 2l
N−1(cid:88)

2

1

1 − λN

l=0

u(x + L

l
N

), N is odd,

(35a)

λlu(x + L

l
N

),

|λ| = 1,

(35b)

n=−∞ ˆf (n)e2πinx/L, we have

where L = b − a is the size of I and N is odd in (35a).

Expand u by Fourier series, i.e., u(x) =(cid:80)∞
N−1(cid:88)
∞(cid:88)
(−1)l−N + 2l
∞(cid:88)
N−1(cid:88)
∞(cid:88)

(cid:1)u(x) =

n=−∞

n=−∞

2π
L inx

ˆf (n)e

l=0

l=0

=

2

−2e2πi n
(1 + e2πi n

N

=

n=−∞

ˆf (n)ein( 2π

L x+2π l
N )

−N + 2l

2

(−ei2π n

N )l

ˆf (n)e

2π
L inx,

(cid:2)λu(x) =

1

1 − λN

N )2

∞(cid:88)

n=−∞

λl

N−1(cid:88)
∞(cid:88)

l=0

ˆf (n)ein( 2π

L x+2π l
N )

N−1(cid:88)

ˆf (n)e

2π
L inx

(λei2π n

N )l

1 − λN

1

∞(cid:88)

=

=

n=−∞
ˆf (n)

n=−∞

1 − λe2πi n

N

l=0

2π
L inx.

e

124

In addition, we can apply the operator on the same function recursively, we have

1

···

1

(1 − λne2πi n

N )νn

ˆf (n)ei 2π

L inx,

N )ν1

2π
L inx,

ˆf (n)

(1 − λ1e2πi n
(cid:16) −2e2πi n

(1 − λe2πi n

N

N )ν

(1 + e2πi n

N )2

e

(cid:17)ν

ˆf (n)e

2π
L inx.

(cid:2)ν1
λ1

··· (cid:2)νn
λn

u(x) =

((cid:2)λ)νu(x) =

((cid:1))νu(x) =

∞(cid:88)
∞(cid:88)
∞(cid:88)

n=−∞

n=−∞

n=−∞

Next, we estimate the two operators. Assuming u ∈ W 3,1(I), then the Fourier coeﬃcient

ˆf (n) satisﬁes:

(cid:12)(cid:12)(cid:12) ˆf (n)
(cid:12)(cid:12)(cid:12) ≤ C

|u|

W k+4,1(I)
1 + |n|3

.

(36)

Since N is odd, then ωn = e2πi n

N (cid:54)= −1,∀n. Hence, (cid:1)u(x) are well deﬁned. We estimate

(cid:1)u(x) by splitting it into blocks of size N as

∞(cid:88)

l=−∞

(cid:1)u(x) =

Sl, where Sl =

lN + N−1

2(cid:88)

n=lN− N−1

2

−2e2πi n
(1 + e2πi n

N

N )2

ˆf (n)e

2π
L inx.

Let’s estimate S0

k+1 ﬁrst. Denote W1(n) = −2e

2

|1+ωn|2 ≤

2

|1+ei3π/4|2

2−√
= 2

2

. For other n, |W1(n)| ≤ |W1( N−1

(1+e

2πi n
N
2πi n
N )2

e

2π

L inx. For |n| ≤ [ 3N
2 )| =

8 ], |W1(n)| =
≤ CN 2

|1+ω(N−1)/2|2

2

125

from Taylor expansions.

N−1

(cid:12)(cid:12)(cid:12) ˆf (n)W1(n)
(cid:12)(cid:12)(cid:12) +
2(cid:88)
(cid:12)(cid:12)(cid:12) ˆf (n)
(cid:12)(cid:12)(cid:12) + CN 2
8 ]−1(cid:88)

n=[ 3N

(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) ˆf (n)W1(n)
(cid:12)(cid:12)(cid:12) ˆf (n)
(cid:12)(cid:12)(cid:12)
2(cid:88)

N−1

8 ]+1

n=[ 3N

8 ]+1

(

N
4

+ 2)|u|W 3,1(I)

|S0| ≤

≤

≤

n=−[ 3N
8 ]

2

2

[ 3N

− N−1

n=−[ 3N

8 ](cid:88)

[ 3N

n=−[ 3N

(cid:12)(cid:12)(cid:12) ˆf (n)W1(n)
(cid:12)(cid:12)(cid:12) +
8 ]−1(cid:88)
(cid:12)(cid:12)(cid:12) ˆf (n)
(cid:12)(cid:12)(cid:12) + CN 2
8 ](cid:88)
(cid:12)(cid:12)(cid:12) + CN 2
(cid:12)(cid:12)(cid:12) ˆf (n)
8 ](cid:88)
|u|W 3,1(I).
 N−1
2(cid:88)

1 + |n|3 +

n=−[ 3N
8 ]

n=−[ 3N
8 ]

− N−1

1 + ( 3N

8 )3

[ 3N

1

2

1

2

1
( 3
8)3

n=− N−1

2

2

2 − √

2

2 − √

≤ C

Then, in a similar way,

(37)

.

N

(cid:12)(cid:12)(cid:12)1 − λN(cid:12)(cid:12)(cid:12) =

|Sl| ≤ C

n=lN− N−1

 lN + N−1
2(cid:88)
 ∞(cid:88)

1

2

1 + |n|3 +

n=−∞

8)3

|u|W 3,1(I).
|u|W 3,1(I) ≤ C|u|W 3,1(I).

1

1 + |n|3 +

1
(|l| + 3

∞(cid:88)

l=−∞

1
(|l| + 3

8)3

Therefore,

|(cid:1)u(x)| ≤ C

Similar to the estimation for (cid:1)u(x) above, we split (cid:2) into blocks of size N ,

∞(cid:88)

l=−∞

(cid:2)λu(x) =

Sl, where Sl =

(l+1)N−1(cid:88)

n=lN

ˆf (n)W2(n), W2(n) =

e

2π
L inx
1 − λe2πi n

W2(n) is singular when λ is close to any n-th root of unity. Assuming

126

O(hδ(cid:48)
) and |λ − 1| = O(hδ/2) with 0 ≤ δ/2 ≤ 1. We can write λ = e±iθ and assume
θ ∈ (0, π) without loss of generality. First, we establish a relation between δ and δ(cid:48). Since
|λ| = |λN| = 1, we have δ, δ(cid:48) ≥ 0. Because 1 − λN = (e2πin/N )N − (eiθ)N = (e2πin/N −

(cid:12)(cid:12)(cid:12)1 − λN(cid:12)(cid:12)(cid:12) ≤ N

(cid:12)(cid:12)(cid:12)ωn − eiθ(cid:12)(cid:12)(cid:12) ,∀n. With the assumption
(cid:12)(cid:12)(cid:12)ωn − eiθ(cid:12)(cid:12)(cid:12) ≥ Chδ(cid:48)+1. Particularly, when n = 0, we have |1 − λ| ≥

eiθ)((cid:80)N−1
(cid:12)(cid:12)(cid:12)1 − λN(cid:12)(cid:12)(cid:12) ∼ Chδ(cid:48)

l=0 (e2πin/N )N−1−l(eiθ)l), thus

, we get

Chδ(cid:48)+1, hence δ/2 ≤ δ(cid:48) + 1.

