DATA-INFUSED FRACTIONAL MODELING AND SPECTRAL NUMERICAL ANALYSIS

FOR ANOMALOUS TRANSPORT AND TURBULENCE

By

Mehdi Samiee

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Mechanical Engineering – Doctor of Philosophy

Computational Mathematics, Science and Engineering – Dual Major

2021

ABSTRACT

DATA-INFUSED FRACTIONAL MODELING AND SPECTRAL NUMERICAL ANALYSIS

FOR ANOMALOUS TRANSPORT AND TURBULENCE

By

Mehdi Samiee

Fractional calculus introduces tractable mathematical tools for accurate description of anomalous
phenomena, regarded as a manifestation of self-similar structures and memory effects. Fractional
approaches in modeling anomalous dynamics have been increasingly employed over the last decade
in a disparate range of applications, where the standard methods have been found experimentally
to be inadequate. The physical significance of employing fractional operators in such models
is indicated by their potentials in describing the inherent abnormal-exponential or heavy-tailed
processes due to their power-law or logarithmic kernels of convolution type. However, it is
also inevitably challenging to provide efficient and yet accurate numerical methods for fractional
differential equations. More concretely, history-dependent and nonlocal characteristics of fractional
operators impose further numerical complexities, ranging from theoretical analysis to large memory
requirement. Therefore, to establish a robust framework for developing fractional models, a great
deal of attention is required in theoretical and numerical studies of fractional calculus. In particular,
developing a fractional model depends on some key elements, given as: stochastic interpretation,
theoretical analysis, numerical methods, and optimization of model parameters. In this dissertation,
we provide a robust framework for fractional modeling of anomalous features in the applied areas
of research such as solute transport in underground waters and homogeneous isotropic turbulent
flows.

Inspired by the stochastic interpretation of fractional diffusion problems in terms of time-
changed Markov processes, we develop a unified Petrov-Galerkin (PG) spectral method for a
class of fractional partial differential equations (FPDEs) with two-sided derivatives and constant
coefficients. We employ the eigenfunctions of the fractional Sturm-Liouville eigen-problems,
called Jacobi poly-fractonomials, as temporal bases and test functions. Furthermore, we formulate

a novel unified fast linear solver for the resulting high-dimensional linear system based on the
solution of generalized eigenproblem of spatial mass matrices with respect to the corresponding
stiffness matrices. On the theoretical side, we prove existence and uniqueness of the solution
to the corresponding bilinear form of the problem and subsequently, provide the corresponding
stability and error analyses. Moreover, we compare the theoretical and computational rates of
spectral convergence by performing several numerical simulations. Motivated by the multifractal
characteristics of anomalous phenomena, we extend the unified PG method to the generalized form
of FPDEs with distributed-order two-sided derivatives and carry out the corresponding stability
and error analyses.

In the large-eddy simulation of homogeneous turbulence, presence of nonlocal triad interactions
and intermittent structures urges developing new nonlocal closure models. Such anomalous effects
become even more pronounced when the filter-width enlarges. Within the proposed framework, we
start modeling of turbulent effects at the Boltzmann transport using fractional calculus. In practice,
we approximate the filtered collision operator with a power-law function which contributes to
the corresponding fractional subgrid-scale (SGS) model in the filtered NS equations through a
meticulous derivation. Inspired by the self-similar cascading of energy from large to small scales
and the exponential decay in the dissipation range, we adopt tempered Lévy-stable distribution as
a statistically-consistent choice at the kinetic level and subsequently, we derive the corresponding
operator in terms of the tempered fractional Laplacian, (Ɗ + 
)
(·), 
 ∈ (0, 1], 
 ࣔ 1
2, and 
 > 0
in the filtered NS equations, termed as tempered fractional SGS (TFSGS) model. The model
parameters appear to be strictly depending on the filter-width and the flow Reynolds number. Such
a tractable fractional operator offers a great flexibility in characterizing nonlocal structures in the
turbulent inertial and dissipation ranges through fractional and tempering parameters. To infer
optimum values of the model parameters, we develop a robust optimization algorithm, involving
two-point structure functions and conventional correlation coefficients. In a priori / a posteriori
statistical analyses, we evaluate capabilities of the TFSGS model in fulfilling the closed essential
requirements and also study the numerical stability of LES solutions in time.

Copyright by
MEHDI SAMIEE
2021

To my parents

v

ACKNOWLEDGEMENTS

First and foremost I am extremely grateful to my supervisor, Professor Zayernouri for his invaluable
advice, continuous support, and illuminating guidance throughout my doctoral program. My sincere
gratitude extends to my committee members, Professor Brereton, Professor Jaberi, and Professor
Murillo, for their thoughtful and meticulous comments and suggestions. I also appreciate Professor
Meerschaert for the informative and helpful conversations during his class and group meetings.

I would like to thank to my dear peers in the FMATH group for a cherished time spent together
in the lab and in social settings especially Dr. Jorge Suzuki for sharing his invaluable thoughts and
experiences. I would like to acknowledge my great collaborators Mr. Ali Akhavan-Safaei and Dr.
Ehsan Kharazmi for the fruitful discussions and being open to exchanging ideas. Additionally, I
would like to express my deepest gratitude to my friends who offered me their gracious supports
and encouragements in good times and tough times during my PhD.

My most special appreciations and heartfelt thanks go to my parents, Muhammad Hussein and
Zahra, and to my kind brother, Milad, for all the love and endless encouragements they have given
me to shape my life. Their sacrifices and support have helped me complete this achievement. As a
small token of my appreciation, this dissertation is dedicated to them.

This dissertation was supported by the department of mechanical engineering and department
of computational mathematics, science and engineering in the college of engineering at Michigan
State University. The high-performance computing resources and services were provided by the
Institute for Cyber-Enabled Research (ICER) at Michigan State University.

vi

TABLE OF CONTENTS

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xi
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x

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LIST OF TABLES .
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LIST OF FIGURES .
LIST OF ALGORITHMS .

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CHAPTER 1

INTRODUCTION .

1.1 Anomalous Transport .
.
1.2 Fractional Calculus .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Preliminaries on Fractional Calculus . . . . . . . . . . . . . . . . . . . . .
Fractional Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Outline of the Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1
1.2.2

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1
1
4
7
8
9

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2.3.1

CHAPTER 2 A UNIFIED SPECTRAL METHOD FOR FPDES WITH TWO-SIDED

DERIVATIVES .
.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Background .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Preliminaries on Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Problem Definition .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Stochastic Interpretation of the FPDEs . . . . . . . . . . . . . . . . . . . . 21
2.4 Petrov-Galerkin Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2
Solution and Test Function Spaces . . . . . . . . . . . . . . . . . . . . . . 29
Petrov-Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.3
2.5 Well-posedness Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Discrete Form Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Space of Basis Functions (

) . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6.1
2.6.2
Space of Test Functions (

) . . . . . . . . . . . . . . . . . . . . . . . . . 37
Implementation of PG Spectral Method . . . . . . . . . . . . . . . . . . . . . . . 37
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.8.1 Computational Considerations . . . . . . . . . . . . . . . . . . . . . . . . 45
2.9 Stability and Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
.
2.10 Error Analysis .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.11 Numerical Tests .
CHAPTER 3 A UNIFIED PG METHOD FOR DISTRIBUTED-ORDER FPDES . . . . . 54
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1 Background .
3.2 Preliminaries on Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.1 Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.2
Solution and Test Function Spaces . . . . . . . . . . . . . . . . . . . . . . 66
3.3.3 Well-posedness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.7
2.8 Unified Fast FPDE Solver .

3.4 Petrov Galerkin Method .

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vii

Space of Basis (

) and Test (

) Functions
. . . . . . . . . . . . . . . . 72
3.4.1
3.4.2
Implementation of the PG Spectral Method . . . . . . . . . . . . . . . . . 72
3.4.3 Unified Fast FPDE Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5 Stability and Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.5.1
3.5.2 Error Analysis . .
.

Stability Analysis .
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3.6 Numerical Tests .

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4.3.1 Boltzmann Transport Equation (BTE)
4.3.2
4.3.3 Derivation of the FSGS model
4.3.4

4.1 Background .
4.2 Preliminaries on Fractional Laplacian . . . . . . . . . . . . . . . . . . . . . . .
4.3 Governing Equations

CHAPTER 4 A FRACTIONAL SUBGRID-SCALE MODEL FOR TURBULENT FLOWS 84
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
. 87
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
. . . . . . . . . . . . . . . . . . . . 89
Filtered Boltzmann Transport Equation (FBTE) . . . . . . . . . . . . . . . 91
. . . . . . . . . . . . . . . . . . . . . . . . 94
Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Second-law of Thermodynamics . . . . . . . . . . . . . . . . . . 97
4.3.4.1
Frame Invariance
4.3.4.2
. . . . . . . . . . . . . . . . . . . . . . . . . 98
. . . . . . . . . . . . . . . . . . . 98
4.4.1 Estimation of Fractional Exponent 
 . . . . . . . . . . . . . . . . . . . . . 101
4.4.2 Analysis of the Model Performance
. . . . . . . . . . . . . . . . . . . . . 104
4.4.3 Towards Modeling Nonlocal Effects . . . . . . . . . . . . . . . . . . . . . 107

4.4 A Priori Analysis of the Fractional SGS Model

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CHAPTER 5 A TEMPERED FRACTIONAL APPROACH TO LES OF TURBULENT

FLOWS .
.

5.4 Statistical Analysis

5.1 Background .
5.2 Preliminaries on Tempered Fractional Calculus

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

. 109
. 109
. . . . . . . . . . . . . . . . . . . 113
5.2.1 Tempered fractional Laplacian . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3 Boltzmannian Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3.1 A review on the subgrid-scale modeling . . . . . . . . . . . . . . . . . . . 116
5.3.2 The BGK equation and the closure problem . . . . . . . . . . . . . . . . . 117
5.3.3 Tempered fractional SGS modeling . . . . . . . . . . . . . . . . . . . . . 120
. . . . . . . . . . . . . . . . . . 123
5.3.4 TFSGS formulations for the SGS stresses
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4.1 Optimization strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.5.1 DNS Database and LES Platform . . . . . . . . . . . . . . . . . . . . . . . 127
5.5.2 Optimal estimation of fractional parameters . . . . . . . . . . . . . . . . . 128
5.5.3
Interpretation of two-point structure functions . . . . . . . . . . . . . . . . 131
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
PDF of SGS stresses
5.5.4
. 136
CHAPTER 6 SUMMARY AND FUTURE WORKS . . . . . . . . . . . . . . . . . . . . 138
. 138
. 141

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6.1 Concluding Remarks
6.2 Future Works
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5.6 A Posteriori Analysis .

5.5 A Priori Study .

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viii

.

APPENDICES .
APPENDIX A
APPENDIX B
APPENDIX C
APPENDIX D
BIBLIOGRAPHY .

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DERIVATIONS IN CHAPTER 2 . . . . . . . . . . . . . . . . . . . 143
DERIVATIONS IN CHAPTER 3 . . . . . . . . . . . . . . . . . . . 149
DERIVATIONS IN CHAPTER 4 . . . . . . . . . . . . . . . . . . . 151
DERIVATIONS IN CHAPTER 5 . . . . . . . . . . . . . . . . . . . 155
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ix

LIST OF TABLES

Table 1.1: Variety of the underlying anomalous stochastic processes [1]. . . . . . . . . . . .

5

Table 2.1: Convergence study of the PG spectral method for (1+1)-D diffusion problem,
4 and 
3 = 51
where 

1 = 

1 = 2
5
in (2.105). Here, we denote by ¯
0 the practical rate of the convergence,
numerically achieved. .

10 and 
 = 2. Besides, 
1 = 5 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

20, 
2 = 53

Table 2.2: Here, we set 
1 = 5 1

PG spectral method for (1+1)-D diffusion problem, where 

1 = 

1 = 2
20, where 
 = 5

 = 2. Besides, the limit orders are 
1 = 11

20 and 
 = 1 in (2.106) to study the convergence of the
10 and
20 is fixed.

20 and 
1 = 19

53

sion problems. 



 = 
 
1+
 ×
3

Table 3.1: CPU time, PG spectral method for fully distributed (1+d)-dimensional diffu-

=1(1+ 

) 
2
(1− 

) 
2
+1, where 
 = 10−4,


1 = 3, and the expansion order is 4 × 11
.

. . . . . . . . . . . . . . . . . . . . 83

Table 4.1: Computational parameters and statistical features of a forced HIT problem,

provided by JHTDB [2]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Table 4.2: Micro-scale statistical characteristics of turbulence for the applied initial con-

dition in DNS of decaying HIT.

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

Table 4.3: A priori results for the correlation coefficients, 

 
, of SGS stresses obtained

from the FSGS and SMG models at two different Reynolds numbers, L
 . . . . . 105

Table 4.4: Study of FSGS model in terms of L through a priori analysis at other time
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

instants of Case (I) .

.

. 106

Table 5.1: Flow parameters and statistical properties in the DNS of a forced HIT flow.

. . . 128

Table 5.2: The ensemble-averaged correlation coefficients (


) between SGS stresses
obtained by the filtered DNS data (T
,

) for

 = 1, 2, 3. In the fractional models, 

 

 is set as 0.76, 0.58, and 0.51 for
L
 = 4, 8, 12 respectively.

) and the TFSGS model (T
,
 


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129









Table 5.3: Optimized parameters associated with the TFSGS model in terms of Algorithm

5.1 for differing filter widths. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 132

x

LIST OF FIGURES

Figure 1.1:

[3] Overlay of a migrating MDCK-F NHE cell with its path covered within
480 minutes. The cell frequently changes its shape and direction during
migration. Double-logarithmic plot of the mean squared displacement as a
function of time. Experimental data points for both cell types are symbolized
by triangles and circles.

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

2

Figure 1.2:

[4] Positions of 10 small particles (distinguished by different colors) (left)

 = 0.2, (middle) 
 = 0.35, and (right) 
 = 0.5, for a fixed total time. Note
that 
 denotes the size ratio of small and large particles. The distribution,

(
), of the size, 
, of clusters follow a power-law dependence, 
(
) ∼ 
−2.19.

.

2

Figure 1.3:

Figure 1.4:

[5] (left) Dispersal of bank notes on geographical scales originating from three
different places. (middle) In the quantitative analysis of bank note dispersal,
the measured probability density function, 
(
), of traversing a distance, 
,
in 4 days undergoes a power-law 
(
) ∼ 
−1.59. (right) 
(
) for three classes
of initial entry locations (black triangles for metropolitan areas, diamonds for
cities of intermediate size, and circles for small towns).

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

[6] The trajectories of Lagrangian particles in a hetrogeneous porous media;
(left) probability distributions of normalized Lagrangian longitudinal velocity

Ɗ
 (
) for the different time lags, (right) colored normalized
increments,
mean squared displacement of purely advected particles, 

 =< Ɗ
2 >,
versus normalized time.

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

Ɗ


Figure 1.5:

[7] Ramp-cliff structures in a scalar field at 


 = 650 and 

 = 1. The
vertical colored solid line show the spatial positions for the magnifications of
the scalar fluctuation profiles, plotted in the right.

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

3

3

5

Figure 1.6: Schematic framework of the thesis with main focus on developing fractional
models for anomalous transport, e.g., solute particle transport in underground
waters and SGS motions in turbulent flows. The proposed framework is es-
tablished based on the key elements, listed as: mathematical formulation of
the fractional model starting at the kinetic level, theoretical analysis, devel-
oping numerical schemes for the proposed FPDE approaches, and optimizing
performance of the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 3.1: Temporal/Spatial p-refinement for case I with singularity of order 
 = 10−4.
(Left): 
1 = 3, 
2 = 
3 = 2, and expansion order of N×9. (Middle): 
1 = 2,

2 = 
3 = 2, and expansion order of 3 × M. (Right): 
1 = 3, 
2 = 
3 = 2,
and expansion order of 4 × M.

. . . . . . . . . . . . . . . . . . . . . . . . . . 81

. 10

xi

Figure 3.2: Spatial p-refinement for case II, 
1 = 3, 
 = 0.1, and 
 = 0.9. . . . . . . . . . . 82
Figure 3.3: Spatial p-refinement for case III with singularity of order 
 = 10−4. (Left):
(1 + 2)-dimensional, 
1 = 3, 
2
 = 
2
+1 = 1, where the expansion order is
N × M1 × M2. (Left): (1 + 3)-dimensional, 
1 = 3, 
2
 = 
2
+1 = 1, where
the expansion order is N × M1 × M2 × M3.

. . . . . . . . . . . . . . . . . . 82

Figure 4.1: Time evolution of 
(
(cid:48))/
0, 
(
(cid:48))/
0, and 


(
(cid:48)) during the DNS of decay-

ing HIT .

.

.

.

.

.

.

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

Figure 4.2: 

 versus 
 for 
 = 1.85 × 10−4 in Case (I) and 
 = 10−3 in Case (II)

. . . . . 102

Figure 4.3: Variation of the correlation coefficient, 

, and the regression coefficient, 

,
in terms of the fractional exponent, 
 ∈ (0, 1) using the DNS database of
Case (I) for (a) 
 = 1, 2, 3 at L
 = 4, and (b) 
 = 1 at L
 = 2, 8, 32 . . . . . . . . 103
Figure 4.4: 

 

 versus L
 for the Case (I), of properties are given in Table 4.1 . . . . . . . 103

Figure 4.5:

(a) The surface of 

 

, obtained by the Kriging method, versus L
 and 



using the data points, denoted by ਭ, which are estimated by the a priori tests
of FSGS model for Case (II) and (b) comparison between the curves of 

 


versus L
, which are designated by 


 ≈ 26, 35, 45

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

Figure 4.6: Comparing the correlation coefficients, 

, from the FSGS model using the
optimum fractional exponent, with the corresponding ones implied by the
Smagorinsky model

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

.

Figure 4.7: A priori results for the correlation between the true and model values for the
components of ∇ · T 
, where [∇ · T 
] 



 = 

 (−Ɗ)
 ¯V|
=

 

, yielding
the correlation coefficients, as shown . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 4.8: Comparison of Kriging-constructed surfaces of 

 for 
 = 1, 2, 3, obtained
from a priori study of the FSGS and the SMG models, versus L
 and 



on Case (II) .

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

.

.

.

.

Figure 4.9: A priori results for the PDF of the true and modeled (∇ · T 
)1 regarding 


variations at each L


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

Figure 5.1: variation of the correlation coefficients (a) 
11, (b) 
22, and (c) 
33 versus

 ∈ (0, 1) for L
 = 4, 8, 12 by setting 
 (cid:39) 0 in (5.41). The maximum values
lie in the dashed boxes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xii

Figure 5.2:

(a) Comparing results of the normalized two-point stress-strain rate corre-
lation functions (
Ɗ) and (b) error analysis of the longitudinal dissipation
spectrum (E
) for fractional SGS and Smagorinsky models, where L
 = 8
and 
 represents the radius of Fourier wave numbers. In both plots, the arrows
point to the dissipation range.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Figure 5.3: Comparing results of the normalized two-point stress-strain rate correlation
functions for (a) L
 = 4 and (b) L
 = 12. The inset plots illustrate the scaled
error of longitudinal dissipation spectrum (E
) versus the radius of Fourier
wave numbers (
). The arrows point to the dissipation range.

. . . . . . . . . . 131

Figure 5.4: Two-point velocity-stress correlation function in a stationary HIT flow for
L
 = 8 using box filtering. The segment of balance region (BR) has been
thickened up in (a) for all the graphs. The dissipation and the inertial ranges
have been enlarged in plots (b) with semi-logarithmic scale on the x-axis and
(c) with logarithmic scale on the both axes, respectively. . . . . . . . . . . . . . 133

Figure 5.5: Two-point velocity-stress correlation function in a stationary HIT flow for (a)
L
 = 4 and (b) L
 = 12 using box filtering. The inertial range has been
enlarged in the inset plots with logarithmic scale on the both axes. . . . . . . . . 133

Figure 5.6: Scatter plots of the SGS stresses obtained by the filtered DNS data (T
, 

33 )
) using optimized parameters in Table 5.3

versus the modeled stresses (T
, 
 

for (a) L
 = 4, (b) L
 = 8, and (c) L
 = 12.

Figure 5.7: PDF of the ensemble-averaged(cid:0)∇ · T(cid:1)

33

. . . . . . . . . . . . . . . . . . . 135

tional and the SMG models at (a) L
 = 4, (b) L
 = 8, and (c) L
 = 12. . . . . . 135


=3 for the optimized (tempered) frac-

Figure 5.8: PDF of the ensemble-averaged SGS dissipation for the optimized (tempered)
fractional and the SMG models at (a) L
 = 4, (b) L
 = 8, and (c) L
 = 12.

. . 136

Figure 5.9: Decaying of the resolved turbulent kinetic energy, 



, for the optimized

TFSGS and the SMG models with (a) 403 and (b) 203grid points. . . . . . . . . 137

Figure 5.10: Time-evolution of the resolved enstrophy, E, for the optimized TFSGS and

the SMG models with (a) 403 and (b) 203grid points.

. . . . . . . . . . . . . . 137

xiii

LIST OF ALGORITHMS

Algorithm 5.1: Estimation of the optimal parameters for a specific L . . . . . . . . . . . . . . 126

xiv

CHAPTER 1

INTRODUCTION

1.1 Anomalous Transport

Anomalous transport represents a proper kinetic description of dynamical systems with much
more complex behavior than the standard diffusion. In classical physics, standard diffusion equation
governs the probability distribution function (PDF) of the underlying Brownian motion, in which
(Ɗ
)2 ∼ 
, where Ɗ
 denotes the displacement of the scalar particle in a given time, 
, and (·)
denotes ensemble averaging.
In contrast, for describing anomalous diffusion a large variety of
stochastic processes have been proposed over the last decade, in which the characteristics grow
non-linearly in time, i.e., (Ɗ
)2 ∼ 

.
If 
 < 1, the phenomenon undergoes sub-diffusion and
if 
 > 1, it is called super-diffusion. Modern non-intrusive experimental techniques disclosed
significance of anomalous dynamics in a disparate range of dynamical systems such as turbulence
[8, 1, 9, 10], porous media [11, 12, 13, 14, 15, 16], geoscience [17], bioscience [18, 19, 20], and
viscoelastic material [21, 22, 23, 24]. Some of the important evidences are discussed as follows.

Biology: By analyzing the trajectories of wild-type and mutated epithelial (transformed Madin-
Darby canine kidney) cells, Dieterich et al.
[3] presented the anomalous characteristics of cell
migration in an experimental study. In figure 1.1, they depicted the contours and the path of a
migrating MDCK-F NHE cell monitored for 480 minutes. At first sight, the underlying stochastic
process of particles by a mean squared displacement, which is proportional to 
2 at short times
corresponding to ballistic motion and 
 for long time intervals designating normal diffusion. More
recently, Sentjabrskaja et al. [4] investigated anomalous dynamics such as proteins moving through
cells in a colloidal model system, in which dilute small spheres move in a glassy matrix of large
spheres. As illustrated in figure 1.2, they monitored the evolution of the position of small particles
and reported the anomalous behavior of small spheres, which is strongly connected to slow dynamics
of the matrix of large particles.

1

Figure 1.1: [3] Overlay of a migrating MDCK-F NHE cell with its path covered within 480 minutes.
The cell frequently changes its shape and direction during migration. Double-logarithmic plot of
the mean squared displacement as a function of time. Experimental data points for both cell types
are symbolized by triangles and circles.

Figure 1.2: [4] Positions of 10 small particles (distinguished by different colors) (left) 
 = 0.2,
(middle) 
 = 0.35, and (right) 
 = 0.5, for a fixed total time. Note that 
 denotes the size ratio
of small and large particles. The distribution, 
(
), of the size, 
, of clusters follow a power-law
dependence, 
(
) ∼ 
−2.19.

Human transportation: Brockmann [5] presented the super-diffusive statistics of human travel
by analyzing the geographical circulation of individual bank notes. In figure 1.3, they illustrated
qualitative features of bank notes which is described by a power-law distribution. For more
information the reader is referred to [25].

Heterogeneous porous media: Many laboratory experiments confirmed the existence of non-
Gaussian processes, which govern the PDF of particle dispersion in porous media. Recently, Anna
et al. [6] conducted an experimental study on the intermittent structures of fluid velocities and
its relation to the anomalous dispersion of solute particles. In figure 1.4, they plotted the super-

2

Figure 1.3: [5] (left) Dispersal of bank notes on geographical scales originating from three different
places. (middle) In the quantitative analysis of bank note dispersal, the measured probability density
function, 
(
), of traversing a distance, 
, in 4 days undergoes a power-law 
(
) ∼ 
−1.59. (right)

(
) for three classes of initial entry locations (black triangles for metropolitan areas, diamonds for
cities of intermediate size, and circles for small towns).

Figure 1.4: [6] The trajectories of Lagrangian particles in a hetrogeneous porous media; (left)
probability distributions of normalized Lagrangian longitudinal velocity increments,

Ɗ
 (
) for
the different time lags, (right) colored normalized mean squared displacement of purely advected
particles, 

 =< Ɗ
2 >, versus normalized time.

Ɗ


diffusive transport of the particles. Moreover, Yang and Wang [26] conducted a numerical study on
the anomalous transport of solute particles using lattice Boltzmann method at high Peclet numbers.
Supported by their results and other experimental findings in [27, 28], anomalous transport is

3

indeed ubiquitous in porous media.

Turbulence: One hallmark of fully-developed turbulent flows is the emergence of non-Gaussian
statistics in the velocity field using Gaussian initial conditions. Wilczek et al.
[29] discussed
some aspects of the relation between non-Gaussianity, coherent structures and phase correlations in
turbulence. The connection between intermittent behavior and the presence of organized structures,
embedded within turbulent background, have been discussed numerically and experimentally over
the past decade (see e.g., [30, 31, 32]). Camussi and Verzicco [33] elaborated on the relation
between anomalies and coherent structures. Rosales and Meneveau [34] proposed the multiscale
minimal Lagrangian map (MMLM) approach to generate synthetic turbulent vector fields. They
used the MMLM method in [35] to focus mainly on the intermittency and scaling properties
of the synthetic velocity field in the inertial range.
In this study, they presented an analysis
of multifractal characteristics of energy dissipation for the description of strongly intermittent
structures. Moreover, by measuring scalar intermittent moments of high orders, Iyer et al.
[7]
provided unequivocal evidence on the scalar intermittency and multifractal statistical process. In
figure 1.5, they illustrated the generic existence of scalar cliffs. Such ramp-cliff structures suggested
that the local spatial barriers to scalar mixing have a significant impact on scalar intermittency.
Moreover, Shraiman and Siggia [8] reviewed recent progresses in the statistical description of
passive scalar turbulence, centered on the anomalous scaling properties and the appearance of
coherent structures in the scalar field. For more information on anomalous dynamics of turbulent
flows, the reader is referred to [31, 36].

More concretely, anomalous phenomena undergo a large variety of stochastic processes than
a simple power-law, which describe the continuous time random walk (CTRW) of particles at the
kinetic level. We summarized some of the processes in Table 1.1.

1.2 Fractional Calculus

Fractional calculus seamlessly generalizes the notion of standard integer-order calculus to its
fractional-order counterpart, leading to a broader class of mathematical models, namely fractional

4

Figure 1.5: [7] Ramp-cliff structures in a scalar field at 


 = 650 and 

 = 1. The vertical colored
solid line show the spatial positions for the magnifications of the scalar fluctuation profiles, plotted
in the right.

Table 1.1: Variety of the underlying anomalous stochastic processes [1].

Correlation Function

(cid:10)Ɗ
2(cid:11) ∼ 
2
(cid:10)Ɗ
2(cid:11) ∼ 
 
,
(cid:10)Ɗ
2(cid:11) ∼ 
2/
,
(cid:10)Ɗ
2(cid:11) ∼ ln
 

(cid:10)Ɗ
2(cid:11) ∼ 
2/

(−

)

0<
≤2 (
=2)

0<
<1

0<


Stochastic Process
Normal Diffusion
Sub-Diffusion

Super-Diffusion
(ballistic motion)

Super slow Sub-Diffusion

Tempered Super-Diffusion
(Truncated Lèvy Process)

Application

Homogeneous media

Particle-laden flows, Biological media

& Porous media

Plasma turbulence &

Chaotic motion in a turbulent flow

Sinai diffusion

(
 = 4)

Scalar dispersion in shear flows, Plasma
& Scalar dispersion in bounded flows

ordinary differential equations (FODEs) and fractional partial differential equations (FPDEs) [37,
38, 39, 40, 41]. Despite the considerable progresses in developing nonlocal models using standard
methods for anomalous phenomena, fractional calculus appears to be a mathematical tractable

5

tool, which deals with integro-differential equations with mainly singular kernels of power law or
logarithm type. As alternative approaches in the recent decades, the fractional derivatives have
been found helpful and effective in modeling anomalous diffusion processes due to their inherent
potentials in describing nonlocal interactions, self-similar structures, sharp peaks, and memory
effects (see e.g., [42, 43]).
In the following, we provide a brief review of fractional models in
variety of applications.

Fractional Rheology: Regarding the mathematical modeling of irreversible mechanical processes,
fractional calculus is used as a robust tool to provide more precise constitutive models. Suzuki et al.
[22] proposed two fractional-order models for unaxial large strains, which interpolates between the
standard elasto-plastic and visco-elasto-plastic models. In this respect, we refer to these papers: [44]
where a 3D non-local in the stress state model was defined; [45] where a new constitutive equation
was proposed based on a time–space fractional derivative to characterize the visco-elasto-plastic
behavior of metals and alloys. For more details, we refer the reader to [46].
Biology: Complex dynamic processes of biological tissues can be represented by fractional-order
formulations, which incorporates the memory effects. Goran et al. [47] presented a modified single-
dispersion cole model for more accurate description of bioimpedence properties of human skin.
Recently, Baleanue et al. [48] presented a new fractional model for a tumor-immune surveillance
mechanism. This model formulates the interactions between various tumor cell populations and
immune response by a system of FODEs. More information is provided in [49] and the references
therein.
Porous Media: In the standard advection-dispersion equations, solute transport in ground water
has been modeled traditionally based on the underlying Fickian process (equivalently a process of
Brownian motion). Supported by the experimental evidence (see e.g., [50]), solute particles do
not follow a Fickian process particularly in heterogeneous porous media. Accordingly, standard
approaches are found to be inherently insufficient to reproduce long-range interactions and memory
effects. Regarding that, FPDE approaches have been introduced widely to provide more adequate
and accurate models, i.e., fractional advection-dispersion equations, for transport of solute particles

6

[51]. Suzuki et al. [52] developed a fractional heat transfer equation to describe anomalous diffusion
in a fractured reservoir. Sapora et al. [53] investigated the diffusion processes on finite porous
domains using fractional finite difference methods. More details on this topic can be found in [54].

1.2.1 Preliminaries on Fractional Calculus
Here, we obtain some basic definitions from fractional calculus [38, 55]. Denoted by 
D


 
(
),
the left-sided Reimann-Liouville fractional derivative of order 
 in which 
(
) ∈ 

[
, 
] and

 = (cid:100)
(cid:101), is defined as:






D



 
(
) =

1

Ɖ(
 − 
)










(
)

(
 − 
)
+1−




,


 ∈ [
, 
],

(1.1)

ˆ 









(
), is given by
ˆ 


where Ɖ represents the Euler gamma function. The corresponding right-sided Reimann-Liouville
fractional derivative of order 
, 
D







D



 
(
) =

1

Ɖ(
 − 
) (−1)
 









(
)

(
 − 
)
+1−







,


 ∈ [
, 
].

(1.2)

In (1.1) and (1.2), as 
 → 
, the fractional derivatives tend to the standard 
-th order derivative
with respect to 
. We recall from [56, 57] that the following link between the Reimann-Liouville
and Caputo fractional derivatives, where






D


D



 
 (
) =

 
 (
) =






 (
)

 (
)

Ɖ(1 − 
)(
 − 
)
 + 

Ɖ(1 − 
)(
 − 
)
 + 



D


D



 
 (
)

 
 (
),

(1.3)

(1.4)

(1.5)

(1.6)

when (cid:100)
(cid:101) = 1. Generally





D



 
 (
) =





D



 
 (
) =

ˆ 

ˆ 








1

Ɖ(
 − 
)
(−1)

Ɖ(
 − 
)



,



,


 ∈ [
, 
],


 ∈ [
, 
],


(
)(
)

(
 − 
)
+1−



(
)(
)

(
 − 
)
+1−


D


where (cid:100)
(cid:101) = 
.

 
(
) = 

Dirichlet initial and boundary conditions are enforced.

In (1.3) and (1.4), 




D



 
(
) = 
D



 
(
) when homogeneous

7

1.2.2 Fractional Modeling

The advantage of fractional model lies in the straightforward way of modeling the underlying heavy-
tailed, self-similar, and non-Markovian processes for anomalous phenomena. With increasing
number of fractional models in diverse applications, the development of robust mathematical
theories and efficient computational methods has become urgent for approximating the solutions
of FPDEs. History-dependent and nonlocal characteristics of fractional operators impose further
numerical complexities, ranging from theoretical analysis to large memory requirement.

On the theoretical side, the notion of well-posedness analysis is very significant to ensure the
existence and uniqueness of the solution to FPDEs. In particular, well-posedness analysis plays
even a more crucial role when the proper boundary conditions of fractional equations are under
doubt. Additionally, several mathematical rules such as the integration by parts formula cannot
be applied in the standard way for fractional operators. In the context of numerical analysis, the
computational costs of the numerical solutions for FPDEs are heavier than the classical PDEs due
to the complicated forms of fractional operators, involving power-law to logarithmic kernels of
convolution type. Moreover, the huge memory requirement of FPDEs/FODEs is computationally
demanding especially for the temporal fractional problems.

Over the past decade, fractional calculus has gained considerable attention from the engineering
and computational science communities to address some key challenges and approximate solutions
of fractional equations more precisely and faster. It also inspired the engineering communities to
allow the real world scientific applications come true by using fractional approaches. However,
according to the computational challenges and complex formulations of fractional models, further
theoretical and numerical studies are required to establish a robust framework for developing
accurate and efficient models for anomalous transport.

In order to build a framework for developing fractional models, there are some essential

elements, which are listed as follows.

• Stochastic formulation: From the point of view of CTRW, motion of particles are modeled

8

by a proper random walk at the kinetic level for a precise and comprehensive understanding of
the anomalous system. Such stochastic interpretations can give rise to fractional differential
equations such as FPDEs, where the associated fractional exponents are connected with the
anomalous features.

• Theoretical analysis: In dealing with FPDEs, the well-posedness analysis plays a crucial
role in studying the existence and uniqueness of the underlying solutions and preventing
any instability in the numerical method. The primary step in performing such analysis is to
construct solution spaces, equipped with the proper equivalent norms.

• Numerical methods: Developing reliable numerical schemes for approximating the solution
of FPDEs requires a great deal of attention especially in treating fractional operators. It is
of great importance and necessity to propose efficient computational methods, which can
resolve the singularity in a fractional equation accurately.

• Optimization: In order to introduce a mathematical representation of anomalous structures
and improve the accuracy of fractional models in statistical features, it is markedly essen-
tial to construct a mapping function from the underlying physical properties of anomalous
phenomena to the model parameters through using proper optimization algorithms. The key
idea is to use higher-order nonlocal statistical properties to offer a better representation of
anomalous dynamics.

1.3 Outline of the Present Work

The overall goal of this research is to develop fractional models in a robust framework, incor-
porating the aforementioned key elements. In this study, we employ some useful mathematical
and computational tools to extend the framework to more applied areas of research such as solute
transport in underground water and turbulent flows. The proposed framework for developing frac-
tional models are illustrated in figure 1.6. The dissertation is divided to six chapters, which a short
summary of each is presented in the following.

9

Figure 1.6: Schematic framework of the thesis with main focus on developing fractional models
for anomalous transport, e.g., solute particle transport in underground waters and SGS motions
in turbulent flows. The proposed framework is established based on the key elements, listed as:
mathematical formulation of the fractional model starting at the kinetic level, theoretical analysis,
developing numerical schemes for the proposed FPDE approaches, and optimizing performance of
the model.

Chapter 2:

Supported by the macroscopic experiments at the Columbus Air Force Base in Mississippi
in [58], dispersion of particles in a highly heterogeneous aquifer undergoes time-changed Le´vy
processes rather than Brownian motion. Within the framework of CTRW, Baeumer et al. in [59]
established the stochastic interpretation for the FPDEs on a bounded domain in terms of time-
changed Markov processes by an inverse stable subordinator whose index equals the order of the
fractional temporal derivative.
Inspired by that, we consider a class of FPDEs with two-sided

10

derivatives and constant coefficients of the form

0D2



 
 +

[


 

D2







 + 


 

D2







] =






=1






=1

[

 
 
 
 D2
 



 



 + 

 
 
 
 D2
 



 



] − 
 
 + 


in a (1+
)-dimensional space-time hypercube, where 2

, ∈ (0, 1), 2

, ∈ (1, 2), and 2
, ∈ (0, 2),
2
 ࣔ 1, subject to Dirichlet initial and boundary conditions, where 
 = 1, 2, ..., 
.

In this chapter, we make several key contributions to the field of fractional calculus by developing
a unified PG spectral method for the FPDEs and performing the corresponding numerical analysis
(see [60, 61]). These contributions are mainly listed as follows.

• We develop a unified PG spectral method for the FPDEs employing the Legendre polynomials
as the spatial basis/test functions and the eigenfunctions of the fractional Sturm-Liouville
eigen-problems of the first/second kinds in [56], called Jacobi poly-fractonomials, as temporal
basis/test functions, respectively.

• We construct the underlying solution/test spaces, associated with the equivalent norms in-
duced by the corresponding weak form of (1.7). Subsequently, we perform the well-posedness
analysis using the Babuška-Lax-Milgram theorem [62] to guarantee the existence and unique-
ness of solutions of the bilinear form.

• We formulate a novel unified fast linear solver for the resulting high-dimensional linear system
based on the solution of generalized eigen-problem of spatial mass matrices with respect to
the corresponding stiffness matrices, hence, making the complexity of the problem optimal,
i.e., O(
 
+2).

• We provide the stability analysis of the PG method for the finite-dimensional subspaces of
solution/test spaces. Moreover, we perform the corresponding error analysis to find the
spectral accuracy of the PG method.

• We carry out several numerical test cases using non-smooth exact solutions to examine the

CPU time and compare the practical and theoretical convergence rate of the PG method.

11

Chapter 3: In many complex multi-fractal anomalous phenomena described by accelerating super-
diffusion and decelerating sub-diffusion processes [63, 64, 65, 38, 66, 67, 68, 69], a single power-law
scaling is not observed over the whole domain. A faithful description of such anomalous transport
requires exploiting distributed-order derivatives, in which the underlying derivative orders are
distributed over a range of values.

ˆ 




Let define the distributed-order derivative as


D



 
 (
, 
) :=


(
) 0D



 
 (
, 
)

,







(1.7)
where 
 → 
(
) be a continuous mapping in [



, 



] [70] and 
 > 0. We note that by
choosing the distribution function in the distributed-order derivatives to be the Dirac delta function

(
 − 
0), we recover a single (fixed) term fractional derivative, i.e.,

 
 (
, 
)

 = 0D
0

(1.8)
where 
0 ∈ (



, 



). The main purpose of this chapter [71, 72] is to develop and analyze a
Petrov-Galerkin (PG) spectral method to solve a (1 + 
)-dimensional fully distributed-order FPDE
with two-sided derivatives of the form


(
 − 
0) 0D


ˆ 






 (
, 
),










0D2



 
 

 +



(

) [










D2







 + 










D2







] 




ˆ 















(
) 

ˆ 



















ˆ 














=1



 
 D2
 



 







 
(
 
) [

 



 + 

 







 
 D2
 



] 

 
 − 
 
 + 
 ,

=


 



=1

(1.9)
subject to homogeneous Dirichlet boundary conditions and initial condition, where 
 ∈ [0, 
],

 
 ∈ [
 
 , 
 
], 2



 < 2



 ∈ (0, 2], 2



 ࣔ 1, 2



 ࣔ 1, 2




∈ (0, 1],
∈ (1, 2] for 
, 
 = 1, 2, ..., 
. The coefficients 


, 


, 


, 


, and 
 are
and 2




considered to be constant. Besides, the distribution function 0 < 
(
) ∈ 
1([



, 



]) ,
0 < 

(

) ∈ 
1([




]). We carry out the following
main contributions as listed here.

]), and 0 < 
 
(
 
) ∈ 
1([





< 2





< 2





, 





, 





























• We construct the underlying function spaces by extending the distributed Sobolev space in
[70] to higher dimensions in time and space. These solution/test spaces are endowed with
the equivalent norms, which are induced by the corresponding bilinear form of (1.9).

12

• We develop a unified PG spectral method, employing Legendre polynomials and Jacobi
poly-fractonomials [56] as spatial and temporal basis/test functions, respectively. We also
formulate a fast solver for the corresponding weak form of (1.9), which significantly reduces
the computational expenses in high-dimensional problems.

• We prove the existence and uniqueness of the solutions to the weak form of (1.9) in the
underlying solution/test spaces. Additionally, we performed the stability analysis of the PG
method for the finite-dimensional subspaces of solution/test spaces.

• We perform the corresponding error analysis, where the distributed Sobolev spaces enable

us to obtain accurate error estimate of the scheme.

• We carry out several numerical test cases using non-smooth exact solutions to examine the

CPU time and compare the practical and theoretical convergence rate of the PG method.

• To examine the performance and convergence of the developed PG method, we also perform

several numerical simulations using smooth and non-smooth test cases.

Chapter 4:

Supported by several numerical and empirical studies (see e.g., [31, 73, 74, 75, 76]), SGS mo-
tions are characterized by the intermittent structures and anomalous scaling properties in turbulent
flows. Such anomalous features become even more pronounced when the filter-width enlarges.
That inspires the development of new nonlocal closure models, which captures the anomalous
scaling of the structure function exponents. In this chapter, we outline a new framework to the
functional modeling of SGS stresses in large eddy simulation (LES) of turbulent flows, starting
from the kinetic theory.


 





(1.10)
where 
 

 = 
 

(
, 
, 
) represents the equilibrium distribution function and 
 is the relaxation
time. Assuming that the system of gaseous particles is in thermodynamic equilibrium, the equi-








The Boltzmann transport equation [77, 78] is given by

+ 
 · ∇ 
 =(cid:0) 
 


(cid:1)





 ≡ − 
 − 
 



,

13

librium distribution is given by a Maxwell (Gaussian) distribution function [79]. To proceed for
¯
 , and the residual values, 
 (cid:48).
the LES of a turbulent flow, we can decompose 
 to the filtered,
¯
 = 
 ∗ 
 , where 
 is the kernel of any
The overbar represents the spatial isotropic filtering, i.e.
generic spatial filtering. Then, we formulate the filtered Boltzmann transport according to


 ¯





+ 
 · ∇ ¯
 = − ¯
 − 
 

(Ɗ)




,

(1.11)

and 
 and ¯
 represent the agitation speed and the filtered velocity field,

where ¯Ɗ := |
− ¯
|2

2
respectively.

Within the proposed framework, we treat the resource of turbulent small scale motions by
modeling 
 

(Ɗ) − 
 

( ¯Ɗ) with a Lévy-stable distribution. Subsequently, by ensemble-averaging
(1.11), we derive the corresponding filtered Navier-Stokes equations, where the divergence of SGS
stresses emerges as a single-parameter fractional Laplacian, (−Ɗ)
(·), 
 ∈ (0, 1], of the filtered
velocity field. The only model parameter, i.e., the fractional exponent, appears to be strictly
depending on the filter-width and the flow Reynolds number.

Here, we briefly highlight the main contributions of this work as follows:

• We develop a new functional approach to model the SGS stresses by employing the fractional
Laplacian of the filtered velocity within the Boltzmann transport framework. The fractional
exponent in the model arises from the heavy-tailed behavior of the SGS stresses.

• We show that the model is frame invariant and constrain it to a set of conditions to preserve

the second-law of thermodynamics.

• We perform the a priori studies to assess performance of the model primarily by the corre-
lation and regression coefficients utilizing the results of direct numerical simulation (DNS)
for three-dimensional forced and decaying homogeneous isotropic turbulence (HIT) prob-
lems. We also investigate statistical characteristics of the proposed model, as a hallmark of
fractional operators, in a range of filter widths.

14

On the basis of the theoretical background and the a priori analyses provided in this study, the
proposed framework show a remarkable potential for developing more sophisticated SGS models
in LES of turbulent flows by leveraging proper mathematical tools in fractional calculus.

Chapter 5: Presence of nonlocal triad interactions and intermittent structures in homogeneous
turbulence give a key role to multi-point statistical functions in formulating an ideal LES model
[80, 81, 76]. Such nonlocality, caused by filtering pressure terms in the NS equations, leave urges for
developing nonlocal and fractional LES models [82]. Within the fractional framework, introduced
in chapter 4, we take a step forward in modeling the real physics of turbulent energy cascade by
employing a tempered Lévy-stable distribution at the kinetic level. Subsequently, we derive the
corresponding operator in terms of the tempered fractional Laplacian, (Ɗ + 
)
(·), for 
 ∈ (0, 1)
and 
 > 0 at the filtered NS equations, where the model parameters depend on the filter width. The
significance of this approach lies in:

• We treat the source of turbulent small-scale motions at the kinetic level, by employing a
tempered heavy-tailed distribution in approximating 
 

. This leads us to the tempered
fractional operator in the filtered NS equations as a proper choice for modeling a power-law
like behavior in the mid-range and a Gaussian tail in real-physics anomalous phenomena.

• The proposed TFSGS model sets the ground for fulfilling essential statistical conditions as
a relatively best approximation of an ideal LES model through fractional and tempering
parameters. To achieve an optimized edition of the TFSGS model, we devise an optimization
strategy, which involves conventional one-point correlation coefficients, two-point structures,
and the SGS dissipation.

• The optimized TFSGS model presents reasonably accurate predictions of two-point structure
functions for a range of filter widths while maintaining the expected correlations between the
modeled and true SGS stresses.

15

• The corresponding fractional LES solutions present a stable prediction of energy and enstro-

phy decays in the a posteriori analysis.

Chapter 6: The dissertation is concluded with the discussed findings and possible future works
extended from the current investigation.

16

A UNIFIED SPECTRAL METHOD FOR FPDES WITH TWO-SIDED DERIVATIVES

CHAPTER 2

2.1 Background

Due to the history dependence and non-local character of fractional differential equations
(FDEs), their discretization becomes computationally challenging. Numerical methods, developed
to discretize FPDEs, can be categorized in two major classes: i) local methods, e.g., finite difference
method (FDM), finite volume method (FVM), and finite element method (FEM), and ii) global
methods, e.g., single and multi-domain spectral methods (SM).

Local schemes have been studied extensively in the literature. Lubich introduced the discretized
fractional calculus within the spirit of FDM [83]. Sugimoto employed a FDM scheme for approx-
imating fractional Burger’s equation [84, 85]. Meerschaert and Tadjeran [86] developed finite
difference approximations to solve one-dimensional advection-dispersion equations with variable
coefficients on a finite domain. Tadjeran and Meerschaert [87] employed a practical alternating
directions implicit (ADI) method to solve a class of fractional partial differential equations with
variable coefficients in bounded domain. Hejazi et al.
[88] developed a finite-volume method
utilizing fractionally shifted grunwald formula for the fractional derivatives for space-fractional
advection-dispersion equation on a finite domain. To solve the two-dimensional two-sided space-
fractional convection diffusion equation, Chen and Deng [89] proposed a practical alternating
directions implicit method. Zeng et al. [90] constructed a finite element method and a multistep
method for unconditionally stable time-integration of sub-diffusion problem. In addition, Zhao et
al. developed second-order FDM for the variable-order FPDEs in [91]. Li et al. [92] proposed
an implicit finite difference scheme for solving the generalized time-fractional Burger’s equation.
Recently, Feng et al. [93] proposed a second-order Crank-Nicolson scheme to approximate the
Riesz space-fractional advection-dispersion equations (FADE). Recently, Zayernouri and Matza-
vinos [94] have developed an explicit fractional adams/Bashforth/Moulton and implicit fractional

17

Adams-Moulton finite difference methods, applicable to high-order time-integration of nonlinear
FPDEs and amenable for formulating implicit/explicit (IMEX) splitting methods.

Regarding global methods, Sugimoto [84, 85] used Fourier SM in a fractional Burger’s equa-
tion. Shen and Wang [95] constructed a set of Fourier-like basis functions for Legendre-Galerkin
method for non-periodic boundary value problems and proposed a new space-time spectral method.
Sweilam et al. [96] considered Chebyshev Pseudo-spectral method for solving one-dimensional
FADE, where the fractional derivative is described in Caputo sense. Chen et al. [97] developed an
approach for high-order time integration within multi-domain setting for time-fractional diffusion
equations. Mokhtary developed a fully discrete Galerkin method to numerically approximate initial
value fractional integro-differential equations [98]. More recent works in this area can be found in
[99, 100].