In addition, |W2(n)| = |λ − ωn|−1 ≤ Ch−(δ(cid:48)+1). With the assumption that 0 ≤ δ/2 ≤ 1,
N . Let n1 = (cid:98)n0/2(cid:99), n2 = 2n0 − n1,
n0+1
|u|W 2,1(I). For other n, |w2(n)| ≤ |w2(n1)| ≤

N ≤ θ < 2π
n0

1

1+n2
1

|S0| ≤ Ch−δ/2

(cid:12)(cid:12)(cid:12) ˆf (n)
(cid:12)(cid:12)(cid:12) ≤ C 1
there ∃ n0 ∼ O(hδ/2−1) s.t. 2π
then for n1 ≤ n ≤ n2,
2|sin(πn1/N−θ/2)| ≤ Ch−δ/2. Thus,
n1−1(cid:88)
(cid:12)(cid:12)(cid:12) ˆf (n)
(cid:12)(cid:12)(cid:12)
N−1(cid:88)
h−δ/2
N−1(cid:88)
h−δ/2
N−1(cid:88)
h−δ/2
N−1(cid:88)

 + Ch−(δ(cid:48)+1)
1 + |n|2 + h−(δ(cid:48)+1)(n2 − n1 + 1)
1 + |n|2 + h−(δ(cid:48)+1)hδ/2−1h2−δ
|u|W 2,1(I).
1 + |n|2 + h−δ(cid:48)−δ/2

n=n2+1

≤ C

≤ C

≤ C

n=0

1

1

1

+

n=0

n=0

n=0

n=n1

n2(cid:88)

(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) ˆf (n)
|u|W 2,1(I)
|u|W 2,1(I)

1 + n2
1

1

Using similar approaches, for l (cid:54)= 0,

h−δ/2

(l+1)N−1(cid:88)

n=lN

|Sl| ≤ C

1

1 + |n|2 + h−δ(cid:48)+δ/2

1

|n1/N + l|2

|u|W 2,1(I).

127

Summing up, we reach the estimation

h−δ/2

∞(cid:88)

n=−∞

1 + |n|2 + h−δ(cid:48)−δ/2 + h−δ(cid:48)+δ/2 (cid:88)

1

l∈N,l(cid:54)=0

|u|W 2,1(I)

1
|l|2

(38)

|(cid:2)λu(x)| ≤ C

≤ Ch−δ(cid:48)−δ/2|u|W 2,1(I).

Corollary .0.7. When λi, i ≤ n is a complex number with |λi| = 1, independent of h, above

estimates yields following results:

(cid:12)(cid:12)(cid:12)(cid:2)ν1

λ1

(cid:12)(cid:12)(cid:12) ≤ C|u|

··· (cid:2)νn
λn

u(x)

1+(cid:80)n

i=1 νi,1

,

(I)

|((cid:1)u(x))ν| ≤ C|u|W 1+2ν,1(I).

(39)

W

Estimates for Mj,m

Let’s recall the deﬁnition

Mj,m = (Aj + Bj)−1(GL−

j,m + HL+

j,m),

∀m ∈ Z+,∀j ∈ ZN ,

where G, H, Aj, Bj, L−

j,m, L+

j,m, Γj, Λj are deﬁned in Table 2.1.

Aj + Bj = G[L−

j,k−1, L−

j,k] + H[L+

j,k−1, L+

j,k] =

1

0

1
2

 M+ +

 α1 −β2

−β1 −α1



1

0

0

1
hj

 M−,

0

1
hj

128

 [L−

j,k−1 ± L+

j,k−1, L−

j,k ± L+
j,k]

0

where

M± =

=

1


0 hj

1 ± (−1)k−1

1 ± (−1)k

k(k − 1)(1 ± (−1)k) k(k + 1)(1 ± (−1)k+1)

 .

Therefore,

(Aj + Bj)−1 =

M−1−

1
D1



−α1

β1hj − k(k−(−1)k)

2

 ,

β2 −

hj

2k(k+(−1)k)

α1hj

where D1 =

(−1)khj

2k(k+(−1)k)

((−1)kΓj + Λj).

In what follows, we estimate Mj,m when scale-invariant ﬂux parameters are used in (40),

and when α2

1 + β1β2 = 1

4 in (41).

• Scale-invariant ﬂux parameters.

129

D1 is bounded by deﬁnitions of Γj, Λj and mesh regularity condition. Then

˜β2hh−1

j −

1

2k(k+(−1)k)

α1



 1

2 + α1
− ˜β1h−1hj

 ,

− ˜β2hh−1
2 − α1

1

j

(Aj + Bj)−1G

M−1−

1
D1

˜β1h−1hj − k(k−(−1)k)

2

(Aj + Bj)−1H

1

0

1

0

 =

0

1
hj
−α1

 =

0

1
hj
−α1





˜β1h−1hj − k(k−(−1)k)

2

˜β2hh−1

j −

1

2k(k+(−1)k)

α1

 1

2 − α1
˜β1h−1hj



˜β2hh−1

j

1
2 + α1

M−1−

1
D1

and




Mj,m =(Aj + Bj)−1G

1

0



0

1
hj

+ (−1)m(Aj + Bj)−1H


1

0

m(m + 1)

1

0

1
hj





 .

1

−m(m + 1)

If the mesh is uniform, the three formulas above are independent of mesh size h.

For nonuniform mesh, by mesh regularity condition, ∃σ1, σ2, s.t., σ1 ≤ h−1hj ≤ σ2,

therefore,

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(Aj + Bj)−1G

1

0

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)∞


0

1
hj

≤ C,

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(Aj + Bj)−1H

1

0



(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)∞

0

1
hj

≤ C, (cid:13)(cid:13)Mj,m

(cid:13)(cid:13)∞ ≤ C.

(40)

130

• α2

1 + β1β2 = 1
4.

Λj = 0. D1 =

hj Γj

2k(k+(−1)k)

and above formulas simpliﬁes to

0

1
hj
−α1 − 1
2 +
β1hj − k(k−(−1)k)

β1hj

k(k+(−1)k)
− α1k(k − (−1)k) α1 − 1

1

β2h−1
j −
2 + β2h−1

2k(k+(−1)k)
j k(k − (−1)k)

 ,

1

1

0

 =

0

2

 =

0

1
hj
α1 − 1

(Aj + Bj)−1G

M−1−

1

2D1

(Aj + Bj)−1H





M−1−

1

2D1

2 −
β1hj − k(k−(−1)k)

2

β1hj

k(k+(−1)k)
+ α1k(k − (−1)k) α1 + 1

1

β2h−1
j −
2 − β2h−1

2k(k+(−1)k)
j k(k − (−1)k)

Thus, we have the following estimation

1 +

(cid:18)
|β1|,

2 +α1|
| 1

h

|β2|
h2

,

| 1
2−α1|
,
|Γj|

h

(cid:19)

 .

max

(cid:107)Mj,m(cid:107)∞ ≤

 .