Moreover, Zayernouri and Karniadakis [56, 101] introduced a new family of basis/test functions,
called (tempered)Jacobi poly-fractonomials, known as the explicit eigenfunctions of (tempered)
fractional Strum-Liouville problems in bounded domains of the first and second kind. Following
this new spectral theory, they have developed a number of single- and multi-domain spectral
methods [102, 103, 55]. Recently, Zaho et al. [104] developed a spectral method for the tempered
fractional diffusion equations (TFDEs) using the generalized Jacobian functio [105]. Mao and
Shen [106] developed Galerkin spectral methods for solving multi-dimensional fractional elliptic
equations with variable coefficients.

The main contribution of the present work is to construct a unified Petrov-Galerkin spectral
method and a unified fast solver for the weak form of linear FPDEs with constant coefficients in
(1+d) dimensional space-time hypercube of the form






=1

0D2



 
 +

[


 

D2







 + 


 

D2







] =





[

 
 
 
 D2
 



 



=1
−
 
 + 
 ,


 + 

 
 
 
 D2
 



 



]

(2.1)

where 2

, ∈ (0, 1), 2

, ∈ (1, 2), and 2
, ∈ (0, 2), 2
 ࣔ 1, subject to Dirichlet initial and
boundary conditions, where 
 = 1, 2, ..., 
. Compared to the problem considered in [45], we

18

extend the one-sided spatial derivatives to two-sided ones, also, we include an advection term in
order to consider the drift effects. Employing different (Legendre polynomial) spatial basis/test
functions and the additional advection term then would not allow employing the fast linear solver
developed in [45]. Accordingly, we formulate a new fast linear solver for advection-dispersion
problems. We additionally aim to perform the well-posedness and stability analysis in any (1+d)
dimensions in [2], while in [45], only the stability of 1-D problem has been carried out. Furthermore,
we briefly presented the stochastic interpretation of FADE on bounded domain which sheds light
on the well-posedness of the problem from the perspective of the probability theory.

The outline of this chapter is as follows: in section 2.2, we introduce some definitions from
fractional calculus. In section 2.3, we present the strong form definition of the problem. In section
2.4, we construct the solution/test spaces and develop the PG method. Next, We prove the well-
posedness of the weak form and perform the stability analysis in section 2.5. In section 2.6 to 2.8,
we develop a unified fast linear solver and obtain the closed-form solution in terms of the genralized
eigenvalues and eigenvectors of the corresponding mass and stiffness matrices. In section 2.11,
the performance of the PG method is examined via several numerical simulations for low-to-high
dimensional problems with smooth and non-smooth solutions.

2.2 Preliminaries on Fractional Calculus

Here, we obtain some basic definitions from fractional calculus [38, 55]. Denoted by 
D



 
(
),
the left-sided Reimann-Liouville fractional derivative of order 
 in which 
(
) ∈ 

[
, 
] and

 = (cid:100)
(cid:101), is defined as:






D



 
(
) =

1

Ɖ(
 − 
)










(
)

(
 − 
)
+1−




,


 ∈ [
, 
],

(2.2)

ˆ 





19

where Ɖ represents the Euler gamma function. The corresponding right-sided Reimann-Liouville
fractional derivative of order 
, 
D







D



 
(
) =

1

Ɖ(
 − 
) (−1)
 









(
)

(
 − 
)
+1−







,


 ∈ [
, 
].

(2.3)





(
), is given by
ˆ 


In (2.2) and (2.3), as 
 → 
, the fractional derivatives tend to the standard 
-th order derivative
with respect to 
. We recall from [56, 57] that the following link between the Reimann-Liouville
and Caputo fractional derivatives, where






D


D



 
 (
) =

 
 (
) =






 (
)

 (
)

Ɖ(1 − 
)(
 − 
)
 + 

Ɖ(1 − 
)(
 − 
)
 + 



D


D



 
 (
)

 
 (
),

when (cid:100)
(cid:101) = 1. Generally

(2.4)

(2.5)

(2.6)

1








D



 
 (
) =


D


Ɖ(
 − 
)
(−1)

Ɖ(
 − 
)
In (2.4) and (2.5), 




 
 (
) =

ˆ 

ˆ 









(
)(
)

(
 − 
)
+1−



(
)(
)

(
 − 
)
+1−


D




,



,


 ∈ [
, 
],


 ∈ [
, 
],

(2.7)

 
(
) when homogeneous


 
(
) = 
D


where (cid:100)
(cid:101) = 
.

 
(
) = 

Dirichlet initial and boundary conditions are enforced.


D


To analytically obtain the fractional differentiation of our basis function, we employ the follow-

ing relations [56] as:



−1I



 {(1 + 
) 


,





(
)} =

Ɖ(
 + 
 + 1)
Ɖ(
 + 
 + 
 + 1) (1 + 
) 
+
 

−
,
+





(
),

(2.8)

and






I


1 {(1 − 
)


,





(
)} =

Ɖ(
 + 
 + 1)
Ɖ(
 + 
 + 
 + 1) (1 − 
)
+
 

+
,
−


(2.9)
(
) denotes the standard Jacobi Polynomials of order

(
),




where 0 < 
 < 1, 
 > −1, 
 > −1 and 

, 

n and parameters 
 and 
. It is worth mentioning that
ˆ 


 (
)









I



 { 
 (
)} =

1
Ɖ(
)

(
 − 
)1−


and






I



 { 
 (
)} =

1
Ɖ(
)




ˆ 







,


 ∈ [
, 
],


 (
)

(
 − 
)1−




,


 ∈ [
, 
].

By substituting 
 = 
 and 
 = −
, we can simplify equations (2.8) and (2.9), thereby we have:



−1I



 {(1 + 
)−
 

,−





(
)} =

Ɖ(
 − 
 + 1)
Ɖ(
 + 1) 

(
),


 ∈ [−1, 1]

(2.10)

20

and





1 {(1 − 
)−
 
−
,


I





(
)} =

Ɖ(
 − 
 + 1)
Ɖ(
 + 1) 

(
),


 ∈ [−1, 1].

(2.11)

Accordingly, we have the fractional derivative of Legendre polynomial by differentiating (2.10) and
(2.11) as

−1D



 

(
) =

Ɖ(
 + 1)
Ɖ(
 − 
 + 1) 
 
,−





(
) (1 + 
)−


and

where 

(
) = 
 0,0




1 

(
) =

Ɖ(
 + 1)
Ɖ(
 − 
 + 1) 
 −
,


(
) (1 − 
)−
,

D

(
) represents Legendre polynomial of degree n.




2.3 Problem Definition

(2.12)

(2.13)

Let 
 : R
+1 → R for some positive integer 
 and  = [0, 
]×[
1, 
1]×[
2, 
2]×· · ·×[

, 

],

where

0D2



 
 +









=1

(cid:2)


 

D2


(cid:2)

 
 
 
 D2
 



 







 + 


 

D2







(cid:3)

(cid:3) + 
 
 = 
 ,

−


 + 

 
 
 
 D2
 


(2.14)
, and 

 
 are all constant. Besides, 2

 ∈ (0, 1), 2
 
 ∈ (1, 2), and 2
 ∈ (0, 2),
and 
, 



2
 ࣔ 1, for 
 = 1, 2, · · · , 
. This equation is subject to the following Dirichlet initial and boundary
conditions as:

, 


 , 

 



=1


 



|
=0 = 0,

|
=0 =









|
 
 =
 
 = 
|
 
 =
 
 = 0,


 ∈ (0, 1/2),
|
=0 = 0,

 
 ∈ (1/2, 1),


 ∈ (1/2, 1),


 = 1, 2, · · · , 
.

2.3.1 Stochastic Interpretation of the FPDEs

Following [59], we provide a brief stochastic interpretation of the FPDEs in (2.14) that further
sheds light on the well-posedness of the problem from the perspective of probability theory. Let

21

suppose that in (2.14), 
 ≡ 0 and 
 = 0 and 0 < 2
 < 1 and that 

 = −∞ and 

 = +∞ for

 = 1, 2, · · · . Then (2.14) governs [59] a time-changed Lévy process 
(

) on R
 whose Fourier
transform is E[
−

·
(
)] = 


(
) with the Fourier symbol


(
) = −

[


(


)2

 + 


(−


)2

] +

[


(−


)2

 + 


(


)2

].

(2.15)






=1






=1

Recalling that in one dimension the Lévy process 
(
) with Fourier Transform E[
−


(
)] =



0(
) where 
0(
) = 

(

)
 + 

(−

)
 for 
 > 0 and 1 < 
 ≤ 2, 
 ≥ 0, 
 ≥ 0, and

 + 
 = 1 is a stable Lévy process with index 
 and skewness 
 − 
 [38, 59].

Then 
(
) has 
 independent components, each of which is the sum of two independent stable
Lévy processes with index 2

 and 2

, respectively. Furthermore, for a bounded domain '
,
letting 
(cid:48)(
) denote the modification of the process 
(
) that vanishes the first time it leaves the
domain, (2.14) governs the time-changed Markov process 
(cid:48)(

). You can find complete details
in [38].

2.4 Petrov-Galerkin Mathematical Formulation

We introduce the underlying solution and test spaces with their proper norms. Moreover, we
provide some lemmas in order to prove the well-posedness of the problem in addition to constructing
the spatial basis/test functions and performing the discrete stability and convergence analysis of the
PG spectral method.

2.4.1 Mathematical Framework
We first recall the definition of the Sobolev space for real 
 ≥ 0 from [70, 107]. Let



(R) =(cid:8)
 ∈ 
2(R)| (1 + |
|2) 


2F (
)(
) ∈ 
2(R)(cid:9),

endowed with the norm (cid:107)
(cid:107)

(R) = (cid:107)(1 + |
|2) 

transform of 
. For bounded domain 
 = (0, 
), we define



(
) =(cid:8)
 ∈ 
2(
) | ∃ ˜
 ∈ 

(R) 
.
. ˜
|
 = 
(cid:9),

(2.16)
2 
(
)(
)(cid:107) 
2(R), where F (
) is the Fourier

(2.17)

22

inf

˜
∈

(R), ˜
|
 =


associated with (cid:107)
(cid:107)

(
) =
0 (
) be the spaces of
smooth functions with compact support in (0, 
] and [0, 
), respectively. Then, denoted by 
 

(
)
and 
 

(
) are the closure of 0
∞(
) and 
∞
0 (
) with respect to the norm (cid:107) · (cid:107)

(
) in (0, 
] and
[0, 
), respectively. Here, we recall from [107, 108] that

(cid:107) ˜
(cid:107)

(R). Let 0
∞(
) and 
∞

| · |

(
) ≡ | · |
 

(
) ≡ | · |
 

(
),

(2.18)

where ” ≡ ” denotes equivalence relation and |·|
 

(
) = (cid:107) 0D



(·)(cid:107) 
2(
).
Take ' = (
, 
). 

(') denotes the usual Sobolev space associated with the real index 
 ≥ 0
2 on the bounded interval ', and equipped with the norm (cid:107) · (cid:107)

('). In [109], it has


 (·)(cid:107) 
2(
), |·|
 

(
) = (cid:107) 
D


and 
 ࣔ 
 − 1
been shown that the following norms are equivalent:

(cid:107) · (cid:107)

(') ≡ (cid:107) · (cid:107)
 

(') ≡ (cid:107) · (cid:107)
 

(') ≡ | · |∗

(cid:17) 1



('),

(cid:16)(cid:107) 
D


(cid:16)(cid:107) 
D


(2.19)

(cid:17) 1

2 ,


2(')+(cid:107)·(cid:107)2


2(')


2(')

2 , (cid:107)·(cid:107)
 

(') =


 (·)(cid:107)2
2 . It follows from Lemma 5.2 in [108] that

 (·)(cid:107) 1


 (·)(cid:107) 1

2

2
 

(') = (cid:107) 
D

2. Then, the norms (cid:107) · (cid:107)
 

(') and (cid:107) · (cid:107)
 

(') are


2(') (cid:107) 
D


2

2(').

(2.20)



(') = |(
D


where (cid:107)·(cid:107)
 

(') =
and | · |∗


 (·)(cid:107)2

 (·), 
D



(') ≡ | · | 1
| · |∗


2(')+(cid:107)·(cid:107)2

 (·))'| 1

 

(') | · | 1
Lemma 2.4.1. Let 
 ≥ 0 and 
 ࣔ 
 − 1
equivalent to (cid:107) · (cid:107)
 

(') in space 
∞
(cid:16)(cid:107) 
D


(cid:107) · (cid:107)
 

(') =

2

0 ('), where

 (·)(cid:107)2

Proof. See Appendix A.

In the usual Sobolev space, for 
 ∈ 

(') we define |
|∗



(') = |(
D



 
, 
D







('), assuming
Denoted by 
 



sup

|(
D


∈

(')
0 (') and 
 




 
, 
D

0 (') are the closure of 
∞


)'| 1




2 > 0 ∀
 ∈ 

(').

and (cid:107) · (cid:107)
 

(')in ', respectively, where 
∞
support in '.

0 (') with respect to the norms (cid:107)·(cid:107)
 

(')
0 (') is the space of smooth functions with compact

23


2(') + (cid:107) 
D



 (·)(cid:107)2


2(') + (cid:107) · (cid:107)2


2(')

(cid:17) 1

2 .

(2.21)


)| 1
' ∀
 ∈
2

Lemma 2.4.2. For 
 ≥ 0 and 
 ࣔ 
 − 1
seminorms are equivalent to | · |∗
of 
∞

2, 
 




('), where 
 



0 ('), 
 


0 ('), 
 



0 ('), and 



0 ('), and 




0 (') are equal and their
0 (') denotes the closure

0 (') with compact support on ' with respect to the norms (cid:107) · (cid:107)
 

(') and (cid:107) · (cid:107)
 

(').

0 (') and 
 


0 (') and 




Proof. In [108, 109], it has been shown that the spaces 
 


0 (') is equal with 
 


similar steps, we can show that 



seminorms are equivalent.

 
, 
D


 
, 
D


 
)'

(cid:12)(cid:12) > 0, Lemma 2.4.2 directly respectively
(cid:12)(cid:12) ≥ 
2 |
|
 

(') |
|
 

('), where 
1 and 
2 are positive constants and inde-

(cid:12)(cid:12) and(cid:12)(cid:12)(
D

(cid:12)(cid:12) ≥ 
1 |
|
 

(') |
|
 

(') and

Assuming(cid:12)(cid:12)(
D

results in(cid:12)(cid:12)(
D

(cid:12)(cid:12)(
D


0 (') are equal. Following
0 (') and the corresponding


)'

)'


, 
D



 
)'














, 
D

pendent of 
.

Let '1 = (
1, 
1), '
 = (

, 

) × '
−1 for 
 = 2, · · · , 
, and X1 = 


0 ('1) with the associated

1

norm (cid:107) · (cid:107)
 

1('1). Accordingly, we construct X
 such that

X2 = 


...

X
 = 


(cid:26)

(cid:107) · (cid:107)X
 =

(cid:107) · (cid:107)2


2
0




0

(cid:16)(
2, 
2); 
2('1)(cid:17) ∩ 
2((
2, 
2);X1),
(cid:16)(

, 

); 
2('
−1)(cid:17) ∩ 
2((

, 

);X
−1),

2(cid:16)(

 ,

);X
−1

 


(cid:16)(

 ,

);
2('
−1)(cid:17) + (cid:107) · (cid:107)2
(cid:17) + (cid:107) · (cid:107)2

2 for 
 = 1, · · · , 
. Then

2('
) + (cid:107) 

D





 (·)(cid:107)2

(·)(cid:107)2

.

2

(cid:17)(cid:27) 1
(cid:27) 1

2

(cid:26) 



(cid:16)(cid:107) 

D


(cid:26)

(cid:107) · (cid:107)X2 ≡

(cid:107) · (cid:107)2


 

2

(cid:107) · (cid:107)X
 ≡

(2.24)
Proof. X1 is endowed with the norm (cid:107) · (cid:107)X1, where (cid:107) · (cid:107)X1 ≡ (cid:107) · (cid:107)

1('1) (see Lemma 2.4.1).
Moreover, X2 is associated with the norm


2('
)


2('
)


=1





.

(cid:16)(
2,
2);
2('1)(cid:17) + (cid:107) · (cid:107)2


2(cid:16)(
2,
2);X1

(cid:17)(cid:27) 1

2

,

(2.25)

associated with the norm

Lemma 2.4.3. Let 

 > 0 and 

 ࣔ 
 − 1

(2.22)

(2.23)

24

where

(cid:107)
(cid:107)2


 

2

ˆ 
2

| 
2D
2

2


|2 

2


2


|2 

2 +

| 
2D
2
2
(cid:17)


1

2('2) + (cid:107) 
2D
2
2 (
)(cid:107)2


2
|
|2 

2
(
)(cid:107)2

+


1

(cid:16)(
2,
2);
2('1)(cid:17) =

(cid:16)ˆ 
2
ˆ 
1
ˆ 
2
= (cid:107) 
2D
2

2
(cid:16)ˆ 
1
(cid:17) =
= (cid:107) 
1D
1

1


2


|2 

1 +
| 
1D
1
1

1

2('2) + (cid:107) 
1D
1
1

(cid:107)2


2('2) + (cid:107)
(cid:107)2
ˆ 
1


2('2),

(cid:17)

|
|2 

1



2


1

(cid:27) 1

2

.

(2.26)

ˆ 
1


|2 

1 +

| 
1D
1

1

1

2('2) + (cid:107)
(cid:107)2

(cid:107)2
(cid:17) + (cid:107) · (cid:107)2


2('2).


2('
−1)


2('
−1) + (cid:107) 

D





 (·)(cid:107)2


2('
−1)

and
(cid:107)
(cid:107)2

Now, we assume that


2

ˆ 
2


2(cid:16)(
2,
2);X1
(cid:26) 
−1


 


(cid:16)(

 ,

);
2('
−1)(cid:17)
(cid:16)ˆ 



(cid:16)(cid:107) 

D



(cid:107)
(cid:107)2
ˆ

(·)(cid:107)2

(cid:107) · (cid:107)X
−1 ≡


=1





Then,

=
'
−1
= (cid:107) 

D









(
)(cid:107)2

and


2(cid:16)(

 ,

);X
−1

(cid:107)
(cid:107)2


|2 


 +
| 

D






2('
) + (cid:107) 

D



−1

(cid:17) =

−1



=1

=


=1

'


(cid:16)ˆ
(cid:16)(cid:107) 

D



(cid:1)





'1

25

ˆ 




|2 


 +

| 

D






2('
) + (cid:107)
(cid:107)2


2('
),






 (
)(cid:107)2


|2
'
 +

ˆ

'


| 

D






| 

D






(cid:17)

|
|2 




ˆ 








'
−1


|2
'


ˆ

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


(cid:107)2


2('
)


2('
).

|
|2
'


'


Therefore, (2.24) arises from (2.26).
In Lemma 2.8 in [109], it is shown that if 
, 
 ∈ 



(cid:0) 
1D2



1


, 
(cid:1)

'1

= (cid:0) 
1D



1


, 
1D


1

we generalize this lemma for the corresponding (1+d)-D case.






2('
) + (cid:107) 

D



(cid:107)2

, 
(cid:1)

, and (cid:0) 
1D2


0 ('1) for 0 < 2
 < 2 and 2
 ࣔ 1, then
. Here,

1


, 
1D


1

= (cid:0) 
1D



(cid:1)

'1

'1


1

Lemma 2.4.4. If 0 < 2

 < 2 and 2

 ࣔ 1 for 
 = 1, · · · , 
, and 
, 
 ∈ X
, then(cid:0) 

D2


(cid:0) 

D



, and(cid:0) 

D2



=(cid:0) 

D




, 
(cid:1)


, 

D







(cid:1)


(cid:1)


, 

D






'


'


'


















.


, 
(cid:1)

=

'


Proof. See Appendix A.

Additionally, in the light of Lemma 2.4.2, we can prove that

|(cid:0) 

D








, 

D










'


≡ |
|


(cid:1)

(cid:1)
| + |(cid:0) 

D


(cid:16)(

 ,

);
2('
−1)(cid:17)(|(cid:0) 

D


(cid:16)(

, 

); 
2('
−1)(cid:17)

sup




0


∈
 




0

where we assume

|

'



, 

D







 


(cid:16)(

 ,

);
2('
−1)(cid:17) |
|

 


(cid:16)(

 ,

);
2('
−1)(cid:17) , (2.27)

(cid:1)
|+|(cid:0) 

D



(cid:1)


, 

D







, 

D






'


'










|) >

(cid:0)


 =(cid:0)


 
, 
(cid:1)

. Next, we study the property of the fractional time deriva-

0 for any 
 ∈ 


tive in the following lemmas.
Lemma 2.4.5. If 0 < 2
 < 1 (1 < 2
 < 2) and 
, 
 ∈ 

(
), when 
|
=0(= 


0D2

Proof. See [70].
Lemma 2.4.4 and 2.4.5 will help us obtain the corresponding weak form of (2.14). Let 2
 ∈ (0, 1)
and  = 
 × '
. We define


, where 
 = (0, 
).



 |
=0) = 0, then


 
(cid:1)


 
, 
D


0D


:=


 | (cid:107)
(
, ·)(cid:107) 
2('
) ∈ 

(
), 
|
=0 = 
|

=

 = 
|

=

 = 0, 
 = 1, · · · , 


(2.28)
which is equipped with the norm (cid:107)
(cid:107)
 

(
;
2('
)). For real 0 < 2
 < 1, 
 

(
; 
2('
)) is
associated with the norm (cid:107) · (cid:107)
 

(
;
2('
)), which is defined as

(cid:13)(cid:13)(cid:13) (cid:107)
(
, ·)(cid:107) 
2('
)

(cid:13)(cid:13)(cid:13)
 

(
).

(cid:107)
(cid:107)
 

(
;
2('
)) =

0

(cid:16)





; 
2('
)(cid:17)

(cid:110)

(cid:111)

,

26

Therefore, we have

(cid:107)
(cid:107)
 

(
;
2('
)) =

=

=

(cid:13)(cid:13)(cid:13) (cid:107)
(
, ·)(cid:107) 
2('
)
(cid:18)(cid:0)ˆ
(cid:26)ˆ 

(cid:16)(cid:107) 0D


'


 (
)(cid:107)2

0

(cid:13)(cid:13)(cid:13)
 

(
)

 
|2 
'
(cid:1) 1

2

| 0D


2() + (cid:107)
(cid:107)2


2()

(cid:19)2
(cid:17) 1

2 .



 +

(cid:27) 1

2

ˆ 


ˆ

0

'


|
|2 
'
 



(2.29)

Similarly, we define

0

(cid:16)





; 
2('
)(cid:17)

:=

(cid:110)


 | (cid:107)
(
, ·)(cid:107) 
2('
) ∈ 

(
), 
|
=
 = 
|

=

 = 
|

=

 = 0, 
 = 1, · · · , 


(cid:111)

,

which is equipped with the norm (cid:107)
(cid:107)
 

(
;
2('
)). Following (2.29),

(cid:13)(cid:13)(cid:13)
 

(
)

(cid:13)(cid:13)(cid:13) (cid:107)
(
, ·)(cid:107) 
2('
)
(cid:16)(cid:107) 
D


 
(cid:1)


 (
)(cid:107)2


2() + (cid:107)
(cid:107)2
0

(
; 
2('
)), then

(cid:17) 1

2 .


2()

0D



 
, 
D


 ∀
 ∈ 


0

(
; 
2('
))

(cid:107)
(cid:107)
 

(
;
2('
)) =

=


 
, 
(cid:1)

 =(cid:0)

0D2


Lemma 2.4.6. If 0 < 2
 < 2, 2
 ࣔ 1 and 
 ∈ 


(2.30)

(2.31)

(2.32)

(2.33)

(cid:0)

 
, 
(cid:1)

(cid:0)

.
Proof. Following Lemma 2.4.5,
ˆ 

= (cid:0)

0D2


 =

ˆ

0D2

'


 
, 
D



 
(cid:1)

0
0D


.