(41)

Proof of Lemma 2.2.2

=(cid:80)k−2

By Deﬁnition 2.2.1, P (cid:63)

h u|Ij

m=0 uj,mLj,m + ´uj,k−1Lj,k−1 + ´uj,kLj,k. We solve the two

coeﬃcients ´uj,k−1, ´uj,k on every cell Ij according to deﬁnition (2.16).

If assumption A0 is satisﬁed, it has been shown in Lemma 2.2.1 that (2.16) is equivalent

131

to (2.20). Substitute u and ux by (2.23), we obtain the following equation

´uj,k−1

 = (Aj + Bj)

uj,k−1

 +

´uj,k

uj,k

(Aj + Bj)

∞(cid:88)

m=k+1

uj,m(GL−

j,m + HL+

j,m),

(42)

the existence and uniqueness of the system above is ensured by assumption A0, that is,

det(Aj + Bj) = 2(−1)kΓj (cid:54)= 0. And (2.26) is proven by multiplying (Aj + Bj)−1 on both

sides of above equality.

If any of the assumption A1/A2/A3 is satisﬁed, (2.16) can be written as

k(cid:88)

G

k(cid:88)

´uj,mL−

m + H

´uj+1,mL+

m = G

uj,mL−

m + H

uj+1,mL+
m,

m=0

m=0

m=0

m=0

∞(cid:88)

∞(cid:88)

where we used (2.23) and G + H = I2 in above equality. Since ´uj,m = uj,m when m ≤ k − 2,

´uj,k−1

 + B

´uj+1,k−1

 =

A

´uj,k

´uj+1,k

∞(cid:88)

m=k−1

uj,mGL−

m + uj+1,mHL+
m.

In order to solve for ´uj,k−1, ´uj,k, we group above coupled equations for all j in a 2N × 2N

´uN−1,k−1

=

,

(43)

linear system as follows,

M



η1

θ1
···

ηN−1

θN−1

ηN

θN





´u1,k−1

´u1,k
···

´uN−1,k

´uN,k−1

´uN,k



132

where

 = G

ηj

θj

∞(cid:88)

m=k−1

uj,mL−

m + H

∞(cid:88)

m=k−1

uj+1,mL+
m,

and M = circ(A, B, 02,··· , 02), denoting a 2N × 2N block-circulant matrix with ﬁrst two
rows as (A, B, 02,··· , 02), with 02 as a 2 × 2 zero matrix. We can calculate that

det A = det B =

−2k
h

(α2

1 + β1β2 − 1
4

) (cid:54)= 0.

(44)

It is clear that the existence and uniqueness of P (cid:63)

h is equivalent to det M (cid:54)= 0. By a direct
computation, det M = det AN det(I2− QN ), where I2 denotes the 2× 2 identity matrix, and

Q = −A−1B =

(−1)k+1

Λ

c1 + c2

b1 − b2

 ,

b1 + b2
c1 − c2

with

c1 = β1 +

k2(k2 − 1)

h2

β2 − 2k2

h

(α2

1 + β1β2 +

1
4

) := Γ,

k
h

(2α1),

c2 =
b1 = −β1 − k2(k2 + 1)
b2 = −2k3

(α2

h2 β2 +

h2
2k
h

β2 +

2k2
h

(α2

1 + β1β2 +

1
4

),

1 + β1β2 +

1
4

).

The eigenvalues of Q are

(−1)(k+1)

Λ

(Γ +

λ1 =

(cid:112)

Γ2 − Λ2),

(Γ −(cid:112)

Γ2 − Λ2).

(−1)(k+1)

Λ

λ2 =

133

(45)

(46)

(47)

(48)

(49)

Since det Q = det B/ det A = 1, we have the relations λ1λ2 = 1 and

1 − b2
b2

2 = Γ2 − Λ2 − c2
2.

(50)

Below we discuss the existence and uniqueness of P (cid:63)

h based on the types of eigenvalues

of Q.

A1 If |Γ| > |Λ|, then λ1,2 are real and diﬀerent. Therefore, we can perform eigenvalue

decomposition of Q,

Q = T DT−1,

where

and



T =

− b1+b2

√
c2+

Γ2−Λ2

1
b1−b2
√

c2+

1

Γ2−Λ2
√
Γ2−Λ2
√
Γ2−Λ2

where det T = 2
c2+

where

T =

λ1

 ,

0

0 λ2

D =

 , T−1 =

1

det T



1
√
Γ2−Λ2

− b1−b2
c2+

 ,

√
b1+b2
c2+

Γ2−Λ2

1

(51)

, except for the case when (b1 − b2)(b1 + b2) = 0 and c2 < 0,

 1

b1−b2
2c2

 , T−1 =



− b1+b2
2c2

1

1

− b1−b2
2c2

b1+b2
2c2

1

 .

(52)

134

In both situations, we have

det M = det AN det(I2 −

λN

1

0

0

λN
2

) = det AN det(

1 − λN

1

0

).

0

1 − λN

2

det M (cid:54)= 0 if and only if (λ1)N (cid:54)= 1 and (λ2)N (cid:54)= 1, which is true since |λ1| (cid:54)= 1 and |λ2| (cid:54)= 1.

A2 If |Γ| = |Λ|, then λ1 = λ2 = (−1)k+1 Γ

Λ and we have two repeated eigenvalues.

Perform Jordan decomposition:c1 + c2

b1 − b2

 = T

c1

0

T −1,

1

c1

b1 + b2
c1 − c2

and

T =

 c2
 ,
2b1 0
 ,

1
b1 − b2 0

T =

if b1 (cid:54)= b2,

if b1 = b2.

(53)

0

1

We deﬁne

c1

0

 =

λ1

0

1

c1

(−1)k+1

Λ

λ1

 , Q = T J T −1,

J =

(−1)k+1

Λ

135

then

Qj = T J jT −1, J j =

 ,

1 κj
0 λj
1

λj
T −1,

2 −κN

0

2

I2 − QN = T

jΓj−1.

where κj = (−1)(k+1)j

Λj

In both situations, det M (cid:54)= 0 if and only if (λ1)N (cid:54)= 1, meaning that we require N to
be odd and further, if k is odd, we require Γ = −Λ; if k is even, we require Γ = Λ. In both
cases, λ1 = λ2 = −1.

A3 If |Γ| < |Λ|, then λ1,2 are complex, |λ1,2| = 1, λ1 = λ2, still Q is diagonalizable, and

similar to A1, det M (cid:54)= 0 turns to (λ1)N (cid:54)= 1 and (λ2)N = (λ1)N (cid:54)= 1, i.e. we require

 Γ

Λ

(cid:115)(cid:18) Γ

Λ

N
(cid:19)2 − 1

(cid:54)= 1.

(−1)(k+1)N

+

Summarize above results, we proved the existence and uniqueness for P (cid:63)

h when any of

the assumptions A0/A1/A2/A3 is satisﬁed.

In order to obtain the exact formula of ´uj,k−1 and ´uj,k, we analyze the inverse of the

matrix M. It is known that the inverse of a nonsingular circulant matrix is also circulant, so

is a block-circulant matrix. In particular,

M−1 = circ(r0, r1,··· , rN−1) ⊗ A−1

136

where ⊗ means Kronecker product for block matrices and rl is a 2 × 2 matrix deﬁned as,

rj = Qj(I2 − QN )−1,

j = 0,··· , N − 1.

(54)

Therefore, if any of the assumptions A1/A2/A3 is satisﬁed,

uj+l,mGL−

m + uj+l+1,mHL+
m

(cid:17)

,

∞(cid:88)

m=k+1

uj+l,m[L−

k−1, L−

k ]−1L−

m

A

=

l=0

´uj,k

uj+l,k

 +
∞(cid:88)

uj+l+1,k

´uj,k−1

uj+l,k−1
 + B
uj+l+1,k−1
rlA−1(cid:16)
N−1(cid:88)
 +
 − Q
(cid:16)uj+l,k−1
uj+l+1,k−1
N−1(cid:88)
(cid:17)
 +
uj,k−1
k ]−1L+
− uj+l+1,mQ[L+
(cid:16) N−1(cid:88)
∞(cid:88)
uj,k−1
 +

− uj+N,mrN [L−
k−1, L−
∞(cid:88)

k−1, L+

uj+l,mV2,m + uj,mr0[L−
(cid:17)
k ]−1L−
(cid:0)uj,mV1,m +

N−1(cid:88)

uj+l,mrlV2,m

uj+l+1,k

uj+l,k

m=k+1

m=k+1

l=1

=

rl

l=0

uj,k

=

=

m

m

uj,k

m=k+1

l=0

k−1, L−

k ]−1L−

m

(cid:1),

where rN = QN (I2 − QN )−1 = r0 − I2 is used in the third equality. And (2.27) is proven.

137

Proof of Lemma 2.2.3

Denote

´uj,k−1 − uj,k−1

´uj,k − uj,k

 ,

Uj =

When assumption A0 is satisﬁed (2.28) is a direct result of (2.26) and (41).

If any of the assumptions A1/A2/A3 is satisﬁed, we have

´uj,k−1 − uj,k−1

´uj,k − uj,k

 =

Uj =

∞(cid:88)

(uj,mV1,m +

m=k+1

N−1(cid:88)

l=0

uj+l,mrlV2,m).

In order to estimate Uj, we ﬁrst compute rl to get its detailed dependence on l. If A1/A3,

Q is diagonalizable, then

rl = T Dl(I2 − DN )−1T−1 =

 T−1 +

1 0

0 0

T

 T−1

0 0

0 1

T

λl
2

1 − λN

2

λl
1

1 − λN

1

λl
1

1 − λN

1

Q1 +

λl
2

1 − λN

2

(I2 − Q1),

=

where

Q1 =

√
2

1
Γ2 − Λ2

when T is given by (51), and

c2 +

√
Γ2 − Λ2

b1 − b2

 ,

b1 + b2
√
Γ2 − Λ2

−c2 +

138

(55)

(56)

 2c2

b1 − b2

 ,

b1 + b2

0

Q1 =

1
2c2

when T is given by (52).

If assumption A2 is satisﬁed,

where

rl = Ql(I2 − QN )−1 =

(−1)l
2

I2 + (−1)l−N + 2l

4Γ

Q2,

T −1 =

0 1

0 0

 c2

b1 + b2
b1 − b2 −c2

 .

Q2 = T

(57)

(58)

(59)

When assumption A1 is satisﬁed, eigenvalues λ1, λ2 are real.

N−1(cid:88)

l=0

1,2

| λl
1 − λN

1,2

| =

1

1 − |λ1,2|

1 − |λ1,2|N
1,2| .
|1 − λN

Without loss of generality, we assume |λ1| < 1 < |λ2|, then

N−1(cid:88)
N−1(cid:88)

l=0

1

| λl
1 − λN
| λl
1 − λN

2

1

2

| ≤

1

1 − |λ1| =

|λ2|
|λ2| − 1

,

| ≤

1

|λ2| − 1

.

And thus

l=0

N−1(cid:88)

l=0

(cid:107)rl(cid:107)∞ ≤ (cid:107)Q1(cid:107)∞
1 − |λ1| +

(cid:107)I2 − Q1(cid:107)∞
1 − |λ2|

.

(60)

139

Then we have

(cid:107)Uj(cid:107)∞ ≤ C(1 +

N−1(cid:88)

l=0

(cid:107)rl(cid:107)∞)|u|

≤ Chk+1|u|

W k+1,∞(I)

W k+1,∞(I)
(cid:107)Q1(cid:107)∞
1 − |λ1| +

(1 +

(cid:107)I2 − Q1(cid:107)∞
1 − |λ2|

),

where (2.25) and the fact that V1,m, V2,m,∀m ≥ 0 are constant matrices independent of h

are used in above inequalities.

When assumption A3 is satisﬁed, eigenvalues λ1, λ2 are complex, with |λ1,2| = 1 and

above estimation does not apply. We perform more detailed computation and use Fourier

analysis to bound Uj by utilizing the smoothness and periodicity of u. If u ∈ W k+2+n,∞(I),

N−1(cid:88)
N−1(cid:88)

l=0

rl

rl

∞(cid:88)
∞(cid:88)

(cid:0)ul+j,m − n(cid:88)
n(cid:88)

s=0

m=k+1

l=0

m=k+1

s=0

Uj =

+

= O(hk+2+n|u|

∞(cid:88)

n(cid:88)

) +

W k+2+n,∞(Ij )
N−1(cid:88)

θshk+1+s

λl
1

(

1 − λN

1

m=k+1

s=0

l=0

θshk+1+su(k+1+s)(x

θshk+1+su(k+1+s)(x

∞(cid:88)

j+l− 1
2

)V2,m +

uj,mV1,m

(cid:1)

j+l− 1
2

))V2,m

∞(cid:88)

m=k+1

uj,mV1,m

λl
2

1 − λN

2

(I2 − Q1))u(k+1+s)(x

j+l− 1
2

)V2,m

m=k+1

Q1 +

∞(cid:88)

m=k+1

+

+

= O(hk+2+n|u|

∞(cid:88)

n(cid:88)

m=k+1

s=0

≤ Chk+1(cid:107)u(cid:107)

W k+2+n,∞(Ij )

) +

uj,mV1,m

θshk+1+s(Q1 (cid:2)λ1

+(I2 − Q1)(cid:2)λ2

j− 1
2
(1 + h−δ(cid:48)−δ/2((cid:107)Q1(cid:107)∞ + (cid:107)I2 − Q1(cid:107)∞)),

)u(k+1+s)(x

W k+3,∞(I)

)V2,m

where (32) is used in the ﬁrst equality and (38) is used in the last inequality.