 
 
 
'
 

 =

ˆ

ˆ 


'


0

0D



 
 
D



 
 
'
 



0

(
; 
2('
)) and 2
 ∈ (0, 1), |(0D



 
, 
D



 
)| ≤

Lemma 2.4.7. For 
 ∈ 

(cid:107)
(cid:107)
 

(
;
2('
)) (cid:107)
(cid:107)
 

(
;
2('
)) ∀
 ∈ 

Proof.

|(0D



 
, 
D



 
)| =

| 0D



 
 
D



 
| 


'


(cid:17)

0

(
; 
2('
)) .
(cid:16)ˆ

ˆ 


'


0

27

By Hölder inequality,

(cid:17) 1

2

(2.34)

2(cid:16)ˆ
(cid:17) 1


 
)|
| 0D

| 0D

| 
D


0

'


|(0D



 
, 
D

ˆ 

ˆ 

ˆ 


≤(cid:16)ˆ
≤(cid:16)ˆ
×(cid:16)ˆ
(cid:16)(cid:107) 0D


0

 
(cid:107)2

'


'


=

0

'


ˆ 

ˆ 

ˆ 


0


 
|2 


'


ˆ

 
|2 


'
 +
ˆ
2(cid:16)(cid:107) 
D

(cid:17) 1


 
|2 


'
 +

'


'


0


 
|2 


'


2

0
|
|2 


'


| 
D


(cid:17) 1
(cid:17) 1

2() + (cid:107)
(cid:107)2

|
|2 


'



 
(cid:107)2

2


2() + (cid:107)
(cid:107)2


2()


2()

(cid:17) 1

2

= (cid:107)
(cid:107)
 

(
;
2('
)) (cid:107)
(cid:107)
 

(
;
2('
)).

Lemma 2.4.8. For any 
 ∈ 

independent of 
 and 
 such that

0

(
; 
2('
)) and 2
 ∈ (0, 1), there exists a constant 
 > 0 and

0ࣔ
∈


sup

0

(
;
2('
))
|(0D


 
, 
D



 
, 
D


|(0D


 
)|
|
|
 

(
;
2('
))

 
)| > 0 ∀
 ∈ 


0

(
; 
2('
)).

≥ 
|
|
 

(
;
2('
)),

(2.35)

assuming

sup


∈ 

0

(
;
2('
))

Proof. Following Lemma 2.4 in [110] and Lemma 3.1 in [111], for any given 
 ∈ 

let V
 = 
I


 
, 
D



 
(cid:1) under the assumption of

(cid:0) 
 (
) 0D


sup





∈ 

0

(
;
2('
))

 (
) = (1−
(
−
)) and 
(
) is the Heaviside function. ClearlyV
 ∈ 

by Hölder inequality we obtain

0

(
; 
2('
))
|(0D


 
)| > 0, where
0

(
; 
2('
)). Therefore,

|V
|2




0

(
;
2('
)) = (cid:107) 
D

≤ (cid:107) 
 (
)(cid:107)2





2() = (cid:107) 
 (
) 0D


 
(cid:107)2


2() = ˜
1(cid:107) 0D



2(),


 
(cid:107)2


2()

(2.36)


 
(cid:1)(cid:17)(cid:107)2
(cid:0) 
 (
) 0D


(cid:16)

I


2() (cid:107) 0D



 
(cid:107)2




28

where ˜
1 = (cid:107) 
 (
)(cid:107)2


2() and independent of 
 and 
. Using V
, we have

 
, 
D



V
)| =


I


(cid:16)


 
(cid:1)(cid:17)(cid:12)(cid:12) 

 
'





|(0D


ˆ
ˆ

'


'


ˆ 

ˆ 


0

0
ˆ





 
 
D


(cid:12)(cid:12) 0D

(cid:0) 
 (
) 0D

(cid:12)(cid:12) 0D


 
(cid:12)(cid:12)2(cid:12)(cid:12) 
 (
)(cid:12)(cid:12) 

 
'
 .
(cid:12)(cid:12) 0D


 
(cid:12)(cid:12)2 

 
'
 = ¯
1 |
|2

ˆ 


=

According to the second mean value theorem for integrals [112],
ˆ

ˆ 


(cid:12)(cid:12) 0D


 
(cid:12)(cid:12)2(cid:12)(cid:12) 
 (
)(cid:12)(cid:12) 

 
'
 ≥ ¯
1

'


0

'


0

where ¯
1 = lim


−→
−| 
 (
)| = 1. Considering (2.36) and (2.37), we obtain
≥ ¯
1
˜
1

≥ |(0D


V
)|
|V
|
 

(
;
2('
))

|(0D


 
)|
|
|
 

(
;
2('
))


 
, 
D



 
, 
D


sup

0ࣔ
∈


0

(
;
2('
))
¯
1
˜
1

where 
 =

> 0 and independent of 
 and 
.




0

(
;
2('
)), (2.37)

|
| 

0

(
;
2('
)),

(2.38)

2.4.2 Solution and Test Function Spaces
Let 2
 ∈ (0, 1) and 2

 ∈ (1, 2) for 
 = 1, · · · , 
. We define the solution space

B
,
1,··· ,

() := 



; 
2('
)(cid:17) ∩ 
2(
;X
),

(cid:107)
(cid:107)B
,
1,··· ,

 () =


 

(
;
2('
)) + (cid:107)
(cid:107)2


2(
;X
)

(cid:111) 1

2 ,

(2.39)

(2.40)

(2.41)

(cid:1)(cid:111) 1

2 .

endowed with the norm

where due to (2.23) and Lemma 2.4.3,

(cid:107)
(cid:107) 
2(
;X
) =

(cid:13)(cid:13)(cid:13) (cid:107)
(
, .)(cid:107)X

(cid:110)(cid:107)
(cid:107)2
(cid:110)(cid:107)
(cid:107)2


2() +

=

Therefore, by (2.29) and (2.41),

0

(cid:16)
(cid:110)(cid:107)
(cid:107)2
(cid:13)(cid:13)(cid:13)
2(
)



(cid:0)(cid:107) 

D








=1

29

(cid:107)
(cid:107)B
,
1,··· ,

 () =


2()+(cid:107) 0D



 (
)(cid:107)2


2()+


2()+(cid:107) 

D





 (
)(cid:107)2


2()



 (
)(cid:107)2


2()

(
)(cid:107)2


2() + (cid:107) 

D






(cid:0)(cid:107) 

D



(
)(cid:107)2






=1

(cid:1)(cid:111) 1

2 .

(2.42)

Likewise, we define the test space



,
1,··· ,

() := 


endowed with the norm

(cid:107)
(cid:107)

,
1,··· ,

 () =
=

0

(cid:16)


; 
2('
)(cid:17) ∩ 
2(
;X
),
(cid:111) 1

(cid:110)(cid:107)
(cid:107)2
(cid:110)(cid:107)
(cid:107)2

 

(
;
2('
)) + (cid:107)
(cid:107)2




2() + (cid:107) 
D


 (
)(cid:107)2
(cid:0)(cid:107) 

D



2() + (cid:107) 

D


(
)(cid:107)2


2(
;X
)

2()





+

2 .


=1



 (
)(cid:107)2


2()

(2.43)

(2.44)

(cid:1)(cid:111) 1

2 .

Remark 2.4.9. If 2
 ∈ (0, 1), our method is essentially Galerkin in the ∞-dimensional space. Yet
in the discretization, we choose two different subspaces as basis and test spaces, leading to the PG
spectral method; that is, 

 ⊂ B
,
1,··· ,

() and 

 ⊂ 

,
1,··· ,

() such that 

 ࣔ 

.

In case 2
 ∈ (1, 2), we define the solution space as

0,0

(cid:16)


; 
2('
)(cid:17) ∩ 
2(
;X
),

B
,
1,··· ,

() := 


where

0,0

(cid:16)





; 
2('
)(cid:17)

(cid:110)

:=


 | (cid:107)
(
, ·)(cid:107) 
2('
) ∈ 

(
),
|
=0 = 
|
=0 = 
|

=

 = 
|

=

 = 0, 
 = 1, · · · , 









which is associated with (cid:107) · (cid:107)B
,
1,··· ,

 (). The corresponding test space is also defined as

0,0

(cid:16)


; 
2('
)(cid:17) ∩ 
2(
;X
),



,
1,··· ,

() := 


where

0,0

(cid:16)





; 
2('
)(cid:17)

(cid:110)

:=


 | (cid:107)
(
, ·)(cid:107) 
2('
) ∈ 

(
),
|
=
 = 
|
=
 = 
|

=

 = 
|

=

 = 0, 
 = 1, · · · , 









30

(2.45)

(cid:111)

,

(2.46)

(cid:111)

,

which is endowed with (cid:107) · (cid:107)

,
1,··· ,

 (). It should be noted that similar to Lemma 2.4.7, for any

 ∈ 


0,0

(cid:0)
; 
2('
)(cid:1) and 2
 ∈ (1, 2), we obtain

 

(cid:16)

|(0D

|
|

0ࣔ
∈ 


sup

0,0

(
;
2('
))

 
, 
D


|(0D


assuming

sup


∈ 

0,0

(
;
2('
))


 
, 
D



 
)|


;
2('
)(cid:17) ≥ |
|


 

(cid:16)


;
2('
)(cid:17) ,


 
)| > 0 ∀
 ∈ 


0,0

(
; 
2('
)).

(2.47)

2.4.3 Petrov-Galerkin Method
Let 
 ∈ B
,
1,··· ,

() and  = (0, 
) × (
1, 
1) × (
2, 
2) × · · · × (

, 

), where 
 is a positive
integer. We define the corresponding bilinear form as


) + 


(

D 







, 

D 







)

(cid:3)


, 

D 







, 
 
 D
 


 



) + 

 
 (
 
 D
 


 



, 
 
 D
 


 



)

(cid:3)


(
, 
) = (0D


+









=1


 
, 
D



 
)

(cid:2)


(

D 


(cid:2)

 
 (
 
 D
 



 






−

=1
+
(
, 
),

(2.48)
, and 


 are all constant. 2
 
 ∈ (0, 1), 2
 
 ∈ (1, 2), and 2
 ∈ (0, 2) for

, 


 , 




where 
, 




 = 1, 2, · · · , 
. Now, the problem reads as: find 
 ∈ B
,
1,··· ,

() such that


(
, 
) = ( 
 , 
), ∀
 ∈ 

,
1,··· ,

(),

(2.49)
where 
(
, 
) is a continuous bilinear form and 
 ∈ (B
,
1,··· ,

)∗(), which is the dual space of
B
,
1,··· ,

(). It should be noted that (0D2


 
) is proven in Proposition
1 in [113] and later in [70] requiring less regularity and constraint.


 
, 
) = (0D



 
, 
D


Remark 2.4.10. In case 
 < 1
2, the solution to the bilinear form in (2.48) does not lead to the
homogeneous initial condition in the strong form. To guarantee the equivalence between the
problem under the strong formulation and the bilinear form, we assume that 
 in (2.14) posses
enough regularity [114].

31

2.5 Well-posedness Analysis

Based upon the Lemmas provided in Section 2.4, we are able to prove the well-posed and further

the stability of the problem (2.49) in the following theorems.
Assumption 1. For any 
 ∈ 

,
1,··· ,

() with 
 ࣔ 0, we assume that



 



∈B
,
1,··· ,

 ()

∈B
,
1,··· ,

 ()
Lemma 2.5.1. (Continuity) The bilinear form in (2.48) is continuous, i.e., for 
 ∈ B
,
1,··· ,

(),

> 0 for 
 = 1, · · · , 
.


, 
 
 D
 


 



, 
 
 D
 


 


> 0 and

)| + |(
 
 D
 


 




 



)|(cid:17)


 
)|(cid:17)


 
, 
D


(cid:16)|(0D

(cid:16)|(
 
 D
 



 


∃
 > 0,

|
(
, 
)| ≤ 
 (cid:107)
(cid:107)B
,
1,··· ,

 ()(cid:107)
(cid:107)

,
1,··· ,

 () ∀
 ∈ 

,
1,··· ,

().

(2.50)

Proof. The proof follows easily using (2.27) and Lemma 2.4.7.

Theorem 2.5.2. The inf-sup condition for the bilinear form, defined in (2.48) when 
 = 1, i.e.,

inf

0ࣔ
∈B
,
1()

sup

0ࣔ
∈

,
1()

|
(
, 
)|

(cid:107)
(cid:107)

,
1()(cid:107)
(cid:107)B
,
1() ≥ 
 > 0,

(2.51)

holds with 
 > 0, where  = 
 × '1 and Assumption 1 holds.

Proof. It is evident that 
 and 
 are in Hilbert spaces (see [108, 109]). By Assumption 1, we can
prove that


 
)| +(cid:16)|(
1D 
1
1

|
(
, 
)| ≡ |(0D



 
, 
D


+(cid:16)|(
1D
1
1


, 
1D
1

1


)| + |(
1D
1

1


)| + |(
1D 
1
1


, 
1D 
1

1

, 
1D
1
1

Next, by (2.27) and Lemma 2.4.8 we obtain

 (
))|

|(0D



 (
), 
D

|
|

,
1()

sup

0ࣔ
∈

,
1()

≡


, 
1D 
1
1


)|(cid:17) + |(
, 
)|.

 

(cid:16)(0,
);
2('1)(cid:17)


 (
), 
D



 (
))|

|(0D

|
|


)|(cid:17)

(2.52)

(2.53)

0ࣔ
∈

,
1()

sup


 

(cid:16)(0,
);
2('1)(cid:17) ,

≥ 
1|
|

32

sup

0ࣔ
∈

,
1()

sup

0ࣔ
∈

,
1()

|(
1D
1
1 (
), 
1D
1

1

(
))| + |(
1D
1

1
|
|

,
1()
|(
1D
1
1 (
), 
1D
1
(
))| + |(
1D
1

1

1
|
|

 

1

'1;
2(
)(cid:17)

(cid:16)

(
), 
1D
1
1 (
))|

≡

(
), 
1D
1
1 (
))|

≥ 
2|
|


 

1

(cid:16)

'1;
2(
)(cid:17) ,

sup

and
dent and Assumption 1 holds. Therefore, for 
 ∈ B
,
1,··· ,

()

0ࣔ
∈

,
1()

≥ (cid:107)
(cid:107) 
2(), where 
1, 
2,and 
3 are positive constants and indepen-

|(
,
)|
(cid:107)
(cid:107)
2()

|
(
, 
)|
(cid:107)
(cid:107)

,
1()
|(0D


sup

0ࣔ
∈

,
1()

≥ ¯


sup

0ࣔ
∈

,
1()


 (
), 
D



 (
))| + |(
1D
1
1 (
), 
1D
1

1

(
))|

(cid:107)
(cid:107)

,
1()

+ ¯


≥ ¯
 ˜


|(
1D
1

1

(cid:110)|
|

0ࣔ
∈

,
1()

sup


 

(cid:16)(0,
);
2('1)(cid:17) + |
|

(
), 
1D
1
1 (
))| + |(
, 
)|

(cid:107)
(cid:107)

,
1()

(cid:16)

'1;
2(
)(cid:17) + (cid:107)
(cid:107) 
2()


 

1

(cid:111)

,

where ˜
 is 


{
1, 
2, 
3, 
4} and ¯
 is a positive constant from (2.52). Accordingly,

sup

0ࣔ
∈

,
1()

|
(
, 
)|
(cid:107)
(cid:107)

,
1()

≥ 
 (cid:107)
(cid:107)B
,
1(),

where 
 = ¯
 ˜
 is a positive constant and independent of 
 , 
1, and 
.

(2.54)

(2.55)

In Theorem 2.5.3, we extend the proof of Theorem 2.5.2 to the corresponding (1+d)-dimensional

problem in (2.48).

Theorem 2.5.3. The inf-sup condition of the bilinear form, defined in (2.48) for any 
 ≥ 1, i.e.,
(cid:107)
(cid:107)

,
1,··· ,

 ()(cid:107)
(cid:107)B
,
1,··· ,

 () ≥ 
 > 0, (2.56)

0ࣔ
∈B
,
1,··· ,

 ()

0ࣔ
∈

,
1,··· ,

 ()

|
(
, 
)|

sup

inf

holds with 
 > 0, where  = 
 × '
 and Assumption 1 holds.

33

Proof. By Assumption 1,

|
(
, 
)| ≡ |(0D



 
)| +


, 

D 







)| + |(

D 










(cid:16)|(

D 








 
, 
D


(cid:16)|(
 
 D
 



 







=1

+


=1

, 
 
 D
 


 



)| + |(
 
 D
 


 



, 
 
 D
 


 



)|(cid:17)

)|(cid:17) + |(
, 
)|.


, 

D 






(2.57)

It follows from (2.27) that for 
 ∈ B
,
1,··· ,

() and 
 ∈ 

,
1,··· ,

() we have

(
))| + |(

D






(
), 

D





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

D







(
)(cid:107) 
2() + (cid:107) 

D







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

D





 (
)(cid:107) 
2()



 (
))|(cid:17)



 (
))|(cid:17)
(cid:17)

Thus


=1










=1
≥ ˜
1



 (
), 

D






(cid:16)|(

D


(cid:16)(cid:107) 

D





(cid:16)|(

D


(cid:16)(cid:107) 

D





(cid:16)(cid:107) 
 
 D
 




×
= ˜
1|
|
2(
;X
) |
|
2(
;X
).



 (
), 

D







=1
≥ ˜
1


=1


=1


 


(
))| + |(

D






(
), 

D





 (
)(cid:107) 
2() + (cid:107) 

D







(
)(cid:107) 
2()
(cid:17)

(
)(cid:107) 
2(),+(cid:107) 
 
 D
 



 
 (
)(cid:107) 
2()

(cid:17)

.

(2.58)

(2.59)

where ˜
1 is a positive constant and independent of 
 and 

. Considering Lemma 2.4.8, there exists
a positive constant ˜
2 > 0 independent of 
 and 
 such that

sup

0ࣔ
∈

,
1()

|(0D



 (
))|


 (
), 
D

|
|

,
1,··· ,

 ()

≡

|(0D

|
|


 (
), 
D



 

(cid:16)(0,
);
2('
)(cid:17)


(
))|

sup

0ࣔ
∈

,
1,··· ,

 ()


 

(cid:16)(0,
);
2('
)(cid:17) .

≥ ˜
2|
|

34

It follows from (2.58) and (2.59) that

sup

0ࣔ
∈

,
1,··· ,

 ()
≥ ¯


sup

0ࣔ
∈

,
1,··· ,

 ()

|
(
, 
)|

(cid:107)
(cid:107)

,
1,··· ,

 ()

 
, 
D


|(0D


 
)|



(cid:107)
(cid:107)

,
1,··· ,

 ()

(cid:16)|(
 
 D
 



 



, 
 
 D
 


 



)| + |(
 
 D
 


 


+ ¯



=1

sup

0ࣔ
∈

,
1,··· ,

 ()

(cid:16) ˜
2(cid:107)
(cid:107)
 

(
;
2('
)) + ˜
1(cid:107)
(cid:107) 
2(
;X
) + (cid:107)
(cid:107) 
2()

(cid:107)
(cid:107)

,
1,··· ,

 ()

(cid:17)

≥ ¯

≥ ¯
 ¯
 (cid:107)
(cid:107)B
,
1,··· ,

 (),


)|(cid:17)


, 
 
 D
 


 


(2.60)

where ¯
 = 


{ ˜
2, ˜
1} and also ¯
 > 0 form (2.57). Therefore,

inf

0ࣔ
∈B
,
1,··· ,

 ()

sup

0ࣔ
∈

,
1,··· ,

 ()

|
(
, 
)|

(cid:107)
(cid:107)B
,
1,··· ,

 ()(cid:107)
(cid:107)

,
1,··· ,

 ()

≥ 
,

(2.61)

where 
 = ¯
 ¯
 is a positive constant and independent of 
 and 
.
Theorem 2.5.4. (well-posedness) For all 0 < 
 < 2, 2
 ࣔ 1, and 1 < 2

 < 2, and 
 = 1, · · · , 
,

there exists a unique solution to (2.49), which is continuously dependent on 
 ∈(cid:0)B
,
1,··· ,

(cid:1)ਭ(),
where(cid:0)B
,
1,··· ,

(cid:1)ਭ() is the dual space of B
,
1,··· ,

() and Assumption 1 holds.

Proof. The continuity and the inf-sup condition, which are proven in Lemmas 2.5.1 and Theorem
2.5.3 respectively, yield the well-posedness of the weak form in (2.49) in (1+d)-dimension due to
the generalized Babuška-Lax-Milgram theorem [62].

2.6 Discrete Form Formulation

We choose proper subspaces of B
,
1,··· ,

() and 

,
1,··· ,

() as finite dimensional 


and 

 with dim(

) = dim(

) = 
. Now, the discrete problem reads: find 

 ∈ 

 such that


(

 , 

) = ( 
 , 

), ∀

 ∈ 

 .

(2.62)

By representing 

 as a linear combination of points/elements in 

, i.e., the corresponding
(1 + 
)-dimensional space-time basis functions, the finite-dimensional problem (2.62) leads to a

35

linear system known as Lyapunov system. For instance, when 
 = 1, we obtain the corresponding
Lyapunov equation in the space-time domain (0, 
) × (
1, 
1) as



 U 



1 + 

1
− 

1



 U 




 U 




1,
 + 

1 

 U 



1,


1,
 − 

1 

 U 



1,
 + 


 U 



1 = 
,

(2.63)

where all are defined in 2.7. To find the general form of Lyapunov equation, we can define 

 

 as

−

1



1,
 − 

1 

1,
 + 

1



1,
 + 

1 

1,
 = 

 



1

.

(2.64)

Considering equation (2.64), we obtain the (1+1)-D space-time Lyapunov system as



 U 



1 + 

 U 

 




1

+ 


 U 



1 = 
.

We present a new class of basis and test functions yielding symmetric stiffness matrices.
Moreover, we compute exactly the corresponding mass matrices, which are either symmetric and
pentadiagonal. In the following, we extensively study the properties of the aforementioned matrices,
allowing us to formulate a general fast linear solver.

2.6.1 Space of Basis Functions (

)

We construct the basis for the spatial discretization employing the Legendre polynomials defined
as



( 
 ) = 

(cid:0)

+1(
) − 

−1(
)(cid:1), 
 = 1, 2, · · ·




 
 ∈ [−1, 1],

(2.65)

where 

 = 2 + (−1)
. The definition reflects the fact that for 
 
 ≤ 1/2 and 1/2 ≤ 
 
 ≤ 1, then
both boundary conditions needs to be presented. Naturally, for the temporal basis functions only
initial conditions are prescribed and the basis function for the temporal discretization is constructed
based on the univariate poly-fractonomials [56] as


 (
) = 

(1 + 
)
 
−
 , 


 



−1 (
),


 = 1, 2, · · ·




 
 ∈ [−1, 1],

(2.66)

36

for 
 ≥ 1. With the notation established, we define the space-time trial space to be

(cid:17)(
 
) : 
 = 1, . . . ,N , 
 
 = 1, . . . ,M 


(cid:111)

,

(2.67)

(cid:110)(cid:16)

(cid:17)(
)

(cid:16)






=1



 = 






 ◦ 


 


where 
(
) = 2
/
 − 1 and 
 
(
) = 2



 
 ◦ 
 


−
 


 
−
 


− 1.

2.6.2 Space of Test Functions (

)

We construct the spatial test functions using Legendre polynomial as well as the basis function in
the Petrov-Galerkin method as


(
) =(cid:101)

 (cid:0)

+1(
) − 

−1(
)(cid:1),

where(cid:101)

 = 2 (−1) 
 + 1. Next, we define the temporal test functions using the univariate poly-


 = 1, 2, · · ·




 
 ∈ [−1, 1],

(2.68)

fractonomials

 



 (
) =(cid:101)

 (1 − 
)
 

 ,−


−1 (
),
(cid:110)(cid:16)





(cid:16)

(cid:17)(
)


 ◦ 

 


◦ 
 



 



=1



 = 





and we construct the corresponding space-time test space as


 = 1, 2, · · ·




 
 ∈ [−1, 1],

(2.69)

(cid:17)(
 
) : 
 = 1, . . . ,N , 
 
 = 1, . . . ,M 


(cid:111)

.

(2.70)

Remark 2.6.1. The choices of 

 in (2.65) and (2.66), also(cid:101)

 in (2.68) and (2.69), result in the

spatial/temporal mass and stiffness matrices being symmetric, which are discussed in Theorems
2.7.1, 2.7.2, and 2.7.3 in more details.