140

When assumption A2 is satisﬁed, by similar computation, if u ∈ W k+2+n,∞(I),

uj,mV1,m

(cid:1)hk+1+su(k+1+s)(x

Q2

j+l− 1
2

)V2,m

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:1) u(k+1+s)(x

)V2,m

j− 1
2

) +

u(k+1+s)(x

j+2l(cid:48)− 1

2

)

∞(cid:88)

Uj = O(hk+2+n|u|
n(cid:88)

∞(cid:88)

+

1
2

θs

m=k+1

s=0

≤ Chk+1|u|

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞(cid:88)

+

m=k+1

s=0

− u(k+1+s)(x

+

2Γ

l=0

) +

m=k+1

W k+2+n,∞(Ij )
N−1(cid:88)

(cid:0)(−1)l + (−1)l−N + 2l
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞(cid:88)
n(cid:88)
θshk+1+s(cid:0)u(k+1+s)(x
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
)(cid:1)V2,m

j+N− 3
2

m=k+1

s=0

W k+1,∞(Ij )
n(cid:88)

1
2

j+2l(cid:48)+ 1

2

θshk+1+s Q2
2Γ
N−1

2(cid:88)

l(cid:48)=0

≤ Chk+1(cid:107)u(cid:107)

W k+4,∞(I)

(1 +

(cid:107)Q2(cid:107)∞
|Γ|

),

where (34) and (39) are used in the last inequality.

The estimates of Uj for assumptions A1, A2 and A3 are ﬁnished, and (2.29), (2.30) and

(2.31) are direct results of the estimation of (cid:107)Uj(cid:107)∞.

Proof of Lemma 3.2.3

Proof. Since P (cid:63)

h u = P

is,

†
hu when A0, the formula for `uj,k−1, `uj,k is the same as (2.26). That

`uj,k−1

 =

uj,k−1

 +

`uj,k

uj,k

∞(cid:88)

m=k+1

uj,mMm.

(61)

Under assumption A1/A2/A3, above formula is well-deﬁned if and only if Mm is not singular.
By the analysis of Mj,m in Appendix, the existence and uniqueness condition is det(A+B) =
2((−1)kΓ + Λ) (cid:54)= 0. Thus, by (40) and (2.25), (3.9) is proven.

141

If any of the assumptions A1/A2/A3 is satisﬁed, then the diﬀerence of two projections

can be written as

W u|Ij

= P (cid:63)

h u|Ij

− P

†
hu|Ij

= (´uj,k−1 − `uj,k−1)Lj,k−1 + (´uj,k − `uj,k)Lj,k.

The properties of P (cid:63)

†
hu yield the following coupled system

A

h u and P

´uj,k − `uj,k

´uj,k−1 − `uj,k−1
 + B
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x
 u

 =
τj
 (u − P

j+ 1
2

(u − P

= G

ux

ιj

´uj+1,k − `uj+1,k

´uj+1,k−1 − `uj+1,k−1
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

 P

†
hu
†
hu)x

(P

−

x

− G

− H

j+ 1
2
†
hu)|−
j+ 3
2
†
hu)x|−

x

x

− (u − P

− (u − P

x

†
hu)|−
†
hu)x|−

x

j+ 1
2

j+ 1
2

 ,

†
hu
†
hu)x

ιj

 =
τj
 P
 ,

(P

∀j ∈ ZN ,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)


+

x

j+ 1
2

j+ 3
2
†
hu (3.8b).

where the second equality was obtained by the deﬁnition of P

Gather the relations above for all j results in a large 2N × 2N linear system with block

circulant matrix M, deﬁned in (43), as coeﬃcient matrix, then the solution is

´uj,k−1 − `uj,k−1

´uj,k − `uj,k

 =

N−1(cid:88)

l=0

rlA−1

τl+j

 ,

ιl+j

j ∈ ZN ,

where by periodicity, when l + j > N , τl+j = τl+j−N , ιl+j = ιl+j−N .

On uniform mesh, by the deﬁnition of Rj,m in (3.12), Rj,m(1) and (Rj,m)x(1) are

independent of j, we denote the corresponding values as Rm(1) and (Rm)x(1) and let

142

R−
m = [Rm(1), (Rm)x(1)]T . By (3.11), we have

x

†
hu)|−
†
hu)x|−

x

j+ 1
2

j+ 1
2

− (u − P

− (u − P

x

†
hu)|−
j+ 3
2
†
hu)x|−

x

j+ 3
2

 =

∞(cid:88)

(uj,m − uj+1,m)R−

m

m=k+1

(u − P

 (u − P
and´uj,k−1 − `uj,k−1
 =

´uj,k − `uj,k

N−1(cid:88)

(cid:0) ∞(cid:88)

rl

(ul+j,m − ul+j+1,m)A−1GR−

m

(cid:1) .
= Sj,

j ∈ ZN . (62)

l=0

m=k+1

Using (33), we can estimate Sj by the same lines as the estimation of Uj in the proof of

Lemma 2.2.3 in Appendix, and (3.10) is proven.

Proof of Lemma 3.3.1

Proof. By error equation, the symmetry of A(·,·) and the deﬁnition of sh, we have

0 = a(e, vh) = a(h, vh) + a(ζh, vh) =

shvhdx +

(cid:90)

(cid:90)

I

(ζh)tvhdx − iA(vh, ζh),

∀vh ∈ V k
h .

(63)

I

Now, we are going to choose three special test functions to extract superconvergence prop-
erties (3.16)-(3.18) about ζh. We ﬁrst prove (3.16). In order to have A(vh, ζh) = (cid:107)(ζh)xx(cid:107)2,
(cid:82)
we choose a function v1 ∈ V k
= αj,k−1Lj,k−1 + αj,kLj,k + (ζh)xx,
v1(ζh)xxdx = (cid:107)(ζh)xx(cid:107)2

h , such that ∀j ∈ ZN , v1|Ij
= 0 and (cid:93)(v1)x|

, ˆv1|

= 0.

L2(Ij )

j+ 1
2

j+ 1
2

Ij

When the assumption A0 holds, the deﬁnition of v1 yields the following local system for

143

each pair of αj,k−1 and αj,k,

(Aj + Bj)

2

xx

xxx

αj,k

(ζh)−

αj,k−1
(cid:12)(cid:12)(cid:12)(cid:12)j+ 1
 = −G
 (ζh)−
1
αj,k−1
 = − (Aj + Bj)−1G
1

− (Aj + Bj)−1H

αj,k

0

0

,

2

xx

xxx

(ζh)+

 (ζh)+
 (ζh)−
 (ζh)+

hj(ζh)−

(cid:12)(cid:12)(cid:12)(cid:12)j− 1
(cid:12)(cid:12)(cid:12)(cid:12)j+ 1
(cid:12)(cid:12)(cid:12)(cid:12)j− 1

xxx

xx

xx

hj(ζh)+

xxx

2

2

− H




0

1
hj

0

1
hj

∀j ∈ ZN .

(64)

,

thus v1 is nontrivial and uniquely deﬁned under assumption A0. By orthogonality of Leg-

endre polynomials, it follows that

(cid:90)

Ij

(cid:90)

(cid:107)v1(cid:107)2

L2(Ij )

= |αj,k−1|2

j,k−1dx + |αj,k|2
L2

j,kdx + (cid:107)(ζh)xx(cid:107)2
L2

L2(Ij )

Ij
j(cid:107)(ζh)xxx(cid:107)2

+ h3

+ (cid:107)(ζh)xx(cid:107)2

L2(Ij )

)

L2(∂Ij )

≤ C(hj(cid:107)(ζh)xx(cid:107)2
≤ C(cid:107)(ζh)xx(cid:107)2

,

L2(Ij )

L2(∂Ij )

where (40), trace inequalities and inverse inequalities are used in above inequality.