2.7 Implementation of PG Spectral Method

We now seek the solution to (2.14) in terms of a linear combination of elements in the space



 of the form



(
, 
) =
N


=1


M1

1=1 · · ·
M




=1 ˆ

,
1,··· ,



37

(cid:104)


 (
)




=1 

 
 (
 
)(cid:105)


 


(2.71)

in . We enforce the corresponding residual



(
, 
1, · · · , 

) = 0D2



 

 +

[


 

D2








 + 


 

D2








]










=1
[

 
 
 
 D2
 



 


−
+ 
 

 − 



=1



 + 

 
 
 
 D2
 



 




]

(2.72)

to be 
2-orthogonal to 

 ∈ 

, which leads to the finite-dimensional variational weak form
(
 
), when 
 = 1, . . . ,N and 
 
 =
in (2.62). Specifically, by choosing 

 =  

1, . . . ,M 
, 
 = 1, 2, · · · , 
, we have


=1 
 



 (
)



N


M1



=1


1=1

(cid:18)


=1



=1

ˆ

,
1,··· ,



{

}
,
{
1}
1,
1 · · · {

}

 ,



M





[


{

}
,
{
1}
1,
1 · · · {


,
}

,

 · · · {

}

 ,








{

}
,
{
1}
1,
1 · · · {


,
}

,

 · · · {

}

 ,

]


 
 {

}
,
{
1}
1,
1 · · · {

 
 ,
}
 
 ,
 
 · · · {

}

 ,



{

}
,
{
1}
1,
1 · · · {

}

 ,



(cid:2)

 
 {

}
,
{
1}
1,
1 · · · {

 
 ,
}
 
 ,
 
 · · · {

}

 ,


(cid:3)

(cid:19)


=1

· · ·

+
+

−

+
+

= 

,
1,··· ,



,

where 

 and 

 denote, respectively, the temporal stiffness and mass matrices whose entries are
defined as

{

}
,
 =

and

(cid:17)(
) 

,

(cid:16)




ˆ 

0 0D

(cid:16)
ˆ 


0


 ◦ 








(cid:17)(
) 
D

(cid:17)(
)(cid:16)


 ◦ 






 ◦ 




(cid:16)
(cid:17)(
) 

.

{

}
,
 =


 ◦ 




38

Moreover, 

 
 and 

 
 , 
 = 1, 2, · · · , 
, are the corresponding spatial stiffness and mass matrices
where the left-sided and right-sided entries of the spatial stiffness matrices are obtained as

{

 
 ,
}
 
 ,
 
 =

{

 
 ,
}
 
 ,
 
 =


 
 D 
 


 



 
 D 
 


 




 
 ◦ 
 




 
 ◦ 
 


ˆ 
 

ˆ 
 



 



 


(cid:16)
(cid:16)

and the corresponding entries of the spatial mass matrix are given by

◦ 
 



 


◦ 
 



 


(cid:16)
(cid:16)


 


(cid:17)(
 
) 
 
 D 
 

(cid:17)(
 
) 
 
 D 
 

(cid:17)(
 
)(cid:16)

◦ 
 



 


(cid:16)


 



 


ˆ 
 


 (
, 
1, · · · , 

)(cid:16)



 
 ◦ 
 


(cid:17)(
)






=1


 ◦ 

 


,


 
 ,
 


(cid:17)(
 
) 

 
 = {

 
 }
 
 ,
 

(cid:17)(
 
) 

 
 = {

 
 }

(cid:17)(
 
) 

 
 .
(cid:16)

(cid:17)(
 
) 
.

◦ 
 



 


{
 
}
 
 ,
 
 =



,
1,··· ,

 =



Moreover, the components of the load vector are computed as

The linear system (2.73) can be exhibited as the following general Lyapunov equation

,

(2.73)

(2.74)

ˆ


=1















=1


=1


=1

(cid:16)

+

+

−

−



 ⊗ 
1 ⊗ 
2 · · · ⊗ 





 ⊗ 
1 ⊗ · · · ⊗ 


,
 ⊗ 

+1 · · · ⊗ 











 

 ⊗ 
1 ⊗ · · · ⊗ 


,
 ⊗ 

+1 · · · ⊗ 





 ⊗ 
1 ⊗ · · · ⊗ 


,
 ⊗ 

+1 · · · ⊗ 











 

 ⊗ 
1 ⊗ · · · ⊗ 


,
 ⊗ 

+1 · · · ⊗ 



(cid:17) U = 
.

+ 
 

 ⊗ 
1 ⊗ 
2 · · · ⊗ 



Let




 × 


,
 + 


 × 


,
 − 


 × 


,
 − 


 × 


,
 = 
 
 






.

(2.75)

39

Considering the fact that all the aforementioned stiffness and mass matrices are symmetric, 


,
,



,
, 


,
, and 


,
 can be replaced by 
 
 



which remains symmetric. Therefore,




(cid:16)

+







 ⊗ 
1 ⊗ 
2 · · · ⊗ 



[ 

 ⊗ 
1 ⊗ · · · ⊗ 

−1 ⊗ 
 
 




=1
+
 

 ⊗ 
1 ⊗ 
2 · · · ⊗ 






(cid:17) U = 
,

(2.76)

⊗ 

+1 · · · ⊗ 

 ]

in which ⊗ represents the Kronecker product, 
 denotes the multi-dimensional load matrix whose
entries are given in (2.73), and U denotes the corresponding multi-dimensional matrix of unknown
coefficients with entries ˆ

,
1,··· ,

.

In the Theorems 2.7.1, 2.7.2, and 2.7.3, we study the properties of the aforementioned matrices.
Besides, we present efficient ways of deriving the spatial mass and the temporal stiffness matrices
analytically and exact computation of the temporal mass and the spatial stiffness matrices through
proper quadrature rules.

Theorem 2.7.1. The temporal stiffness matrix 

 corresponding to the time-fractional order 
 ∈
(0, 1) is a diagonal N × N matrix, whose entries are obtained as

{

}
,
 =(cid:101)

 



Ɖ(
 + 
)
Ɖ(
)

Ɖ(
 + 
)
Ɖ(
)

(cid:17)2
−1

(cid:16) 2




2

2
 − 1



,
,


, 
 = 1, 2, · · · , N .

Moreover, the entries of temporal mass matrices 

 can be computed exactly by employing a
Gauss-Lobatto-Jacobi (GLJ) rule with respect to the weight function (1− 
)
(1+ 
)
, 
 ∈ [−1, 1],
where 
 = 
/2. Moreover, 

 is symmetric.

Proof. See [55].

Theorem 2.7.2. The spatial mass matrix 
 is a pentadiagonal M × M matrix, whose entries are
explicitly given as



,
 = (cid:101)

 



(cid:104)

(cid:105)

.

(2.77)

2

2
 + 3



,
 −

2

2
 + 3



+1,
−1 −

2

2
 − 3



−1,
+1 +

2

2
 − 3



−1,
−1

40

where 
 = 2 
−



−
 − 1 and 
 ∈ (−1, 1). Substituting the spatial basis/test functions, we have

in which

Proof. The (k, r)th-entry of the spatial mass matrix is given by



,
 =

2

2


,
 − (cid:101)



 

 ◦ 
(
) 
 ◦ 
(
) 

 =(cid:0) 
−

´ 

(cid:1)(cid:101)

 

(cid:2)(cid:101)



,
 =(cid:0) 
−


+1,
−1 − (cid:101)

(cid:101)

, 
 =
ˆ 1
−1
(cid:104)
(cid:1)(cid:101)

 


(cid:104)
(cid:1)(cid:101)

 





(
) 
 
(
) 

 =



+1,
−1 −



+1,
−1 −

2
 + 3



,
 −



,
 −

2

2

2

2

2
 + 3

2
 + 3

2
 + 3
as a pentadiagonal matrix. Moreover,

2

Therefore, we have



,
 = (cid:0) 
 − 



,
 = (cid:0) 
 − 


2
= 

,
 .

(cid:1) ´ 1−1 

(
)
(
) 

,
(cid:3),

−1,
+1 + (cid:101)



,


2
2
 + 1



 
 .

2

2
 − 3



−1,
+1 +

2

2
 − 3



,


2

2
 − 3



−1,
+1 +

2

2
 − 3



,


(2.78)

(2.79)

(2.80)

(cid:105)
(cid:105)

Theorem 2.7.3. The total spatial stiffness matrix 

 


computed as:




is symmetric and its entries can be exactly




 × 


,
 + 


 × 


,
 − 


 × 


,
 − 


 × 


,
 = 
 
 






,

(2.81)

where 
 = 1, 2, · · · , 
.

Proof. Regarding the definition of stiffness matrix, we have

{


,
}
,
 =



D 






(cid:16)

(cid:16)


 ◦ 

(

)(cid:17)

 ◦ 

(

)(cid:17)


+1(

) − 

−1(

)(cid:17) ×
(cid:16)


D 


(cid:1)−2

+1(cid:101)

 


ˆ 1





+1(

) − 

−1(

)(cid:17)
−1D 


−1
(cid:104)(cid:101)

(cid:1)−2

+1(cid:101)

 



−1,
+1 +(cid:101)


+1,
−1 −(cid:101)


+1,
+1 −(cid:101)









,





















ˆ 


= (cid:0) 

 − 


(cid:16)
= (cid:0) 

 − 



2
D 


1





2

(cid:105)

,





−1,
−1

41

(cid:16)

(cid:16)



(

)(cid:17)



(

)(cid:17)
D 


1



Ɖ(
 + 1)
Ɖ(
 − 

 + 1) (1 + 

)−

(1 − 

)−

 
−

,











−1D 





Ɖ(
 + 1)

Ɖ(
 − 

 + 1)

(

) 




,−





(

)


.

where(cid:101)

(cid:101)






,
 =

=

ˆ 1
ˆ 1
−1
−1
(cid:101)






,
 can be computed accurately using Guass-Jacobi quadrature rule as





,
 =

Ɖ(
 + 1)

Ɖ(
 − 

 + 1)

Ɖ(
 + 1)

Ɖ(
 − 

 + 1)



 
−

, 






(

)




,−





(

),

(2.82)






=1

in which Q ≥ N + 2 represents the minimum number of GJ quadrature points {

}Q

=1, asso-
ciated with the weigh function (1 − 

)−

(1 + 

)−

, for exact quadrature, and {

}


=1 are
the corresponding quadrature weights. Employing the property of the Jacobi polynomials where


,




(−

) = (−1)


,






,
. Accordingly,








2










+1,
+1 − (−1)(
+
)(cid:101)

(cid:105)
(cid:105)

−1,
+1 +(cid:101)



,
 as (−1)(
+
)(cid:101)

(

), we can re-express(cid:101)

(cid:104)(−1)(
+
+2)(cid:101)

(cid:1)−2

+1(cid:101)

 


{


}
,
 =(cid:0) 

−



−1,
+1 + (−1)(
+
−2)(cid:101)

−(−1)(
+
)(cid:101)

=(cid:101)

 

 (−1)(
+
)(cid:104)(cid:101)


+1,
−1 −(cid:101)


+1,
+1 −(cid:101)

{
 

}
,
 = {
 

}
,
 × (cid:101)

 

(cid:101)

 



(

)(cid:17)


(

)(cid:17)

In fact,(cid:101)

 and 

 are chosen such that (−1)(
+
) is canceled. Furthermore,
(cid:16)


(

)(cid:17)

(

)(cid:17)
(cid:16)

According to (2.83),

{


,
}
,
 =

ˆ 


ˆ 



(−1)(
+
).





−1,
−1





−1,
−1

(cid:16)
(cid:16)




,




,















D 







D 






.

(2.83)

(2.84)





+1,
−1

(2.85)
where {


,
}
,
 = {


,
}
,
 = {


,
}
,
 = {


}
,
 due to symmetry of 


,
 and 


,
. Similar
to (2.85), we get {


,
}
,
 = {


,
}
,
 = {


}
,
; therefore,

−(


 + 


) 


 + (


 + 


) 


 = 

 



,
 and thereby {

 









,
, 


.




(2.86)
}
,
 as the




Hence it can be easily concluded that the stiffness matrix 

sum of two symmetric matrices are symmetric.

42







D 







D 


=



= {


,
}
,
 ,





2.8 Unified Fast FPDE Solver

We formulate a closed-form solution for the Lyapunov system (2.76) in terms of the generalised
eigensolutions that can be computed very efficiently, leading to the following unified fast solver for
the development of Petrov-Galerkin spectral method.
Theorem 2.8.1. Let {(cid:174)
 
 , 
 
 }M 


 
 =1 be the set of general eigen-solutions of the spatial stiffness
matrix 

 



=1 be the set of general
eigen-solutions of the temporal mass matrix 

 with respect to the stiffness matrix 

. (I) if 
 > 1,
then the multi-dimensional matrix of unknown coefficients U is explicitly obtained as


 with respect to the mass matrix 
 
. Moreover, let {(cid:174)
 
, 

 }N



,
1,··· , 

 (cid:174)
 



 ⊗ (cid:174)
 1


1 ⊗ · · · ⊗ (cid:174)
 






,

(2.87)


1=1
where 

,
1,··· , 

 are given by


=1

N


M1


U =

M





=1

· · ·

(cid:104)((cid:174)
 




 

 (cid:174)
 




,
1,··· , 

 =

in which the numerator represents the standard multi-dimensional inner product, and '
,
1,··· ,


are obtained in terms of the eigenvalues of all mass matrices as

,

(2.88)

'
,
1,··· ,





)


1 · · · (cid:174)
 


 
)
 
 
 (cid:174)
 








) + 




 
)(cid:105)

 
)(cid:105)


=1(
 


.

( (cid:174)
 


 (cid:174)
 1

=1(((cid:174)
 



 )


(cid:104)(1 + 
 


M1




,
1 (cid:174)
 



 ((cid:174)
 1


1)
 ,


=1


1=1

'
,
1,··· ,

 =

N


U =

(II) If 
 = 1, then the two-dimensional matrix of the unknown solution U is obtained as

where 

,
1 is explicitly obtained as


 )(((cid:174)
 1
1)
 
1(cid:174)
 1
1)(cid:104)(1 + 
 




 
 (cid:174)
 1
(cid:174)
 



1



,
1 =


) + 


Proof. Let us consider the following generalised eigenvalue problems as

((cid:174)
 



 

 (cid:174)
 



 
 1
1

(cid:105) .


 
 






(cid:174)
 


 
 = 
 


 



 
 (cid:174)
 


 


, 
 
 = 1, · · · ,M 
 ,


 = 1, 2, · · · , 
,

(2.89)

43

and



 (cid:174)
 



 = 




 

 (cid:174)
 


 ,


 = 1, 2, · · · ,N .

(2.90)

Having the spatial and temporal eigenvectors determined in equations (2.90) and (2.89), we can
represent the unknown coefficient matrix U in (2.71) in terms of the aforementioned eigenvectors
as

· · ·



,
1,··· , 

 (cid:174)
 



 ⊗ (cid:174)
 1


1 ⊗ · · · ⊗ (cid:174)
 






,

(2.91)

M1


N



=1

U =

M




1=1



=1

where 

,
1,··· , 

 are obtained as follows. First, we take the multi-dimensional inner product of
(cid:174)
 


 (cid:174)
 1



 on both sides of the Lyapunov equation (2.76) as


1 · · · (cid:174)
 


+



( (cid:174)
 



 (cid:174)
 1


1 (cid:174)
 2


2 · · · (cid:174)
 




 ⊗ 
1 ⊗ · · · ⊗ 




=1 [

 ⊗ 
1 ⊗ · · · ⊗ 
 
−1 ⊗ 

 


+
 

 ⊗ 
1 ⊗ · · · ⊗ 




 ⊗ 
 
+1 · · · ⊗ 

]

 (cid:174)
 1


1 · · · (cid:174)
 




)
.



)(cid:104)
(cid:105)U = ( (cid:174)
 


Then, by replacing (2.89) and (2.90) into (2.88) and re-arranging the terms, we get

M





,
1,··· , 

 × (cid:16) (cid:174)
 



· · ·

M1


N






=1
+


1=1


=

(cid:174)
 



 

 (cid:174)
 


 ((cid:174)
 1


=1
+
 (cid:174)
 



 

 (cid:174)
 


1 (cid:174)
 2

 (cid:174)
 1
= ( (cid:174)
 

Recalling that 
 
 






N


M1



=1


1=1

· · ·

+


1)
 
1(cid:174)
 1

 ((cid:174)
 1

1)
 
1(cid:174)
 1

2 · · · (cid:174)
 



)
.
(cid:174)
 


 
 = (
 

M



 







,
1,··· , 





=1

(cid:174)
 






(cid:16) (cid:174)
 




 

 (cid:174)
 





((cid:174)
 1

1)
 
1(cid:174)
 1

((cid:174)
 



)
 

 (cid:174)
 






1 · · · ((cid:174)
 


1 ((cid:174)
 2


1)
 
2(cid:174)
 2


 
)
 
 
 



2 · · · ((cid:174)
 



 
 ((cid:174)
 
+1
(cid:174)
 



)
 

 (cid:174)
 









 
+1)
 
 
+1(cid:174)
 
+1


 
+1 · · · ((cid:174)
 




)
 

 (cid:174)
 






1 · · ·
(cid:17)


 
 (cid:174)
 



 
) and 

 (cid:174)
 



 = (




 

 (cid:174)
 



 ), we have


 

 (cid:174)
 



 ((cid:174)
 1


1)
 
1(cid:174)
 1


1 ((cid:174)
 2


2)
 
2(cid:174)
 2


2 · · · ((cid:174)
 




)
 

 (cid:174)
 






 ) ((cid:174)
 1

(



 

 (cid:174)
 



1)
 
1(cid:174)
 1

 ) ((cid:174)
 1

 

 (cid:174)
 

(



1)
 
1(cid:174)
 1

2 · · · (cid:174)
 



)
.

1 (cid:174)
 2


=1

 (cid:174)
 


+
= ( (cid:174)
 


 (cid:174)
 1





 
 (cid:174)
 



1 · · · ((cid:174)
 


1 ((cid:174)
 2


 
)
(
 


 


2)
 
2(cid:174)
 2


2 · · · ((cid:174)
 



 
) · · · ((cid:174)
 



)
 

 (cid:174)
 





(cid:17)



)
 

 (cid:174)
 





44

Therefore,



,
1,··· , 

 =

Then, we have


N


=1


 

 (cid:174)
 


(cid:104)((cid:174)
 



 )



M1

1=1 · · ·
M

×(cid:104)(1 + 
 




1 · · · (cid:174)
 



 
)(cid:105) ×(cid:104)(1 + 
 





)


( (cid:174)
 


 (cid:174)
 1

 
)
 
 
 (cid:174)
 



=1(((cid:174)
 








) + 




=1(
 



 
)(cid:105) .



=1 

,
1,··· , 

 ((cid:174)
 



=1(
 



) + 









 
)(cid:105)


 

 (cid:174)
 



1)
 
1(cid:174)
 1

 )(((cid:174)
 1

 (cid:174)
 1
= ( (cid:174)
 


1 (cid:174)
 2


1) · · · (((cid:174)
 



)
.


2 · · · (cid:174)
 




)
 

 (cid:174)
 



)

Due to the fact that the spatial Mass 
 
 and temporal stiffness matrices 
 
 are diagonal (see
Theorems 2.7.2 and 2.7.1), we have ((cid:174)
 



 
) = 0 if

 
 ࣔ 
 
, which completes the proof for the case 
 > 1.


 ) = 0 if 
 ࣔ 
, and also (((cid:174)
 



 
)
 
 
 (cid:174)
 



 

 (cid:174)
 


Following similar steps for the two-dimensional problem, it is easy to see that if 
 = 1, the

relationship for 
 can be derived as



,
1 =

((cid:174)
 



 

 (cid:174)
 



 )(((cid:174)
 1
1)
 
1(cid:174)
 1
1)(cid:104)(1 + 
 



(cid:174)
 



 
 (cid:174)
 1

1


) + 




 
 1
1

(cid:105) .

(2.92)

In 2.8.1, we present a computational method for the fast solver which reduces the computational
cost significantly.

2.8.1 Computational Considerations
Employing the fast solver in (1+
) dimensional problem 
 ≥ 1 reduces the dominant computational
cost of the eigensolver from O(
2(1+
)) to O(
2+
), which becomes even more efficient in higher
dimensional problems. This approach is extensively discussed in [55].

2.9 Stability and Error Analysis

Theorem 2.9.1. The Petrov-Gelerkin spectral method for (2.62) is stable, i.e.,

inf

0ࣔ

∈



sup
0ࣔ
∈



|
(

 , 

)|

(cid:107)

(cid:107)

,
1,··· ,

 ()(cid:107)

(cid:107)B
,
1,··· ,

 ()
|
(

 , 

)| > 0.

sup

holds with 
 > 0 and independent of 
, where

0ࣔ

∈



45

≥ 
 > 0,

(2.93)

Proof. It is clear that the basis /test spaces are Hilbert spaces. Since 

 ⊂ B
,
1,··· ,

() and


 ⊂ 

,
1,··· ,

(), (2.93) follows directly from Theorem 2.5.4.

2.10 Error Analysis

Let 
M(') denote the space of all polynomials of degree ≤ M on ', where ' ⊂ R. 

M(')
0 (') in ' with


.

0(') is the closure of 
∞
denotes 
M(') ∩ 


respect to (cid:107)·(cid:107)
 

('). In this section, 

 = (

, 

) for 
 = 1, ..., 
, '
 = 

×'
−1, and ' 


0(') for any real positive 
, where 




 =




=1

ࣔ 


Theorem 2.10.1. [115] Let 
1 be a real number, where 
1 ࣔ M1 + 1
2, and 1 ≤ 
1. There
('1) such that for any

1M1
exists a projection operator 

(cid:107)
 

1('1) ≤ 
1M
1−
1

 ∈ 

1('1) ∩ 

(cid:107)
(cid:107)

1('1), where 
1
is a positive constant.

0 ('1), we have (cid:107)
 − 

1

from 

1('1) ∩ 


0 ('1) to 


1

1


1

1,M1


1

1,M1

Maday in [115] proved Theorem 2.10.1 using the error estimate provided in [116] for Legendre
and Chebyshev polynomials. Next, this theorem is extended to Jacobi polyfractonomials of first
kind.
Theorem 2.10.2. [55] Let 
0 ≥ (cid:100)2
(cid:101), 
0 ࣔ N + 1

0,N from 

0(
) ∩ 
 

(
) to 

N('1) such that for any 
 ∈ 

0(
) ∩ 
 

(
), we have



2 and 2
 ∈ (0, 2), 2
 ࣔ 1. There exists an operator

(cid:107)
 − 



0,N 
(cid:107)
 

(
) ≤ 
0 N 
−
0(cid:107)
(cid:107)

0(
),

where 
0 is a positive constant.

Li and Xu in [109] performed the error analysis for the space-time fractional diffusion equation,
employing Lagrangian polynomials. Here, employing Theorems 2.10.1 and 2.10.2 and Theorem

.3 from [117], we study the properties of higher-dimensional approximation operators in the
following lemmas.

46

Lemma 2.10.3. Let the real-valued 1 ≤ 
1, 
2, 

 = (

, 

) 
 = 1, 2,  = 
1 × 
2, and 1
1. If 
 ∈ 


0 (
1)), then

1

2 < 
1, 
2 <


2


2,M2

0 (
2, 

1(
1)) ∩ 

2(
2, 


2
(cid:107)
 − 

(cid:107)B
1,
2() ≤

1

1,M1
M
2−
2
(cid:107)
(cid:107)

2(
2,
2(
1)) + M
2−
2
+M
1−
1
(cid:107)
(cid:107)

1(
1,
2(
2)) + M
1−
1
1
+M−
2
(cid:107)
(cid:107)
 

1(
1,

2(
2)),
2

2

2 M−
1
1 M−
2

1

2

where (cid:107) · (cid:107)B
1,
2() =(cid:8)(cid:107) · (cid:107)2


 

1(
1,
2(
2)) + (cid:107) · (cid:107)2

1

(cid:107)
(cid:107)

2(
2,

1(
1)) + M−
1
(cid:107)
(cid:107)

1(
1,

2(
2)
(cid:9) 1


 

2(
1,
2(
1))

2 , and 
 > 0.