Let vh = v1, then (63) becomes

(cid:90)

I

(cid:90)

I

(ζh)tv1dx − i(cid:107)(ζh)xx(cid:107)2.

0 =

shv1dx +

Hence (cid:107)(ζh)xx(cid:107)2 ≤ (cid:107)sh + (ζh)t(cid:107)·(cid:107)v1(cid:107) ≤ C(cid:107)sh + (ζh)t(cid:107)·(cid:107)(ζh)xx(cid:107). Therefore, (3.16) is proven

when assumption A0 is satisﬁed.

Similarly, in order to have A(vh, ζh) = −(cid:80)N

j=1 |[ζh]|2

, we deﬁne v2 ∈ V k

h , such that

j+ 1
2

144

= αj,k−1Lj,k−1 + αj,kLj,k,(cid:82)

∀j ∈ ZN , v2|Ij
[ζh]|
for each pair of αj,k−1 and αj,k,

j+ 1
2

. When assumption A0 is satisﬁed, this deﬁnition yields the following local system

v2(ζh)xxdx = 0, ˆv2|

j+ 1
2

Ij

= 0 and (cid:93)(v2)x|

j+ 1
2

=

(Aj + Bj)

αj,k

αj,k−1
1

 = G


0

0

1
hj

 0

[ζh]

 0

hj[ζh]

2

(cid:12)(cid:12)(cid:12)(cid:12)j+ 1
(cid:12)(cid:12)(cid:12)(cid:12)j+ 1

2

By same algebra as above, we have

αj,k−1

 = (Aj + Bj)−1G

αj,k

 0

[ζh]

(cid:12)(cid:12)(cid:12)(cid:12)j− 1

2

∀j ∈ ZN .

,

+ H

+ (Aj + Bj)−1H

1

0



0

1
hj

 0

hj[ζh]

(cid:12)(cid:12)(cid:12)(cid:12)j− 1

2

.

By (40), it follows directly that

(cid:107)v2(cid:107)2

L2(Ij )

≤ Ch3

j (|[ζh]|2

j+ 1
2

+ |[ζh]|2

j− 1
2

).

Plug v2 in (63), we obtain

N(cid:88)

j=1

|[ζh]|2

j+ 1
2

= i

(cid:90)

I

(cid:90)

I

shv2dx + i

(ζh)tv2dx ≤ (cid:107)sh + (ζh)t(cid:107)(cid:107)v2(cid:107).

Therefore, (3.17) is proven when assumption A0 is satisﬁed.

Finally, in order to have A(vh, ζh) =(cid:80)N
∀j ∈ ZN , v3|Ij
and (cid:93)(v3)x|

= αj,k−1Lj,k−1+αj,kLj,k such that(cid:82)

j+ 1
2

j=1 |[(ζh)x]|2

, we choose v3 ∈ V k

j+ 1
2
v3(ζh)xxdx = 0, ˆv3|

h , such that
= [(ζh)x]|

j+ 1
2
= 0. Follow the same lines as the estimates for v2, we end up with the

j+ 1
2

Ij

estimates

(cid:107)v3(cid:107)2

L2(Ij )

≤ Chj(|[(ζh)x]|2

j+ 1
2

+ |[(ζh)x]|2

j− 1
2

).

145

Plug v3 in (63), we obtain (3.18) when assumption A0 is satisﬁed.

The deﬁnition of v1 yields the following coupled system

Under assumption A1, we need to compute(cid:80)N
(cid:12)(cid:12)(cid:12)(cid:12)j+ 1
αj,k−1
 (ζh)−

αj+1,k−1

 = −G

 + B

xx

A

(ζh)−

xxx

αj,k

αj+1,k

j=1(|αj,k−1|2 + |αj,k|2) to estimate (cid:107)v1(cid:107)2.

 (ζh)+

xx

(ζh)+

xxx

(cid:12)(cid:12)(cid:12)(cid:12)j+ 1

2

j ∈ ZN . (65)

,

− H

2

Write it in matrix form

M α = b, α = [α1,··· , αN ]T ,

where M is deﬁned in (43) and

αj = [αj,k−1, αj,k], b = [b1,··· , bN ]T , bj = −G

 (ζh)−

(ζh)−

xx

xxx

(cid:12)(cid:12)(cid:12)(cid:12)j+ 1

2

− H

 (ζh)+

xx

(ζh)+

xxx

(cid:12)(cid:12)(cid:12)(cid:12)j+ 1

2

.

Left multiply A−1 in (65), we get an equivalent system

M(cid:48)α = b(cid:48), M(cid:48) = circ(I2, A−1B, 02,··· , 02), b(cid:48) = [b(cid:48)

1,··· , b(cid:48)

N ]T , b(cid:48)

j = A−1bj,

and (M(cid:48))−1 = circ(r0,··· , rN−1). By Theorem 5.6.4 in [29] and similar to the proof in

Lemma 3.1 in [10],

M(cid:48) = (F∗

N ⊗ I2)Ω(FN ⊗ I2),

where FN is the discrete Fourier transform matrix deﬁned by (FN )ij = 1√
ω = ei 2π

N . FN is symmetric and unitary and

N

ω(i−1)(j−1),

Ω = diag(I2 + A−1B, I2 + ωA−1B,··· , I2 + ωN−1A−1B).

146

|Γ|
|Λ| > 1 in A1 ensures that the eigenvalues of Q = −A−1B are not 1, thus

The assumption
I2 + ωjA−1B,∀j, is nonsingular and Ω is invertible. Then

|ρ((M(cid:48))−1)| = (cid:107)(M(cid:48))−1(cid:107)2 ≤ (cid:107)F∗

N ⊗ I2(cid:107)2(cid:107)Ω(cid:107)2(cid:107)FN ⊗ I2(cid:107)2 ≤ C.

(66)

Therefore,

N(cid:88)

j=1

1 0

0 1
h

Since A−1G

(cid:107)b(cid:48)

j(cid:107)2
2 ≤ C

(|αj,k−1|2 + |αj,k|2) = αT α = (b(cid:48))T (M(cid:48))−T (M(cid:48))−1(b(cid:48))T

j=1

(cid:107)b(cid:48)

j(cid:107)2
2.