(cid:107)
(cid:107)
 

2(
2,

1(
1))

(2.94)

Proof. If 
 ∈ 



2(
2, 
2(
1)), and 
 ∈ 

1(
1, 
2(
2)). By the real-valued positive constant 
, we have

0 (
1)), then evidently 
 ∈ 

2(
2, 

1(
1)), 
 ∈

1

0 (
2, 

1(
1)) ∩ 

2(
2, 


2


(cid:107)B
1,
2()

(cid:107)
 − 

(cid:16)(cid:107)
 − 

=
≤ (cid:107)
 − 


2

1


2,M2

1,M1

2

1


2,M2

1,M1

2

1


2,M2

1,M1


 

2(
2,
2(
1)) + (cid:107)
 − 

(cid:107)2

(cid:107)
 

2(
2,
2(
1)) + (cid:107)
 − 


2

1


2,M2

1,M1

2

1


2,M2

1,M1


(cid:107)2

2(
2,
 

1(
1))

(cid:107) 
2(
2,
 

1(
1)).

(2.95)

(cid:17) 1

2


 − 

By Theorem 2.10.1, (2.95) can be simplified to

(cid:107)
 

2(
2,
2(
1))

2


2,M2

 + 

2

1

2


2,M2

1,M1

2,M2

(cid:107)
 

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

2

2,M2

2

2,M2

(cid:107)
 − 

1

1,M1
= (cid:107)
 − 

2

2,M2
≤ (cid:107)
 − 

2

2,M2
≤ M
2−
2
(cid:107)
(cid:107)

2(
2,
2(
1)) + (cid:107)(
+(cid:107)
 − 

(cid:107)
 

2(
2,
2(
1))

1

1,M1
≤ M
2−
2
(cid:107)
(cid:107)

2(
2,
2(
1)) + M
2−
2
+M−
1
(cid:107)
(cid:107)
 

2(
2,

1(
1)),
1

2 M−
1

2

2

1

where I is the identity operator.

47


(cid:107)
 

2(
2,
2(
1))

 − 

2


2,M2

)(cid:107)
 

2(
2,
2(
1))


1

1,M1
− I)(
 − 

1

1,M1


(cid:107)
 

2(
2,
2(
1))

(cid:107)
(cid:107)

2(
2,

1(
1))

(2.96)

Since (cid:107)
 − 


1

1,M1


2


2,M2


(cid:107) 
2(
2,
 

1(
1)) = (cid:107)
 − 


1

1,M1


2


2,M2


(cid:107)
 

1(
1,
2(
2)), we

obtain


 − 


(cid:107) 
2(
2,
 

1(
1))

2


2,M2

 + 

2

1

1


2,M2

1,M1

1,M1

(cid:107)
 

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

1

1,M1

1

1,M1

(cid:107)
 − 

1

1,M1
= (cid:107)
 − 

1

1,M1
≤ (cid:107)
 − 

1

1,M1
≤ M
1−
1
(cid:107)
(cid:107)

1(
1,
2(
2)) + (cid:107)(
+(cid:107)
 − 

(cid:107)
 

1(
1,
2(
2))

2

2,M2
≤ M
1−
1
(cid:107)
(cid:107)

1(
1,
2(
2)) + M
1−
1
+M−
2
(cid:107)
(cid:107)
 

1(
1,

2(
2)).
2

1 M−
2

1

1

2


(cid:107)
 

1(
1,
2(
2))

 − 

2


2,M2

)(cid:107)
 

1(
1,
2(
2))


1

1,M1
− I)(
 − 

2

2,M2


(cid:107)
 

1(
1,
2(
2))

(cid:107)
(cid:107)

1(
1,

2(
2))

(2.97)

Accordingly, (2.94) can be derived immediately from (2.97) and (2.96).

In order to perform the error analysis of (1+d)-dimensional PG method, we first study the
approximation properties in three dimensions and then extend it to (1+d)-dimensions. It should
be noted that in the following lemmas, 


+1,

+2,··· ,

+
 (

+1 × · · · × 

+
 , 
2('
+1,··· ,
+

)) =



+1(

+1, 


+2(

+2, · · · , 


+
 (

+
 , 
2('
+1,··· ,
+


 
.

=1
Following Lemma 2.10.3, we introduce

Lemma 2.10.4. Let the real-valued 1 ≤ 

, 

 = (

, 

),  =



=



=1 

, '
 =



))), where '
+1,··· ,
+



 =




ࣔ
+1,··· ,
+











2 < 

 < 1 for 
 = 1, · · · , 
.

If 
 ∈ 



0 (
1, 

2,
3('1

1

3)) ∩ 

1,
3('2


=1 

, ' 

3, 




=1

ࣔ 





0 (
2)) ∩

2

(cid:107)
(cid:107)




,
 
 ,

 ('3)

3 ))) + M−
 



, 





(cid:107)
(cid:107)
 


 (

,



 
 (
 
 ,
2('


, 


3 )))

(2.98)

(cid:1)

and 1


1,
2('2, 


0 (
3)), then

3
(cid:107)
 − 

1

1,M1
≤ M

−


+M−
 






(cid:107)
 


 (

,
2('

3))

 M−
 


 M−







2


2,M2


3


3,M3
3)) + M

−


(cid:107)
(cid:107)


 (

,
2('

(cid:107)
(cid:107)
 


 (

,


 
 (
 
 ,
2(

)))
(cid:107)
(cid:107)


 M−


(cid:0)M

−



 M−
 







,
 
 (

×
 
 ,
2('




2

+ 3



=1

ࣔ


for 
 = 1, 2, 3, 
 = 1, 2, 3 and 
 ࣔ 
, and 
 = 1, 2, 3 and 
 ࣔ 
, 
, where 
 > 0.

48

Proof. see Appendix A.

Lemma 2.10.4 can be easily extended to the d-dimensional approximation operator as

(cid:107)
 − Ò



(cid:107)
 


 (

,
2('


))

≤

M

−






(cid:107)
(cid:107)


 (

,
2('








M−
 


(cid:107)
(cid:107)


=1

ࣔ


M−
 





+M

−















=1

ࣔ


M−
 





(cid:107)
(cid:107)
 


 (

,



 
 (
 
 ,
2('


, 



 )))


)) +




 (

,



 
 (
 
 ,
2('


, 



 )))

(cid:107)
(cid:107)
 


 (

,



, 
,






 ,
 
 (

×
 
 ,
2('

(cid:1)(cid:107)
(cid:107)


 (

,

1,··· ,

 ('

(cid:17)
, '
 =




))).


=1 



))))

(2.99)


=1 

, ' 



))(cid:17) ∩


 =



0(
, 


(

, 

1,··· ,

−1,

+1,··· ,

('


Theorem 2.10.5. Let 1 ≤ 

, 
 = (0, 
), 

 = (

, 

),  = 
 ×(cid:16)







=1

ࣔ




 and 1


=1

ࣔ 


 

(
, 

1,··· ,

('
)), then we have

M−
 











+


=1

ࣔ



=1

ࣔ
, 


+· · · + M

−




 M−


(cid:0) 


2 < 

 < 1 for 
 = 1, · · · , 
. If 
 ∈(cid:16) 
∩
(cid:16)N 
−
0(cid:107)
(cid:107)

0(
,
2('
)) +
+N 
−
0(cid:16) 


M−
 

(cid:16)




(cid:17)(cid:107)
(cid:107)

0(
,

1,··· ,

 ('
))) +
(cid:17)(cid:107)
(cid:107)


 (

,

1,··· ,

 ('




(cid:107)B
,
1,··· ,

 ()

+M

−




0,N Ò





(cid:107)
 − 


N 
−
0M−
 


M−
 


≤ 



=1


=1


=1













(cid:107)
(cid:107)






=1



0(
,


(cid:110)M

−


(cid:111)(cid:17)




,

,
2(
)))





=1

ࣔ
, 


where Ò


 = 


1

1,M1

· · · 






 ,M


and 
 is a real positive constant.

49





 
 (
 
 ,
2('


))) + · · ·
(cid:107)
(cid:107)


 (

,
2('



×
)) + · · ·

(2.100)

Proof. Directly from (2.41) we conclude that

(cid:107)
(cid:107)B
,
1,··· ,

 () ≤ 


By Theorem 2.10.2 we obtain

(cid:16)(cid:107)
(cid:107)
 

(
,
2('
)) +

(cid:17)

.

(cid:107)
(cid:107) 
2(
,
 


 (

,
2('



)))






=1

(cid:107)
 − 



0,N Ò



(cid:107)
 

(
,
2('
)) ≤ N 
−
0(cid:107)
(cid:107)

0(
,
2('
))

+




+ N 
−
0(cid:16) 




=1

N 
−
0M−
 





M−
 






=1

(cid:107)
(cid:107)

0(
,



 
 (
 
 ,
2('
))) + · · ·

(cid:17)(cid:107)
(cid:107)

0(
,

1,··· ,

 ('
))). (2.101)

Accordingly, the property of composite approximation to time-spatial (1+d)-dimensional space-
time approximation operator in (2.100) is obtained immediately using (??) and (2.101).

Remark 2.10.6. Since the inf-sup condition holds (see Theorem 2.9.1), by the Banach-Nečas-
Babuška theorem [118] and Lemma 2.5.1, the error in the numerical scheme is less than or equal
to a constant times the projection error. Accordingly, we conclude the spectral accuracy of the
scheme.

2.11 Numerical Tests

To study the convergence rate of the PG method in (2.48), we perform numerical simulations

and consider the following relative errors in 
2 as

and in the energy norm as

(cid:107)
(cid:107) 
2() =

(cid:107)
 − 



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



(cid:107) 
2()

(cid:107)
(cid:107)B
,
1() =

(cid:107)
 − 



(cid:107)B
,
1()
(cid:107)



(cid:107)B
,
1()

,

50

(2.102)

(2.103)

where 



 is presented in (2.105) and (2.106) in Case I and Case II respectively. Let  =
(0, 
) × (−1, 1). Recalling that


2()+(cid:107) 0D



 (·)(cid:107)2


2()+(cid:107) −1D
1
1(·)(cid:107)2


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

1 (·)(cid:107)2


2()

2 . (2.104)

(cid:107)·(cid:107)B
,
1() :=(cid:8)(cid:107)·(cid:107)2

(cid:9) 1

We particularly consider the time and space-fractional diffusion equation (i.e. 

 = 

 = 0 in (2.14))
in 2-D space-time as we have obtained similar results for advection-dispersion equation in higher
dimensions.
Case I: We choose the exact solution to be





(
, 
) = 
 
1 ×(cid:2)(1 + 
) 
2 − 
(1 + 
) 
3(cid:3),

(2.105)

20, 
2 = 53

4 and 
3 = 51

20, 
2 = 53

4 and 
3 = 51
5.

10 and 
 = 2. Besides, 
1 = 5 1

in (2.14), where 
 = 2
2−
3. In (2.105), we take 
1 = 5 1
Table 2.1: Convergence study of the PG spectral method for (1+1)-D diffusion problem, where


1 = 

1 = 2
5 in (2.105). Here, we denote
by ¯
0 the practical rate of the convergence, numerically achieved.
20 fixed, where we consider the limit orders 
 = 1
20 and 
 = 9
Case I-A: 
1 = 15
fixed, where 
1 = 11
20 and 
1 = 19
20.
Temporal p-refinement Case I-A

 = 9
(cid:107)
(cid:107)B
,
1()
(¯
0 = 13.32)
0.65358
0.07529
0.00079
5.03×10−7
4.81×10−8


 = 1
(cid:107)
(cid:107)B
,
1()
(¯
0 = 12.81)
0.48488
0.04176
3.44×10−5
5.00×10−7
4.82×10−8

(cid:107)
(cid:107) 
2()
(¯
0 = 14.44)
0.56631
0.05431
0.00045
2.59×10−7
6.61×10−9

20. Case I-B: 
 = 5
20

20 and 
1 = 15
20

20 and 
1 = 15
20

3
5
7
9
11

M


20 and 
 = 5
20

20 and 
 = 5
20

M


3
5
7
9
11

(cid:107)
(cid:107) 
2()
(¯
0 = 14.09)
0.45541
0.04003
2.64×10−5
2.81×10−7
1.45×10−8
Spatial p-refinement Case I-B

1 = 19
(cid:107)
(cid:107) 
2()
(cid:107)
(cid:107)B
,
1()
(¯
1 = 8.51)
(¯
1 = 9.36)
0.55657
0.40578
0.01259
0.03097
3.08×10−5
0.000029
2.45×10−6
6.96×10−7
5.33×10−8
5.42×10−7

3
5
7
9
11

M


51

(cid:107)
(cid:107) 
2()
(¯
1 = 9.08)
0.38525
0.01445
1.06×10−5
6.63×10−7
1.56×10−7


1 = 11
(cid:107)
(cid:107)B
,
1()
(¯
1 = 9.18)
0.45329
0.01738
4.68×10−5
1.19×10−6
7.09×10−8

M


3
5
7
9
11







(cid:107)
(cid:107)

0(
,
2('1)).

20 and 9

20, while 
1 = 15

20 fixed and 

1 = 

1 = 2

2 − 
 is the minimum regularity of the exact solution in time direction.

Temporal p-refinement: In Table 2.1 Case I-A, we study the spectral convergence of the method
for the limit fractional orders of 
 = 1
10 in
(2.14) for (1+1)-D diffusion problem. In the temporal p-refinement, we keep the spatial order of
expansion fixed (M
 = 19) such that the error in spatial direction approaches to the exact solution
sufficiently and hence the rate of the convergence is a function of the minimum regularity in time
direction. Theoretically, the rate of convergence is bounded by M
−
0
(cid:107)
(cid:107)

0(
,
2('1)), where

0 = 
1 + 1
In Table
2.1 we observe that ¯
0 in (cid:107)
(cid:107) 
2() and (cid:107)
(cid:107)B
,
1() are greater than 
0 ≈ 511
20. Accordingly,
(cid:107)
(cid:107) 
2() ≤ M−

Spatial p-refinement: We study the convergence rate of the PG method for the limit orders
of 
1 = 11
20 in Table 2.1 Case I-B. The temporal order of expansion is
constant (M
 = 19) to keep the solution sufficiently accurate in time direction. Similar to temporal
p-refinement, we have (cid:107)
(cid:107) 
2() ≤ M−
1

(cid:107)
(cid:107)

1('1,
2(
)), where 
1 =

3 + 1
2 − 
 as the minimum regularity of the exact solution in spatial direction. In agreement with
Theorem 2.10.5, the practical rates of convergence ¯
1 in (cid:107)
(cid:107) 
2() and in (cid:107)
(cid:107)B
,
1() are greater
than 
1 ≈ 5 7
10. Further to the aforementioned cases, we have observed similar results for higher
dimensional problems, including (1+2)-D time- and space-fractional diffusion equation as well.

(cid:107)
(cid:107)B
,
1() ≤ M−
1


(cid:107)
(cid:107)B
,
1() ≤ M−
0




20 and 19

20 while 
 = 5

Case II: We consider the smooth exact solution to be





(
, 
) = 
 
1 ×(cid:104)

sin(cid:0)

(1 + 
)(cid:1)(cid:105)

,

(2.106)

20 and 
 = 1.

in (2.14), where 
1 = 5 1
20 and 19
p-refinement: The convergence rate of the PG method for the limit orders of 
1 = 11
20 is
20 in Table 2.2. The temporal order of expansion is chosen as (M
 = 19) to
investigated while 
 = 5
keep the solution sufficiently accurate in time direction. The results in Table 2.2 show the expected
exponential decay which verifies the PG method for different values of 
1.

52

Table 2.2: Here, we set 
1 = 5 1
method for (1+1)-D diffusion problem, where 

1 = 

1 = 2
are 
1 = 11

20 and 
 = 1 in (2.106) to study the convergence of the PG spectral
10 and 
 = 2. Besides, the limit orders

20 is fixed.

20 and 
1 = 19

20, where 
 = 5

p-refinement

20 and 
 = 5
20


1 = 11
(cid:107)
(cid:107)B
,
1()
0.04756
2.89×10−5
4.44×10−9
4.10×10−11

M

5
9
13
17

(cid:107)
(cid:107) 
2()
0.02655
1.60×10−5
2.46×10−9
5.90×10−12

20 and 
 = 5
20


1 = 19
(cid:107)
(cid:107)B
,
1()
0.05730
2.72×10−4
4.32×10−8
8.88×10−11

(cid:107)
(cid:107) 
2()
0.03147
1.54×10−4
2.44×10−8
9.17×10−12

M

5
9
13
17

53

CHAPTER 3

A UNIFIED PG METHOD FOR DISTRIBUTED-ORDER FPDES

3.1 Background

In many physical processes, which cannot be characterized with a certain single power-law
scaling over the whole domain, distributed-order differential equations (DDEs) can serve as a
framework for accommodating a distribution of power law behavior. More specifically, distributed
order FPDEs are generating considerable interests in terms of accelerating superdiffusion, decel-
erating subdiffusion random processes in multi-physics anomalous phenomena. To model wave
propagation in complex media like viscoelastic media, acoustics, and seismology, Bazhlekov and
Bazhlekov [119] developed a subordination approach to multi-term time fractional diffusion-wave
equations. Besides, Chechkin et al. [69] proposed distributed-order temporal fractional diffusion
equation for describing the (retarding) sub-diffusion random processes which are subordinated
to the Wiener process. A faithful description of such anomalous transport requires exploiting
distributed-order derivatives, in which the derivative order has a distribution over a range of values.
The reader is referred to [120, 64, 65, 114, 38, 63] and the references given therein for more details
on the distributed-order fractional equations.

Numerical methods for FPDEs, which can exhibit history dependence and non-local features
have been recently addressed by developing finite-element methods [121, 122], spectral/spectral-
element methods [123, 124, 125, 126], and also finite-difference and finite-volume methods
[127, 128, 129]. Distributed-order FPDEs impose further complications in numerical analysis
by introducing distribution functions, which require compliant underlying function spaces, as
well as efficient and accurate integration techniques over the order of the fractional derivatives.
In [130, 131, 132, 133], numerical analysis of distributed-order FPDEs was extensively investi-
gated. More recently, Liao et al. [134] studied simulation of a distributed subdiffusion equation,
approximating the distributed order Caputo derivative using piecewise-linear and quadratic inter-

54

polating polynomials. Kharazmi and Zayernouri [135] developed a pseudo-spectral method of
Petrov-Galerkin sense, employing nodal expansions in the weak formulation of distributed-order
fractional PDEs. In [70], Kharazmi et al. also introduced distributed Sobolev space and developed
two spectrally accurate schemes, namely, a Petrov–Galerkin spectral method and a spectral colloca-
tion method for distributed order fractional differential equations. Besides, Tomovski and Sandev
[136] investigated the solution of generalized distributed-order diffusion equations with fractional
time-derivative, using the Fourier-Laplace transform method.

The main purpose of this study is to develop and analyze a Petrov-Galerkin (PG) spectral method
to solve a (1 + 
)-dimensional fully distributed-order FPDE with two-sided derivatives of the form



(

) [










D2







 + 










D2







] 





 + 

 







 
 D2
 



 



] 

 
 − 
 
 + 
 ,

(3.1)

ˆ 














=


=1


(
) 

ˆ 















0D2



 
 

 +


 
(
 
) [

 






ˆ 














=1



 
 D2
 



 







 
 ∈ [
 
 , 
 
],

subject to homogeneous Dirichlet boundary conditions and zero initial condition, where for 
, 
 =
1, 2, ..., 


 ∈ [0, 
],
2



 < 2



 ∈ (0, 2], 2



 ࣔ 1, 2



 ࣔ 1,
2




0 < 
(
) ∈ 
1((



, 



)), 0 < 

(

) ∈ 
1((





)), 0 < 
 
(
 
) ∈ 
1((





∈ (0, 1), 2





∈ (1, 2],

< 2





< 2





, 





)),













, 

















and the coefficients 


, 


, 


, 


, and 
 are constant. We emphasize that (3.1) is reduced to
fractional advection-dispersion-reaction equations when 
(
), 

(

), and 
 
(
 
) are chosen to be
a Dirac delta function, where for 
 = 1 and 
1(
1) = 
(
1 − 1) the two-sided Riesz derivatives
with proper coefficients recover the standard second order dispersion terms. We briefly highlight
the main contributions of this study as follows:

• We consider fully distributed fractional PDEs as an extension of existing fractional PDEs in
[60, 70] by replacing the fractional operators by their corresponding distributed order ones.
We further derive the weak formulation of the problem.

55

• We construct the underlying function spaces by extending the distributed Sobolev space in

[70] to higher dimensions in time and space, endowed with equivalent associated norms.

• We develop a Petrov-Galerkin spectral method, employing Legendre polynomials and Jacobi
poly-fractonomials [56] as spatial and temporal basis/test functions, respectively. We also
formulate a fast solver for the corresponding weak form of (3.1), which significantly reduces
the computational expenses in high-dimensional problems.

• We establish well-posedness of the weak form of the problem in the underlying distributed
Sobolev spaces respecting the analysis in chapter 2 and prove the stability of proposed
numerical scheme. We additionally perform the corresponding error analysis, where the
distributed Sobolev spaces enable us to obtain accurate error estimate of the scheme.

To examine the performance and convergence of the developed PG method in solving different
cases, we also perform several numerical simulations.

This chapter is organized as follows:

in Section 2, we introduce some preliminaries from
fractional calculus. In Section 3, we present the mathematical framework of the bilinear form and
carry out the corresponding well-posedness analysis. We construct the PG method for the discrete
weak form problem and formulate the fast solver in Section 4. In Section 5, we perform the stability
and error analysis in detail. In Section 6, we illustrate the convergence rate and the efficiency of
method via numerical examples.

3.2 Preliminaries on Fractional Calculus

Recalling the definitions of the fractional derivatives and integrals from [56, 38], we denote

(
) the left-sided and the right-sided Riemann-Liouville fractional

D



 
(
) and 




D

by 




derivatives of order 
 > 0,






D



 
(
) =






D



 
(
) =

1

Ɖ(
 − 
)
(−1)

Ɖ(
 − 
)
















ˆ 

ˆ 









(
)


(
)

(
 − 
)
+1−


(
 − 
)
+1−




,



,


 ∈ [
, 
],


 ∈ [
, 
],

(3.2)

(3.3)

56


(
)




(
−
)
+1−
 

 ∈ 

[
, 
], respectively,

(
) represent the left-sided and the right-sided

´ 


(
−
)
+1−
 

,




´ 


(
)
in which 
(
) ∈ 
1[
, 
] and

 
(
) and 

where 
 = (cid:100)
(cid:101). Besides, 


D


D

Caputo fractional derivatives, where
ˆ 

ˆ 



 
 (
) =


D


1














D



 
 (
) =

Ɖ(
 − 
)
(−1)

Ɖ(
 − 
)


(
)(
)

(
 − 
)
+1−



(
)(
)

(
 − 
)
+1−







,



,


 ∈ [
, 
],


 ∈ [
, 
].

(3.4)

(3.5)

(3.6)

(3.7)

The relationship between the RL and the Caputo fractional derivatives is given by






D


D



 
 (
) =

 
 (
) =






 (
)

 (
)


D

Ɖ(1 − 
)(
 − 
)
 + 

Ɖ(1 − 
)(
 − 
)
 + 


D



 
 (
)

 
 (
),






D



 
 (
) = 


when (cid:100)
(cid:101) = 1, see e.g. (2.33) in [38]. In the case of homogeneous boundary conditions, we obtain

D


 (
). The Reimann-
Liouville fractional integrals of Jacobi poly-fractonomials are analytically obtained in [56, 55] in
the standard domain 
 ∈ [−1, 1] as


 
 (
) := 
D



 
 (
) and 




 (
) := 
D



 (
) = 



D



D













−1I



 {(1 + 
) 


,





and






I


1 {(1 − 
)


,





(
)} =

Ɖ(
 + 
 + 1)
Ɖ(
 + 
 + 
 + 1) (1 + 
) 
+
 

−
,
+





(
),

(
)} =

Ɖ(
 + 
 + 1)
Ɖ(
 + 
 + 
 + 1) (1 − 
)
+
 

+
,
−





(
),

(3.8)

(3.9)

where 0 < 
 < 1, 
 > −1, 
 > −1, and 

, 


 and parameters 
 and 
 [137]. Accordingly,




(
) denotes the standard Jacobi polynomials of order



−1D



 

(
) =

Ɖ(
 + 1)
Ɖ(
 − 
 + 1) 
 
,−





(
) (1 + 
)−


and

where 

(
) := 
 0,0








1 

(
) =

Ɖ(
 + 1)
Ɖ(
 − 
 + 1) 
 −
,



D

(
) represents Legendre polynomial of degree 
 (see [137]).

(
) (1 − 
)−
,




(3.10)

(3.11)

57

Let define the distributed-order derivative as
ˆ 






D



 
 (
, 
) :=








(
) 0D



 
 (
, 
)

,

(3.12)

where 
 → 
(
) be a continuous mapping in [



, 



] [70] and 
 > 0. We note that by
choosing the distribution function in the distributed-order derivatives to be the Dirac delta function

(
 − 
0), we recover a single (fixed) term fractional derivative, i.e.,

 
 (
, 
)

 = 0D
0


(
 − 
0) 0D


ˆ 






 (
, 
),

(3.13)










where 
0 ∈ (



, 



).
3.3 Mathematical Formulation

We introduce the underlying solution and test spaces along with their proper norms, and also
provide some useful lemmas to derive the corresponding bilinear form and thus, prove the well-
posedness of problem.

3.3.1 Mathematical Framework
Let 
∞
0 (') represent the space of smooth functions with compact support in ' = (
, 
). Recalling
the definition of Sobolev space for real 
 ≥ 0 from [70, 107], the usual Sobolev space denoted
by 

(') is the closure of 
∞
0 (') on the finite interval ', which is associated with the norm
(cid:107) · (cid:107)

('). For the real index 
 ≥ 0 and 
 ࣔ 
− 1
2 on the bounded interval ' the following norms
are equivalent [109]

(cid:107) · (cid:107)

(') (cid:27) (cid:107) · (cid:107)
 

(') (cid:27) (cid:107) · (cid:107)
 

(') (cid:27) | · |∗



('),

(cid:16)(cid:107) 
D



 (·)(cid:107)2


(') = |(
D



2(') + (cid:107) · (cid:107)2

 (·))'| 1

 (·), 
D


2 ,


2(')
2 . From

(3.14)

(cid:17) 1

where ” (cid:27) ” denotes equivalence relation, (cid:107) · (cid:107)
 

(') =
2 , and | · |∗

 (·)(cid:107)2
(cid:107) · (cid:107)
 

(') =
Lemma 5.2 in [108], we have


2(') + (cid:107) · (cid:107)2


2(')

(cid:17) 1

(cid:16)(cid:107) 
D


 

(') (cid:27) | · | 1
| · |∗

2

58


 

(') | · | 1

2
 

(') = (cid:107) 
D



 (·)(cid:107) 1

2


2(') (cid:107) 
D



 (·)(cid:107) 1

2

2('),

(3.15)


 (·)(cid:107) 
2('), and | · |
 

(') = (cid:107) 
D


where | · |
 

(') = (cid:107) 
D

3.1 in [61], the norms (cid:107) · (cid:107)
 

(') and (cid:107) · (cid:107)
 

(') are equivalent to (cid:107) · (cid:107)
 

(') in space 
∞
where


(·)(cid:107) 
2('). According to Lemma
0 ('),

(cid:107) · (cid:107)
 

(') =


 (·)(cid:107)2


2(') + (cid:107) 
D



 (·)(cid:107)2


2(') + (cid:107) · (cid:107)2


2(')

(3.16)

(cid:16)(cid:107) 
D


(cid:17) 1

2 .

In the usual Sobolev space, for 
 ∈ 

(') we define


 
, 
D


|
|∗


(') = |(
D

|(
D

2 +|(
D



(') = 0. Denoted by 
 




 
)'| 1
2 + |(
D


)'| 1

where we assume

∈

(')
excludes the solutions to |
|∗
with respect to the norms (cid:107) · (cid:107)
 

(') and (cid:107) · (cid:107)
 

(')in ', respectively.


 
, 
D


, 
D



 
)'| 1

 
)'| 1
0 (') and 
 




 
, 
D


sup







2 , ∀
 ∈ 

('),

2 > 0 ∀
 ∈ 

('), which
0 (') are the closure of 
∞
0 (')

Recalling from [70], 


(R) represents the distributed Sobolev space on R , which is associated

with the following norm

(cid:107) · (cid:107)


(R) =

(cid:32)ˆ 












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


2F (·)(
)(cid:107)2


2(R) 



,

(3.17)

where 0 < 
(
) ∈ 
1( [



, 



] ), 0 < 



 < 



 < 1 (1 < 



 < 



 ≤ 2). Sub-
sequently, we denote by 


(
) the distributed Sobolev space on the bounded open interval

 = (0, 
), which is defined as 


(
) = {
 ∈ 
2(
)| ∃˜
 ∈ 


(R) 
.
. ˜
|
 = 
} with the the
equivalent norms (cid:107) · (cid:107)
,


(
) and (cid:107) · (cid:107)
,


(
) in [70], where

(cid:33) 1

2

2

(cid:33) 1
(cid:33) 1

2

,

.

and

(cid:107) · (cid:107)
,


(
) =

(cid:107) · (cid:107)
,


(
) =

(cid:107) · (cid:107)2


2(
) +

(cid:107) · (cid:107)2


2(
) +

ˆ 











ˆ 












(
) (cid:107) 0D



 (·)(cid:107)2


2(
) 




(
) (cid:107) 
D



(·)(cid:107)2


2(
) 



(cid:32)
(cid:32)

In each realization of a physical process (e.g., sub- or super-diffusion), the distribution function

(
) can be obtained from experimental observations, while the theoretical setting of the problem
remains invariant. More importantly, choice of distributed Sobolev space and the associated norms
provide a sharper estimate for the accuracy of proposed PG method.

59

Let '1 = (
1, 
1), '
 = (

, 

) × '
−1 for 
 = 2, · · · , 
. We define X1 = 

 
1('1) with the

associated norm (cid:107) · (cid:107)

 
1('1), where
ˆ 





(cid:32)

(cid:107)·(cid:107)

 
1('1) =

(cid:107) · (cid:107)2

(cid:18)

Subsequently, we construct X
 such that

...

(·)(cid:107)2


1

1





1


1(
1)


2(
) +

(cid:107) 
1D
1
1(·)(cid:107)2


2('1) + (cid:107) 
1D
1
X2 = 

 
2(cid:16)(
2, 
2); 
2('1)(cid:17) ∩ 
2((
2, 
2);X1),
X
 = 

 

(cid:16)(

, 

); 
2('
−1)(cid:17) ∩ 
2((

, 

);X
−1),
(cid:17)(cid:27) 1
(cid:26)

2(cid:16)(

 ,

);X
−1
(cid:17)



 

(cid:16)(

 ,

);
2('
−1)(cid:17) + (cid:107) · (cid:107)2


(

)(cid:16)(cid:107) 

D



2 for 
 = 1, · · · , 
. Then


2('
) + (cid:107) 

D





 (·)(cid:107)2


2('
)

(cid:107) · (cid:107)2

(·)(cid:107)2





2




associated with the norm

(cid:107) · (cid:107)X
 =

(cid:26) 



ˆ 






=1









(cid:107) · (cid:107)X


(cid:27)

Lemma 3.3.1. Let 

 > 0 and 

 ࣔ 
 − 1

(cid:19)

(cid:33) 1

2

.



1
(3.18)


2('1)

(3.19)

.

(3.20)

(cid:27) 1

2

.




 + (cid:107) · (cid:107)2


2('
)

(3.21)
(cid:17)} 1
(cid:27) (cid:107) · (cid:107)

 
1('1). X2 is associated with
2 , which is proved to be


2(cid:16)(
2,
2);X1
ˆ 
2


|2 

2 +


2

Proof. Considering (3.18), X1 is endowed with (cid:107) · (cid:107)X1
(cid:107) · (cid:107)X2 = {(cid:107) · (cid:107)2


 
 
2

(cid:16)(
2,
2);
2('1)(cid:17) + (cid:107) · (cid:107)2
(cid:16)(
2,
2);
2('1)(cid:17)
(cid:16)ˆ 
2
ˆ 
1
(cid:17)

2(
2)(cid:16)(cid:107) 
2D
2

| 
2D
2
2


2(
2)



1 

2

(
)(cid:107)2


1


2


2

|
|2 

2

=



 
2

(cid:107)
(cid:107)2
ˆ 




ˆ 
2
ˆ 





2





2


2

+

=

2





2

60


|2 

2

| 
2D
2

2
(cid:17)


2('2)


2('2) + (cid:107) 
2D
2
2 (
)(cid:107)2



2 + (cid:107)
(cid:107)2


2('2)

(3.22)


|2 

1 +

ˆ 
1


1

| 
1D
1
1


|2 

1

| 
1D
1

1
(cid:17)


(cid:107)2

2('2) + (cid:107) 
1D
1
1


(cid:107)2


2('2)



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


2('2).

(3.23)

(·)(cid:107)2


2('
−1)+(cid:107) 

D





 (·)(cid:107)2


2('
−1)

(cid:17)




+(cid:107)·(cid:107)2


2('
−1)

(cid:27) 1

2

,

then the inductive step is attained according to

and

Next, providing that

(cid:107)·(cid:107)X
−1

(cid:27)




=

=






1


1


2


=1











2 

1


1(
1)

|
|2 

1

ˆ 






1
1





1


2(cid:16)(
2,
2);X1
(cid:17)
(cid:107)
(cid:107)2
(cid:16)ˆ 
1
ˆ 




ˆ 
2
(cid:17)
1
ˆ 
1





1

1(
1)(cid:16)(cid:107) 
1D
1
+
ˆ 




(cid:26) 
−1



(

)(cid:16)(cid:107) 

D




 

(cid:16)(

 ,

);
2('
−1)(cid:17)
(cid:16)ˆ 


ˆ 




(

)(cid:16)(cid:107) 

D


(cid:17) =
ˆ 


ˆ
'
−1
ˆ 


ˆ

−1

ˆ 




'
−1

(cid:107)
(cid:107)2
ˆ
'
−1
ˆ 















2(cid:16)(

 ,

);X
−1




(
)(cid:107)2

|
|2 


 +

=

=

+




















=


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









2('
).

and

(cid:107)
(cid:107)2



(

)(cid:0)| 

D







ˆ 
















|2 + | 

D






(cid:17)


|2(cid:1) 


 




(cid:17)


'
−1



 (
)(cid:107)2


2('
)




 + (cid:107)
(cid:107)2


2('
),

(3.24)



(

)(cid:0)| 

D








|2 + | 

D






(cid:17)


'
−1 







ˆ 






2('
) + (cid:107) 

D


(cid:16) 
−1



(

)(cid:16)(cid:107) 

D




=1
|
|2
'
−1 

















|2(cid:1) 



(cid:17)


(cid:107)2

2('
) + (cid:107) 

D








(cid:107)2


2('
)






(3.25)

Therefore, (3.21) arises from (3.22), (3.23), (3.24), and (3.25) by induction and the proof is
complete.

61

The following assumptions allow us to prove the uniqueness of the bilinear form by exclud-
| = 0 for 
 = 1, · · · , 
 and


, 

D







, 

D






'


'



(cid:1)

| + |(cid:0) 

D








(cid:1)






 
, 
D



 
)| = 0.

|(0D

Assumption 2. For 
 ∈ X


ing the solutions to |(cid:0) 

D




(

)(cid:16)|(cid:0) 

D



 =



when 
 = 1, · · · , 
, and '


ˆ 





sup

∈X


















(cid:1)

'


| + |(cid:0) 

D








, 

D







, 

D






|(cid:17)


(cid:1)

'





 > 0, ∀
 ∈ X


(
 
 , 
 
).


=1

ࣔ


Assumption 3. For 
 ∈ 
,


(
; 
2('
))
0 ∀
 ∈ 
,


(
; 
2('
)).

sup

0ࣔ
∈
,


(
;
2('
))

´ 








 
(
) |(0D



 
, 
D



 
)|

 >

X
, then(cid:0) 

D2


ˆ 





'

Consequently, we derive








and









ˆ 
















In Lemma 2.4.3 in chapter 2, it is shown that if 1 < 2

 < 2 for 
 = 1, · · · , 
 and 
, 
 ∈
.


(cid:1)


, 

D






'






= (cid:0) 

D



(cid:1)

(cid:1)

'


'










.

'



, 

D







, 

D


























'


'





 =










, 

D







, 
(cid:1)
= (cid:0) 

D



, 
(cid:1)


(

)(cid:0) 

D2



, 
(cid:1)


(

)(cid:0) 

D2




(

)(cid:16)|(cid:0) 

D


ˆ 






 

(cid:16)(

,

);
2('


)(cid:17) |
|


(cid:1)
, and(cid:0) 

D2



, 
(cid:1)


(

)(cid:0) 

D


ˆ 






(

)(cid:0) 

D



(cid:1)
| + |(cid:0) 

D




 

(cid:16)(

,

);
2('


)(cid:17) ,








(cid:27) |
|


, 

D






ˆ 
















 =

'


'























|(cid:17)


(cid:1)

'







(3.28)


, 

D






Additionally, in the light of chapter 2, we have

(3.26)

(3.27)

for 
 = 1, · · · , 
, where Assumption 2 holds.

Next, we study the property of the fractional time-derivative in the following lemmas.

62

Lemma 3.3.2. If 0 < 2



 < 2



 < 1 (1 < 2



 < 2



 < 2) and 
, 
 ∈ 
,


(
), when

|
=0(= 



ˆ 






(
)(cid:0)



 |
=0) = 0, then
ˆ 






(
)(cid:0)


 
, 
(cid:1)














 

 =

0D2


where 
 = (0, 
), 0 < 
(
) ∈ 
1(cid:16)[



, 



](cid:17)

 =(cid:0)

 
, 
(cid:1)

Proof. It follows from [70] that for 
, 
 ∈ 

(
), when 
|
=0(= 


0, we have


 
(cid:1)


 
, 
D


(cid:0)

.

0D2


0D



 
(cid:1)

0D



 
, 
D



 

,

(3.29)



 |
=0) = 0 and 
|
=
(= 





 |
=
) =


 .

(3.30)

Let 0 < 2



 < 2



 < 1 (1 < 2



 < 2



 ≤ 2), and  = 
 × '
, where 
 = (0, 
) and

Then (3.29) arises from (3.30).

'
 =



,


(cid:16)
(cid:110)

:=


=1(

, 

). We define


; 
2('
)(cid:17)

which is endowed with the norm (cid:107) · (cid:107)

(cid:107)
(cid:107)
,


(
;
2('
)) =

Similarly, we define


; 
2('
)(cid:17)


,


(cid:16)
(cid:110)

:=


 | (cid:107)
(
, ·)(cid:107) 
2('
) ∈ 
,


(
), 
|
=0(=

|
=0) = 
|

=

 = 
|

=

 = 0, 
 = 1, · · · , 


,









,


(cid:16)
(cid:13)(cid:13)(cid:13) (cid:107)
(
, ·)(cid:107) 
2('
)
(cid:16)ˆ 






;
2('
)(cid:17), where we have

(cid:13)(cid:13)(cid:13)
,


(
)

=








(
) (cid:107) 0D



 (
)(cid:107)2


2() 

 + (cid:107)
(cid:107)2


2()

(3.31)

(cid:111)

(cid:111)

(cid:17) 1

2 .

(3.32)

(3.33)

(cid:17) 1

2 .

(3.34)


 | (cid:107)
(
, ·)(cid:107) 
2('
) ∈ 
,


(
), 
|
=
(=








|
=0) = 
|

=

 = 
|

=

 = 0, 
 = 1, · · · , 


,

which is equipped with the norm (cid:107) · (cid:107)
,


(
;
2('
)). Following (3.32),

(cid:107)
(cid:107)
,


(
;
2('
)) =

=

(cid:13)(cid:13)(cid:13) (cid:107)
(
, ·)(cid:107) 
2('
)
(cid:16)ˆ 





(cid:13)(cid:13)(cid:13)
,


(
)


(
) (cid:107) 
D



 (
)(cid:107)2


2() 

 + (cid:107)
(cid:107)2


2()







63

Lemma 3.3.3. For 
 ∈ 
,


(
; 
2('
)) and 0 < 2



 < 2



 < 1 (1 < 2



 <
´ 





 
)| 

 ≤ (cid:107)
(cid:107)
,


(
;
2('
)) (cid:107)
(cid:107)
,


(
;
2('
)) ∀
 ∈




 
(
) |(0D

2



 < 2),

,


(
; 
2('
)).


 
, 
D


Proof. From Lemma 2.4.7 in chapter 2 we have


 
)| ≤ (cid:16)(cid:107) 0D


|(0D



 
, 
D



 
(cid:107)2


2() + (cid:107)
(cid:107)2


2()

2(cid:16)(cid:107) 
D

(cid:17) 1


 
(cid:107)2


2() + (cid:107)
(cid:107)2


2()

(cid:17) 1

2 .

Followingly, by Hölder inequality

=







ˆ 




ˆ 




≤(cid:16)ˆ 




(cid:16)ˆ 

















=








(
) |(0D

ˆ

(
)
ˆ 

ˆ

'


'


0


 
, 
D

ˆ 



 
)|


| 0D

0

(
) | 0D


 
(cid:107)2


(
) (cid:107) 0D



2() 

 + (cid:107)
(cid:107)2


 
|2 


'



 
| | 
D



 
|

 
'
 



2(cid:16)ˆ 




(cid:17) 1
(cid:17) 1
×(cid:16)ˆ 






2()







2







= (cid:107)
(cid:107)
,


(
;
2('
)) (cid:107)
(cid:107)
,


(
;
2('
)).

2

(cid:17) 1
(cid:17) 1

2

(3.35)

ˆ

ˆ 


'


0


(
) | 
D



 
|2 


'



(
) (cid:107) 
D



 
(cid:107)2


2() 

 + (cid:107)
(cid:107)2


2()

Lemma 3.3.4. For any 
 ∈ 
,


(
; 
2('
)) and 0 < 2



 < 2



 < 1 (1 < 2



 <
2



 ≤ 2) there exists a constant 
 > 0 and independent of 
 such that

´ 








 
(
) |(0D



 
, 
D

|
|
,


(
;
2('
))


 
)|



≥ 
|
|
,


(
;
2('
)),

(3.36)

sup

0ࣔ
∈
,


(
;
2('
))

under Assumption 3.

let V
 = 
(
 − 
)(cid:0)
 − 
|
=
(cid:1) assuming that

Proof. Following Lemma 2.4 in [110] and Lemma 2.4.8 in chapter 2, for any 
 ∈ 
,


(
; 
2('
))

 
|
=
)| > 0, where 
(
)

´ 








 
(
) |(0D



 
, 
D


64

is the Heaviside function. Evidently, V
 ∈ 
,


(
; 
2('
)). From Hölder inequality, we obtain


,


(
;
2('
))

(
) (cid:107) 
D





(cid:18)


(
 − 
)(cid:0)
 − 
|
=
(cid:1)(cid:19)
(cid:18)

(
 − 
)(cid:0)
 − 
|
=
(cid:1)(cid:19)
(cid:18) 
 
(
 − 
)

(
 − 
) 
(cid:0)
 − 
|
=
(cid:1)
(cid:19)
(cid:18)















I1−


I1−


I1−












(cid:107)2

2() 


(cid:107)2

2() 



(cid:0)
 − 
|
=
(cid:1) + 
(
 − 
) 
(cid:0)
 − 
|
=
(cid:1)





(cid:107)2

2() 




(
) (cid:107) 




(
) (cid:107) 




(
) (cid:107) 



=

=







(cid:107)V
(cid:107)2
ˆ 




ˆ 




ˆ 




ˆ 




ˆ 























=







=

=

(cid:19)

(cid:107)2

2() 



(3.37)

Regarding (12) in [61],


(
) (cid:107) 
D



 
(cid:107)2


2() 

.
ˆ 





(cid:107)V
(cid:107)2
Hence, (cid:107) 
D

ˆ 






,


(
;
2('
)) (cid:27)

2() (cid:27) (cid:107) 0D


V
(cid:107)2

 
, 
D



(
) |(0D









 
(cid:107)2


(
) (cid:107) 0D







2(). Therefore,
ˆ 




ˆ 












V
)| 

 =


 
(cid:107)2


2() 

 = (cid:107)
(cid:107)2


,


(
;
2('
)).

ˆ

'


ˆ

ˆ 

| 0D

ˆ 

| 0D


0

0


(
)


(
)


 
| | 
D



V
|

 
'
 




 
|2

 
'
 



(3.38)

≥ ˜

= |
|2






'


,


(
;
2('
)),

where ˜
 > 0 and independent of 
. Considering (3.37) and (3.38), we obtain

´ 








 
(
) |(0D



 
, 
D

|
|
,


(
;
2('
))


 
)|



sup

0ࣔ
∈
,


(
;
2('
))

´ 








 
(
) |(0D



 
, 
D

|V
|
,


(
;
2('
))

≥
≥ ˜
 |
|
,


(
;
2('
)).


V
)|



(3.39)

Lemma 3.3.5. If 0 < 2



 < 2



 < 1 (1 < 2



 < 2



 ≤ 2) and 
, 
 ∈ 
,


(
; 
2('
)),
then


 
(cid:1)

0D



 
, 
D




,



(3.40)


(
)(cid:0)


 
, 
(cid:1)

0D2


ˆ 











ˆ 






(
)(cid:0)



 =









65

where 0 < 
(
) ∈ 
1(cid:16)[



, 



](cid:17)

.

Proof. By Lemma 3.3.2,


(
)(cid:0)
ˆ 





ˆ 




ˆ








 
, 
(cid:1)
0D2

ˆ 


(
)

=

'








0

ˆ 





ˆ


(
)



 =








'


 
| 

 

 
'
 =

 
| | 
D


| 0D


ˆ 

| 0D2

ˆ 





0








(
)(cid:0)


 
| |
| 

 
'
 




 
(cid:1)

0D



 
, 
D




.



(3.41)

3.3.2 Solution and Test Function Spaces
Take 0 < 2



 < 2



 < 1 (1 < 2



 < 2



 ≤ 2) and 1 < 2





 = 1, · · · , 
. We define the solution space




< 2








≤ 2 for

associated with the norm

Considering Lemma 3.3.1,

(cid:107)
(cid:107) 
2(
;X
) =

=

(cid:107)
(cid:107)B
,
1,··· ,

 () =


; 
2('
)(cid:17) ∩ 
2(
;X
),

B
,
1,··· ,

() := 
,


(cid:16)
(cid:110)(cid:107)
(cid:107)2

,


(
;
2('
)) + (cid:107)
(cid:107)2
(cid:13)(cid:13)(cid:13) (cid:107)
(
, .)(cid:107)X

(cid:13)(cid:13)(cid:13)
2(
)
(cid:26) 




(

)(cid:16)(cid:107) 

D


ˆ 




(cid:27) 1

(
)(cid:107)2

















2(
;X
)


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

2

.


2('
)

(cid:111) 1

2 .


2() + (cid:107) 

D





 (
)(cid:107)2


2()

(3.42)

(3.43)

(3.44)

(cid:17)






Therefore, from (3.32) and (3.44),

(cid:107)
(cid:107)B
,
1,··· ,

 () =

(cid:110)(cid:107)
(cid:107)2




2() +
ˆ 








+


=1









ˆ 













(

)(cid:16)(cid:107) 

D




(
) (cid:107) 0D

(
)(cid:107)2






 (
)(cid:107)2


2() 




2() + (cid:107) 

D





 (
)(cid:107)2


2()

(cid:17)

(3.45)

(cid:111) 1

2 .






66

Similarly, we define the test space

equipped with the norm

(cid:107)
(cid:107)

,
1,··· ,

 () =


; 
2('
)(cid:17) ∩ 
2(
;X
),

(cid:111) 1

2


2(
;X
)

(
) (cid:107) 
D



 (
)(cid:107)2


2() 





,
1,··· ,

() := 
,


(cid:16)
(cid:110)(cid:107)
(cid:107)2
(cid:110)(cid:107)
(cid:107)2

 

(
;
2('
)) + (cid:107)
(cid:107)2




2() +
ˆ 





ˆ 







(

)(cid:16)(cid:107) 

D









+

=





=1









(
)(cid:107)2


2() + (cid:107) 

D





 (
)(cid:107)2


2()





(3.47)
by Lemma (3.3.1) and (3.32). Take  = 
 × '
 for a positive integer 
. The Petrov-Galerkin
spectral method reads as: find 
 ∈ B
,
1,··· ,

() such that


(
, 
) = 
(
), ∀
 ∈ 

,
1,··· ,

(),

(3.46)

(cid:17)






(cid:111) 1

2

(3.48)

(3.49)



 


(cid:17)

(cid:17)


(
) (0D



 
, 
D



 
)



where the functional 
(
) = ( 
 , 
) and


(
, 
) =

+







ˆ 











ˆ 




ˆ 














=1




−









=1


+ 
(
, 
)



(

)(cid:16)

 
(
 
)(cid:16)




(

D 







, 

D 







) + 


(

D 







, 

D 







)








 
 (
 
 D
 


 



, 
 
 D
 


 



) + 

 
 (
 
 D
 


 



, 
 
 D
 


 



)

following (3.26), (3.27) and Lemma 3.3.5 and 
, 



0 < 2



 < 2



 < 1 (1 < 2



 < 2



 ≤ 2), 0 < 2




2





≤ 2 for 
, 
 = 1, 2, · · · , 
.