2(cid:107)b(cid:48)(cid:107)2

2 ≤ C

≤ (cid:107)(M(cid:48))−1(cid:107)2

N(cid:88)
 are constant matrices, we have
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2
(cid:12)(cid:12)(cid:12)(cid:12)j+ 1
(cid:12)(cid:12)(cid:12)(cid:12)j+ 1

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
 (ζh)+

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2

h(ζh)+

xxx

xx

+

2

2

xx

xxx



L2(∂Ij+1)

+ (cid:107)(ζh)xx(cid:107)2
+ h2(cid:107)(ζh)xxx(cid:107)2
+ (cid:107)(ζh)xx(cid:107)2

L2(∂Ij )

L2(∂Ij )

L2(∂Ij )

L2(∂Ij+1)

)

L2(∂Ij+1)

),

0 1
h

1 0
 , A−1H

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
 (ζh)−

h(ζh)−
≤ C((cid:107)(ζh)xx(cid:107)2
+ h2(cid:107)(ζh)xxx(cid:107)2
≤ C((cid:107)(ζh)xx(cid:107)2

where inverse inequality is used to obtain the last inequality. Finally, we obtain the estimate

N(cid:88)

j=1

(cid:107)v1(cid:107)2 =

N(cid:88)

j=1

|αj,k−1|2(cid:107)Lj,k−1(cid:107)2

L2(Ij )

+

147

|αj,k|2(cid:107)Lj,k(cid:107)2

L2(Ij )

+ (cid:107)(ζh)xx(cid:107)2

≤ (cid:107)(ζh)xx(cid:107)2 + Ch

≤ (cid:107)(ζh)xx(cid:107)2 + Ch

N(cid:88)
N(cid:88)

j=1

j=1

(|αj,k−1|2 + |αj,k|2)

((cid:107)(ζh)xx(cid:107)2

+ (cid:107)(ζh)xx(cid:107)2

L2(∂Ij+1)

)

L2(∂Ij )

≤ (cid:107)(ζh)xx(cid:107)2 + Ch(cid:107)(ζh)xx(cid:107)2

L2(∂IN )

≤ C(cid:107)(ζh)xx(cid:107)2,

where inverse inequality is used to obtain the last inequality. Then the estimates for (3.16)

hold true. (3.17) and (3.18) can be proven by the same procedure when assumption A1 is

satisﬁed, and the steps are omitted for brevity.

Remark .0.1. When assumption A2 or A3 is satisﬁed, the eigenvalues of Q are −1 or two
complex number with magnitude 1, then a constant bound for ρ((M(cid:48))−1) as in (66) is not

possible. Therefore, we cannot obtain similar results for assumption A2 and A3.

Proof for Lemma 3.3.2

Proof. Since wq ∈ V k

h , we have

k(cid:88)

m=0

wq|Ij

=

cq
j,mLj,m.

Let vh = D−2Lj,m, m ≤ k − 2 in (3.19a), we obtain

j,m = −i
cq

2m + 1

hj

h2
j
4

∂twq−1D−2Lj,mdx.

(cid:90)

Ij

(67)

(68)

Since D−2Lj,m ∈ P m+2

c

(Ij), by the property u − P (cid:63)

h u ⊥ V k−2

h

in the L2 inner product

148

sense, we have

c1
j,m =



(cid:82)

(cid:90)

Ij
h2
j
4

−i 2m+1
hj

h2
j
4

−i

2m + 1

hj

∂t(u − P (cid:63)

h u)D−2Lj,mdx = 0, m ≤ k − 4,

∂t((uj,k−1 − ´uj,k−1)Lj,k−1

Ij
+(uj,k − ´uj,k)Lj,k)D−2Lj,mdx,

m = k − 3, k − 2.

(69)

By induction using (67), (68), (69), for 0 ≤ m ≤ k − 2 − 2q, cq

j,m = 0.

Furthermore, the ﬁrst nonzero coeﬃcient can be written in a simpler form related to

uj,k−1 by induction.

When q = 1, we compute c1

j,k−3 by (69) and the deﬁnition of w0. That is

(cid:16)hj

(cid:17)2

j,k−3 = −i
c1

2(k − 3) + 1

hj

2

= Ch2

j ∂t(uj,k−1 − ´uj,k−1).

∂t(uj,k−1 − ´uj,k−1)

D−2Lj,k−3Lj,k−1dx

(cid:90)

Ij

Suppose cq−1

k+1−2q = Ch2q−2

j

∂q−1

t

j,k−1−2q = −i
cq

2(k − 1 − 2q) + 1

hj

2

(uj,k−1 − ´uj,k−1), then
(cid:90)
(cid:16) hj

(cid:17)2

∂tcq−1

j,k+1−2q

Ij

D−2Lj,k−1−2qLj,k+1−2qdx

= Chj

2q∂q

t (uj,k−1 − ´uj,k−1).

The induction is completed and the second formula in (3.23) is proven when r = 0.

Next, we begin estimating the coeﬃcient cq
j,m, k − 1 − 2q ≤ m ≤ k − 2,

the estimates for cq

j,m. By Holder’s inequality and (68), we have

(cid:12)(cid:12)(cid:12)cq

j,m

(cid:12)(cid:12)(cid:12) ≤ Ch2− 1

s(cid:107)∂twq−1(cid:107)Ls(Ij ).

149

To estimate the coeﬃcients cq

j,k−1, cq

j,k, we need to discuss them under diﬀerent assump-

tions.

If assumption A0 is satisﬁed, meaning (3.19b) and (3.19c) can be decoupled and

therefore wq is locally deﬁned by (3.19). By (3.20) and following the same algebra of solving

the k-th and (k + 1)-th coeﬃcients in (42),

cq

j,k−1
cq
j,k

 = − k−2(cid:88)

m=0

Mj,mcq

j,m.

By (40), for all j ∈ ZN ,

(cid:12)(cid:12)(cid:12)cq

(cid:12)(cid:12)(cid:12)2

(cid:12)(cid:12)(cid:12)cq
(cid:12)(cid:12)(cid:12)cq

+

(cid:12)(cid:12)(cid:12) ,

j,k

j,k

(cid:12)(cid:12)(cid:12)2 ≤ C
(cid:12)(cid:12)(cid:12)) ≤ C

k−2(cid:88)

m=k−2q−3

(cid:12)(cid:12)(cid:12)cq

j,m

max

k−2q−3≤m≤k−2

(cid:12)(cid:12)(cid:12)2 ≤ Ch3(cid:107)∂twq−1(cid:107)2
(cid:12)(cid:12)(cid:12) ≤ Ch2(cid:107)∂twq−1(cid:107)L∞(Ij ).
(cid:12)(cid:12)(cid:12)cq

L2(Ij )

j,m

.

j,k−1

(cid:12)(cid:12)(cid:12)cq

max(

j,k−1

If any of the assumptions A1/A2/A3 is satisﬁed, (3.20) deﬁnes a coupled system. From

the same lines for obtaining (2.27) in Appendix, the solution for cq

j,k−1, cq

j,k is

C. We have the estimate for cq

l=0

Under assumption A1 and scale invariant ﬂux assumption, (60) implies(cid:80)N−1
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
cq

j,m, m = k − 1, k, that is
N−1(cid:88)

j+l,m| ≤ Ch2(cid:107)∂twq−1(cid:107)L∞(IN ).

(cid:107)rl(cid:107)∞) max|cq

≤ C(1 +

j,k−1
cq
j,k



l=0

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)∞

l=0 (cid:107)rl(cid:107)∞ ≤

150

 = −

cq

j,k−1
cq
j,k

= −

N−1(cid:88)
(cid:16)

l=0

k−2(cid:88)
k−2(cid:88)

m=k−1−2q

m=k−1−2q

rlA−1(GL−
mcq
N−1(cid:88)

j+l,m + HL+

mcq

j+l+1,m)

(cid:17)

(70)

.

cq
j,mV1,m +

cq
j+l,mrlV2,m

Under assumption A2 or A3,(cid:80)N−1

l=0 (cid:107)rl(cid:107)∞ is unbounded. Thus we use Fourier analysis to

bound the coeﬃcients utilizing the smoothness and periodicity by similar steps of estimating
Uj in the proof of Lemma 2.2.3. In the rest of the proof, we make use of two operators (cid:2)
and (cid:1), which are deﬁned in (35b) and (35a).

When assumption A3 is satisﬁed, Q = −A−1B has two imaginary eigenvalues λ1, λ2 with
(I2− Q1), where Q1 is a constant matrix independent

|λ1| = |λ2| = 1. rl =

Q1 +

λl
1
1−λN
1

λl
2
1−λN
2

of h, and deﬁned in (56) and (57). We perform more detailed computation of the coeﬃcients.
In (69), plug in (2.27), for m = k − 3, k − 2, when ut ∈ W k+2+n,∞(I),

[Lj,k−1, Lj,k]∂t

uj,pV1,p +

uj+l,prlV2,p

(cid:16)
∞(cid:88)
N−1(cid:88)

p=k+1

l=0

N−1(cid:88)
(cid:17)

l=0

uj,pF 1

p,m +

uj+l,prlF 2

p,m

(cid:17)

D−2Lj,mdx

c1
j,m = i

2m + 1

h

= i

2m + 1

2

h2
4

h2
4

(cid:90)

Ij

(cid:16)
∞(cid:88)
(cid:16) n(cid:88)
∞(cid:88)

p=k+1

∂t

p=k+1

s=0

= i

2m + 1

N−1(cid:88)

8

(cid:0) λl

1

+

1 − λN

1

∞(cid:88)

Q1 +

n(cid:88)

l=0
2m + 1

8

p=k+1

s=0

= i

µshk+3+su

j− 1
2

(x

(k+1+s)
t

(I2 − Q1)(cid:1) n(cid:88)

λl
2

2

1 − λN

µshk+3+s(cid:0)u

(k+1+s)
t

(x

j− 1
2

)F 1

)F 1

p,m + O(hk+n+3|ut|

W k+2+n,∞(I)

)

(cid:17)

µshk+3+su

(k+1+s)
t

(x

)F 2

p,m

j+l− 1
2

s=0

p,m

(cid:1) + O(hk+3+n|ut|

W k+2+n,∞(I)

),

)u

(k+1+s)
t

(x

)F 2

p,m

j− 1
2

[Lj,k−1, Lj,k]Vν,pD−2Lj,mdx, ν = 1, 2, are constants independent of h

+ (Q1 (cid:2)λ1

where F ν

p,m = 2
h

+(I2 − Q1)(cid:2)λ2
(cid:82)

Ij

and (32) is used in the third equality.

Plug the formula above into (70), by similar computation, we have

 = −i

c1

j,k−1

c1
j,k

k−2(cid:88)

∞(cid:88)

n(cid:88)

m=k−3

p=k+1

s=0

µshk+1+s(cid:0)u

(k+1+s)
t

2m + 1

8

h2

)F 1

p,mV1,m

(x

j− 1
2

151

+(I2 − Q1)(cid:2)λ2
+(I2 − Q1)(cid:2)λ2

+ (Q1 (cid:2)λ1
+ (Q1 (cid:2)λ1
+ O(hk+2+n|ut|

W k+2+n,∞(I)

).

)u

(k+1+s)
t

(x

j− 1
2

)(F 2

p,mV1,m + F 1

p,mV2,m)

(k+1+s)
)2u
t

(x

)F 2

p,mV2,m

j− 1
2

(cid:1)

By (39), we have

(Q1 (cid:2)λ1

+(I2 − Q1)(cid:2)λ2

(k+1+s)
)νu
t

(x

j− 1
2

) ≤ C|ut|

W k+2+s+ν,1(I)

≤ C|u|

W k+4+s+ν,1(I)

.

Therefore,

|c1
j,m| ≤ C2hk+3, m = k − 3, k − 2, and |c1

j,m| ≤ C3hk+3, m = k − 1, k.

By induction and similar computation, we can obtain the formula for cq

j,m. For brevity,

we omit the computation and directly show the estimates

j,m| ≤ C3qhk+1+2q,
|cq

k − 1 − 2q ≤ m ≤ k.

(−1)l

2

When assumption A2 is satisﬁed, Q = −A−1B has two repeated eigenvalues. rl =
I2 + (−1)l −N +2l
4Γ Q2, where Q2/Γ is a constant matrix. For m = k − 3, k − 2, when
j,m by the same procedure as previous case and obtain

)F 2

p,m

(x

j− 1
2

152

ut ∈ W k+2+n,∞(I), we compute c1
n(cid:88)

∞(cid:88)

2m + 1

c1
j,m = i

h2

8

µshk+1+s(cid:0)u

+

1
2

((cid:2)−1 +

Q2
Γ

s=0

p=k+1
(cid:1))u

(k+1+s)
t

(k+1+s)
t

)F 1

(x

j− 1
2

p,m

(cid:1) + O(hk+3+n|ut|

W k+2+n,∞(I)

).

Plug formula above into (70), we have

 = −i

c1

j,k−1

c1
j,k

k−2(cid:88)

m=k−3

∞(cid:88)

n(cid:88)

µshk+1+s(cid:0)u

p=k+1

s=0

2m + 1

8

h2

(k+1+s)
t

(x

j− 1
2

)F 1

p,mV1,m

+

+

1
2
1
4

((cid:2)−1 +

((cid:2)−1 +

Q2
Γ
Q2
Γ

By (39), we have

(cid:1))u
(k+1+s)
t

(x

j− 1
2

)(F 2

p,mV1,m + F 1

(cid:1))2u

(k+1+s)
t

(x

)F 2

p,mV2,m

j− 1
2

p,mV2,m)

(cid:1) + O(hk+2+n|ut|

W k+2+n,∞(I)

).

((cid:2)−1 +

Q2
Γ

(cid:1))νu

(k+1+s)
t

(x

j− 1
2

) ≤ C|ut|

W k+2+s+2ν,1(I)

≤ C|u|

W k+4+s+2ν,1(I)

and

|c1
j,m| ≤ C2hk+3, m = k − 3, k − 2, and |c1

j,m| ≤ C4hk+3, m = k − 1, k.

By induction and similar computation, we can obtain the formula for cq

j,m. For brevity,

we omit the computation and directly show the estimates

|cq
j,m| ≤ C4qhk+1+2q,

k − 1 − 2q ≤ m ≤ k.

All the analysis above works when we change deﬁnition of wq to ∂r

t wq (and change

t wq−1 accordingly) in (3.19). Summarize the estimates for cq

j,m under all

(wq−1)t to ∂r+1
three assumptions, for 1 ≤ q ≤ (cid:98) k−1

|∂r
j,m| ≤ C2r,qhk+1+2q,
t cq

(cid:107)∂r

2 (cid:99), we have
N(cid:88)

t wq(cid:107) ≤ C(

j=1

153

k(cid:88)

m=k−2q−1

|∂r
j,m|2hj)
t cq

1

2 ≤ C2r,qhk+1+2q.

Then (3.23), (3.24) is proven. And (3.25) is a direct result of above estimate and (3.22).

154

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