, 


 , 










, and 


 are all constant. Besides,
<

< 1 and 1 < 2





< 2











Remark 3.3.6. In the case 
 < 1
equivalence between the weak and strong formulations, see [23] for more details.

2, additional regularity assumptions are required to ensure

67



 and 

 are chosen as the finite-dimensional subspaces ofB
,
1,··· ,

() and 

,
1,··· ,

(),

respectively. Then, the PG scheme reads as: find 

 ∈ 

 such that


(

 , 

) = 
(

), ∀
 ∈ 

 ,

(3.50)


(
) (0D



 

 , 
D



 

)



where


(

 , 

) =

+

−







ˆ 











ˆ 




ˆ 














=1















=1
+ 
(

 , 

).



(

)(cid:104)

 
(
 
)(cid:104)




(

D 








 , 

D 








) + 


(

D 








 , 

D 








)








 
 (
 
 D
 


 




 , 
 
 D
 


 




) + 

 
 (
 
 D
 


 




 , 
 
 D
 


 




)



 


(cid:105)

(cid:105)

(3.51)

Representing 

 as a linear combination of elements in 

, the finite-dimensional problem (3.51)
leads to a linear system, known as Lyapunov system, introduced in Section 3.4.

3.3.3 Well-posedness Analysis

The following assumption permit us to prove uniqueness of the weak form in (3.48) in Theorem
3.3.9.
Assumption 4. For all 
 ∈ 

,
1,··· ,

()


 
, 
D



(
) |(0D



 
(
 
)(cid:16)|(
 
 D
 



 


ˆ 




ˆ 




















|(
, 
)| > 0,



 



∈B
,
1,··· ,

 ()



 



∈B
,
1,··· ,

 ()



 



∈B
,
1,··· ,

 ()
when 
 = 1, · · · , 
.


)|(cid:17)



 
 > 0,


 
)|

 > 0,


, 
 
 D
 


 



)| + |(
 
 D
 


 



, 
 
 D
 


 


68

Lemma 3.3.7. (Continuity) Let Assumption 4 hold. The bilinear form in (3.49) is continuous, i.e.,
for 
 ∈ B
,
1,··· ,

(),

∃
 > 0,

|
(
, 
)| ≤ 
 (cid:107)
(cid:107)B
,
1,··· ,

 ()(cid:107)
(cid:107)

,
1,··· ,

 () ∀
 ∈ 

,
1,··· ,

().

(3.52)

Proof. It follows from (3.28) and Lemma 3.3.3.

Theorem 3.3.8. Let Assumption 4 holds. The inf-sup condition of the bilinear form (3.49) for any

 ≥ 1 holds with 
 > 0, i.e.,

inf

0ࣔ
∈B
,
1,··· ,

 ()

sup

0ࣔ
∈

,
1,··· ,

 ()

|
(
, 
)|

(cid:107)
(cid:107)

,
1,··· ,

 ()(cid:107)
(cid:107)B
,
1,··· ,

 () ≥ 
 > 0,

(3.53)

where  = 
 × '
.
Proof. For 
 ∈ B
,
1,··· ,

() and 
 ∈ 

,
1,··· ,

() under Assumption 4,

ˆ 





|
(
, 
)| (cid:27) |(
, 
)| +
ˆ 




ˆ 





















=1

+




+


=1


















(
) |(0D




(

)(cid:16)|(

D 



 
(
 
)(cid:16)|(
 
 D
 



 







 
, 
D



 
)|




, 

D 







)| + |(

D 







, 

D 







, 
 
 D
 


 



)| + |(
 
 D
 


 



, 
 
 D
 


 



)|(cid:17)

)|(cid:17)






Following (3.28) and Theorem 2.5.2 in chapter 2,
















ˆ 








=1
≥ ˜
1
ˆ 






=1





(

)(cid:16)|(

D


(cid:104)ˆ 






(

)(cid:16)(cid:107) 

D














+













(
), 

D





 (
))|(cid:17)


(

)(cid:16)(cid:107) 

D









 (
), 

D








(

)(cid:16)(cid:107) 

D



(
))| + |(

D





ˆ 





(cid:17)
ˆ 























(

)(cid:16)(cid:107) 

D





 (
)(cid:107) 
2()
(cid:17)









(
)(cid:107) 
2()

(
)(cid:107) 
2()
(cid:105)
(cid:17)






.



 (
)(cid:107) 
2()









69



 
 .

(3.54)

(cid:17)






(3.55)

Thus,






=1
≥ ˜
1

ˆ 
















ˆ 







ˆ 






=1



(

)(cid:16)|(

D




(

)(cid:16)(cid:107) 

D



 
(
 
)(cid:16)(cid:107) 
 
 D
 






 













×
= ˜
1|
|
2(
;X
) |
|
2(
;X
).












=1



 (
), 

D






(
))| + |(

D






(
), 

D










 (
)(cid:107) 
2() + (cid:107) 

D







(
)(cid:107) 
2()
(cid:17)

(
)(cid:107) 
2(),+(cid:107) 
 
 D
 



 
 (
)(cid:107) 
2()



 (
))|(cid:17)
(cid:17)








 


(3.56)

where ˜
1 is a positive constant and independent of 
. Considering Lemma 3.3.4, there exists a
positive constant ˜
2 > 0 and independent of 
 such that

´ 








 
(
) |(0D



 (
), 
D

|
|
,


(
;
2('
))


(
))|



≥ ˜
2|
|
,


(
;
2('
)).

(3.57)

(cid:27)

(3.58)

(3.59)

sup

0ࣔ
∈

,
1,··· ,

 ()

Furthermore, for 
 ∈ B
,
1,··· ,

()

sup

0ࣔ
∈

,··· ,

 ()

and

sup

0ࣔ
∈

,··· ,

 ()






=1

´ 















sup

0ࣔ
∈

,
1,··· ,

 ()

´ 








 
(
) |(0D

´ 








 
(
) |(0D



 (
), 
D

|
|
,


(
;
2('
))

 (
), 
D

|
|B
,
1,··· ,

 ()


(
))|




(
))|




 


(cid:107)
(cid:107) 
2(
;X
)


 
(
 
)(cid:16)|(
 
 D
 


)|(cid:17)

 
(
 
)(cid:16)|(
 
 D
 


)|(cid:17)



 




 


(cid:107)
(cid:107)

,
1,··· ,

 ()


 



, 
 
 D
 


 



)|


, 
 
 D
 


 



)|

|(
 
 D
 


 


+


, 
 
 D
 


 




(cid:107)
(cid:107) 
2(
;X
)
´ 













=1




(cid:27)

sup

0ࣔ
∈

,
1,··· ,

 ()

|(
 
 D
 


 



, 
 
 D
 


 


+

(cid:107)
(cid:107)

,
1,··· ,

 ()

.

70

Therefore, from (3.56), (3.57), (3.58), and (3.59) we have

sup

0ࣔ
∈

,
1,··· ,

 ()
≥ ¯


0ࣔ
∈

,
1,··· ,

 ()




|
(
, 
)|

|(
, 
)| +

(cid:107)
(cid:107)

,
1,··· ,

 ()
´ 








 
(
) |(0D

(cid:107)
(cid:107)

,
1,··· ,

 ()

, 
 
 D
 


 

(cid:107)
(cid:107)

,
1,··· ,

 ()




(cid:16)(cid:107)
(cid:107) 
2() + |
|
,


(
;
2('
)) + |
|
2(
;X
)


 
(
 
)(cid:16)|(
 
 D
 


sup
´ 












(cid:17)


=1


 


+

≥ ¯
 ¯



 
, 
D



 
)|




)| + |(
 
 D
 


 



, 
 
 D
 


 



)|(cid:17)



 


,

(3.60)

where ¯
 = 


{ ˜
2, ˜
1}. Accordingly,



 


0ࣔ
∈B
,
1,··· ,

 ()



 


0ࣔ
∈

,
1,··· ,

 ()

|
(
, 
)|

(cid:107)
(cid:107)

,
1,··· ,

 ()

≥ 
 (cid:107)
(cid:107)B
,
1,··· ,

 (),

(3.61)

where 
 = ¯
 ¯
 is a positive constant and independent.

Theorem 3.3.9. (Well-Posedness) For 0 < 2



 < 2



 < 1 (1 < 2



 < 2



 ≤ 2),
≤ 2, and 
 = 1, · · · , 
, there exists a unique solution to (3.50), which is
1 < 2





continuously dependent on 
 ∈(cid:0)B
,
1,··· ,

(cid:1)ਭ(), where(cid:0)B
,
1,··· ,

(cid:1)ਭ() is the dual space of

< 2











B
,
1,··· ,

().

Proof. In virtue of the generalized Babuška-Lax-Milgram theorem [62], the well-posedness of the
weak form in (3.48) in (1+ 
) dimensions is guaranteed by the continuity and the inf-sup condition,
which are proven in Lemma 3.3.7 and Theorem 3.3.8, respectively.

3.4 Petrov Galerkin Method

To construct a Petrov-Galerkin spectral method for the finite-dimensional weak form problem
in (3.50), we first define the proper finite-dimensional basis/test spaces and then implement the
numerical scheme.

71

3.4.1 Space of Basis (

) and Test (

) Functions
As discussed in chapter 2, we take the spatial basis, given in the standard domain 
 ∈ [−1, 1]

as 

(
) = 

(cid:0)

+1(
) − 

−1(
)(cid:1), 
 = 1, 2, · · · , where 

(
) is the Legendre polynomials

of order 
 and 

 = 2 + (−1)
. Besides, employing Jacobi poly-fractonomials of the first
kind [56, 101], the temporal basis functions are given in the standard domain 
 ∈ [−1, 1] as

(
) = 

(1 + 
)
 
−
,


−1 (
), 
 = 1, 2, . . ..



We also let 
(
) = 2
/
 − 1 and 
 
(
) = 2

− 1 to be temporal and spatial affine mappings


−
 


 
−
 


from 
 ∈ [0, 
] and 
 
 ∈ [
 
 , 
 
] to the standard domain [−1, 1], respectively. Therefore,

(cid:110)(cid:16)

(cid:17)(
)

(cid:16)






=1



 = 






 ◦ 


 




 
 ◦ 
 


(cid:17)(
 
) : 
 = 1, 2, · · · ,N , 
 
 = 1, 2, · · · ,M 


(cid:111)

.

(cid:110)(cid:16)

Similarly, we employ Legendre polynomials and Jacobi polyfractonomials of second kind in the
standard domain to construct the finite dimensional test space as

(cid:17)(
)
(cid:16)




−1 (
), 
 = 1, 2, · · · and 
(
) =(cid:101)

(cid:0)


 (
) =(cid:101)

(1 − 
)
 

,−


(cid:17)(
 
) : 
 = 1, 2, · · · ,N , 
 
 = 1, 2, · · · ,M 



+1(
) − 


(cid:111)

−1(
)(cid:1), 
 =

where 


1, 2, · · · . The coefficient(cid:101)

 is defined as(cid:101)

 = 2 (−1) 
 + 1.



 = 






 ◦ 




◦ 
 



 



=1

,

Since the univariate basis/test functions belong to the fractional Sobolev spaces (see [56])
and 0 < 
(
) ∈ 
1((



, 



)), 0 < 
 
(
 
) ∈ 
1((




)) for 
 = 1, · · · , 
, then


 ⊂ B
,
1,··· ,

() and 

 ⊂ 

,
1,··· ,

(). Accordingly, we approximate the solution in
terms of a linear combination of elements in 

, which satisfies initial and boundary conditions.

, 











3.4.2

Implementation of the PG Spectral Method

The solution 

 of (3.50) can be represented as

N


M1


M



· · ·


=1


1=1



=1



(
, 
) =

ˆ

,
1,··· ,



72

(cid:104)


(
)






 
 (
 
)(cid:105)






=1

(3.62)


 (
)






(cid:16)

in  and also we take 

 = 

by replacing 

 and 

 in (3.50), we obtain the following Lyapunov system


=1 
 


(
 
), 
 = 1, 2, . . . ,N, 
 
 = 1, 2, . . . ,M 
. Accordingly,


 ⊗ 
1 ⊗ 
2 · · · ⊗ 

 +




[

 ⊗ 
1 ⊗ · · · ⊗ 
 
−1 ⊗ 

 




 ⊗ 
 
+1 · · · ⊗ 

]


=1
+ 


 ⊗ 
1 ⊗ 
2 · · · ⊗ 



(cid:17)U = 
,

(3.63)

(3.64)

in which ⊗ represents the Kronecker product, 
 denotes the multi-dimensional load matrix whose
entries are given as

ˆ


 (
, 
1, · · · , 

)(cid:16)



,
1,··· ,

 =



(cid:17)(
)

(cid:16)






=1


 ◦ 

 


◦ 
 



 


(cid:17)(
 
) 
,





 





 




, 





= 

 






 





 




 





 − 

 


 



 + 

 
 


 



 − 

 
 


 

, 


. The matrices 


, 


and 
 
 



 and 

 denote the temporal
stiffness and mass matrices, respectively; 

, and 
 
 denote the spatial stiffness and
mass matrices. The entries of spatial mass matrix 
 
 are computed analytically, while we employ
proper quadrature rules to accurately compute the entries of temporal mass matrix 

 as discussed
in chapter 2. The entries of 



 are also computed based on Theorem 3.1 (spectrally/exponentially
accurate quadrature rule in 
-dimension) in [70]. Likewise, we present the computation of 

 


in
Lemma B.1.1 in Appendix B.




Remark 3.4.1. The choices of coefficients in the construction of finite dimensional basis/test
functions lead to symmetric mass/stiffness matrices, which help formulating the following fast
solver.

3.4.3 Unified Fast FPDE Solver

In order to formulate a closed-form solution to the Lyapunov system (3.63), we follow [55] and
develop a fast solver in terms of the generalized eigen-solutions.

Theorem 3.4.2. Take {(cid:174)
 


 

matrix 

 




 with respect to the mass matrix 
 
. Besides, let {(cid:174)




 
 =1 as the set of general eigen-solutions of the spatial stiffness

=1 be the set of general


 }N


, 




 
 }M 

, 
 


73

eigen-solutions of the temporal mass matrix 

 with respect to the stiffness matrix 


unknown coefficients matrix U is obtained as


 . Then the

· · ·



,
1,··· , 

 (cid:174)




 ⊗ (cid:174)
1


1 ⊗ · · · ⊗ (cid:174)







,

(3.65)

N


M1


U =

M




=1


1=1



=1

where

and



,
1,··· , 

 =

(cid:104)((cid:174)





 




 (cid:174)



'
,
1,··· ,

 =

Proof. See theorem 2.8.1 in chapter 2.


1 · · · (cid:174)



 
 (cid:174)
 



 


( (cid:174)
 


 (cid:174)
1

=1((cid:174)
 




)


(cid:104)(1 + 
 





)



 
)(cid:105)






) + 




=1(
 



 
)(cid:105)

.

'
,
1,··· ,



,

(3.66)

Remark 3.4.3. The naive computation of all entries in (3.66) leads to a computational complexity of

(N2+2
), including construction of stiffness and mass matrices. By performing sum-factorization
[55], the operator counts can be reduced to 
(N2+
).

3.5 Stability and Error Analysis

The following theorems provide the finite dimensional stability and error analysis of the pro-

posed scheme, based on the well-posedness analysis from Section 3.3.3.

3.5.1 Stability Analysis

Theorem 3.5.1. Let Assumption 4 hold. The Petrov-Galerkin spectral method for (3.51) is stable,
i.e.,



 


0ࣔ

∈





 

0ࣔ
∈



|
(

 , 

)|

(cid:107)

(cid:107)

,
1,··· ,

 ()(cid:107)

(cid:107)B
,
1,··· ,

 ()

≥ 
 > 0,

(3.67)

holds with 
 > 0 and independent of 
.
Proof. Regarding 

 ⊂ B
,
1,··· ,

() and 

 ⊂ 

,
1,··· ,

(), (3.67) follows directly from
Theorem 3.3.8.

74

Remark 3.5.2. The bilinear form (3.51) can be expanded in terms of the basis and test functions
to obtain the lower limit of 
, see [55].








=1

ࣔ 


< 2






 =



3.5.2 Error Analysis
Denoting by 
M(') the space of all polynomials of degree ≤ M on ' ⊂ R, 


M(') := 
M(') ∩



('), where 0 < 
(
) ∈ 
1((



, 



)) and 


(') is the distributed Sobolev space
associated with the norm (cid:107) · (cid:107)


(').
In this section, we take 
0 = (0, 
), 

 = (

, 

) for

 = 1, ..., 
, '
 = 

 × '
−1, and ' 



. Besides, 0 < 2



 < 2



 < 1 (1 < 2



 <
≤ 2 for 
 = 1, · · · , 
. Where there is no confusion, the symbols 

,
2



 ≤ 2), 1 < 2




) will be dropped from the notations.
'
, and ' 

Theorem 3.5.3. [115] Let 
1 be a real number, where 
1 ࣔ M1 + 1
2, and 1 ≤ 
1. There
('1) such that for any

1M1
exists a projection operator 

(cid:107)
 

1('1) ≤ 
1M
1−
1
(cid:107)
(cid:107)

1('1), where 
1

 ∈ 

1('1) ∩ 

is a positive constant.
Theorem 3.5.4. [70] Let 
0 ≥ 1, 
0 ࣔ N + 1

,


(
) to 




 and the intervals of (



, 



) and (





2. There exists an operator 


0 ('1), we have (cid:107)
 − 

1


0,N from 

0(
) ∩

from 

1('1) ∩ 


0 ('1) to 


1


1

1,M1


1

1,M1

, 





N('1) such that for any 
 ∈ 

0(
) ∩ 
,


(
), we have
(cid:107)
 − 


ˆ 






 

(
) ≤ 
0 N−2
0


0,N 
(cid:107)2








(
) N2
(cid:107)
(cid:107)

0(
) 

,







1

where 
0 is a positive constant and 0 < 
(
) ∈ 
1((



, 



)).

In the following, employing Theorems 3.5.3 and 3.5.4 and also Theorem 2.10.3 in chapter 2,
we study the properties of higher-dimensional approximation operators in the following Lemmas.
(
1) from
Theorem 3.5.5. Let 
1 ≥ 1, 
1 ࣔ M1 + 1


1(
1) ∩ 
,

 
1(
1) to 



1

1,M1
(
1) such that for any 
 ∈ 

1(
1) ∩ 
,

 
1(
1), we have

1M1

,

 
1(
1) ≤ M−2
1

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

1(
1) 

1,

2. There exists a projection operator 

(cid:107)
 − 


1

1,M1


1(
1) M2
1
1

ˆ 





1

1





1

where 0 < 
1(
1) ∈ 
1((




1

, 





1

)).

75

Proof. From Theorem 3.5.3 for 
 ∈ 

1 ∩ 


1 we have

(cid:107)
 

1('1) ≤ M
1−
1

(cid:107)
 − 


1

1,M1

1

(cid:107)
(cid:107)

1('1).

Therefore, for 
 ∈ 

1(
1) ∩ 
 
 
1(
1) we have

ˆ 





(cid:107)
 − 

1

1,M1
ˆ 





≤

1





1


(cid:107)2


,

 
1(
1) =

1(
1) M2
1−2
1

1

1





1
(cid:107)
(cid:107)2


1(
1) (cid:107)
 − 


1('1) 

1 = M−2
1

1


(cid:107)2

1

1,M1
ˆ 





1





1


 

1('1) 

1

1(
1) M2
1
1

(cid:107)
(cid:107)

1(
1) 

1.

Lemma 3.5.6. Let the real-valued 1 ≤ 
1, 
2 and  = 
1 × 
2. If 
 ∈ 
,

 
2(
2, 

1(
1)) ∩


2(
2, 
,


(cid:107)
 − 
M−2
2


(cid:107)2B 
1,
2() ≤


2


2,M2

(cid:17)

0 (
1)), then

1

1

1,M1
ˆ 




2





ˆ 




2


2(
2)(cid:16)M2
2

1(
1)(cid:16)M2
1

2

1

1





1

2

1

+M−2
1
+M−2
2

2

where (cid:107)·(cid:107)B 
1,
2() =(cid:8)(cid:107)·(cid:107)2

(cid:107)
(cid:107)

 
1(
1,

2(
2)) + M−2
1

 
1(
1,
2(
2))+(cid:107)·(cid:107)2

1

2 M−2
1
(cid:107)
(cid:107)

2(
2,
2(
1)) + M2
2
1 M−2
2
(cid:107)
(cid:107)

1(
1,
2(
2)) + M2
1
(cid:107)
(cid:107)

 
2(
2,

1(
1)),
(cid:9) 1

1

2


 
2(
1,
2(
1))



2

(cid:107)
(cid:107)

2(
2,

1(
1))
(cid:107)
(cid:107)

1(
1,

2(
2))

(cid:17)



1

(3.68)

2 , 0 < 
1(
1) ∈ 
1([




1

, 





1

]),

]).

, 





and 0 < 
2(
2) ∈ 
1([




2
Proof. For 
 ∈ 
,

 
2(
2, 

1(
1)) ∩ 

2(
2, 
 
1(
1)), evidently 
 ∈ 

2(
2, 

1(
1)), 
 ∈


2(
2, 
2(
1)), and 
 ∈ 

1(
1, 
2(
2)). Besides, from the definition of (cid:107) · (cid:107)B 
1,
2() we have

2

(cid:107)
 − 

=(cid:8)(cid:107)
 − 


1

1,M1

1

1,M1


2


2,M2

2


2,M2


(cid:107)B 
1,
2()



 
1(
1,
2(
2)) + (cid:107)
 − 

(cid:107)2


1

1,M1


2


2,M2


(cid:107)2



 
2(
2,
2(
1))

76

(cid:9) 1

2 .

Following Lemma 2.10.3 in chapter 2 and Theorem 3.5.5, (cid:107)
− 
can be simplified to


1

1,M1


2


2,M2


(cid:107)2



 
2(
2,
2(
1))


(cid:107)2

 − 


2


2,M2

2

2,M2



 
2(
2,
2(
1))

2

1


2,M2

1,M1


(cid:107)2



 
2(
2,
2(
1))



2(
2,
2(
1)) 

2


 
2(
2,
2(
1)) + (cid:107)
 − 


2(
2,
2(
1)) 

2


1

1,M1


(cid:107)2



 
2(
2,
2(
1))

(3.69)


2(
2) M2
2
2

(cid:107)
(cid:107)2



2(
2,

1(
1)) 

2 + M−2
1

1

(cid:107)
(cid:107)2



 
2(
2,

1(
1)),
(3.70)

(cid:107)
 − 
= (cid:107)
 − 
≤ (cid:107)
 − 
≤ M−2
2

2
+ (cid:107)(
≤ M−2
2

2


(cid:107)2


 − 


1

2


1,M1

2,M2


 
2(
2,
2(
1))

 + 

2

2

1

2,M2

2,M2

1,M1


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

(cid:107)2

2

2,M2
ˆ 





2(
2) M2
2
2
2





2
− I)(
 − 

2

2,M2

(cid:107)
(cid:107)2


)(cid:107)2


1

1,M1

2(
2) M2
2
2

(cid:107)
(cid:107)2

ˆ 





2





2

+ M−2
2

2 M−2
1

1

ˆ 





2





2


(cid:107)2


 − 

where I is the identity operator. Furthermore,
(cid:107)
 − 
= (cid:107)
 − 
≤ (cid:107)
 − 
≤ M−2
1


2

1


2,M2

1,M1

2(
2,
 
1(
1))

 + 

1

1

1

1,M1

1,M1

1,M1

(cid:107)2


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

1

1,M1
ˆ 





1(
1) M2
1
1
1





1
− I)(
 − 

1

1,M1

(cid:107)
(cid:107)2


)(cid:107)2

1
+ (cid:107)(
≤ M−2
1

1

ˆ 





1





1


2

2,M2

1(
1) M2
1
1
ˆ 





+ M−2
2

2 M−2
1

1


1(
1) M2
1
1

1





1


(cid:107)2

 − 


2


2,M2

1

1,M1



 
1(
1,
2(
2))

2

1


2,M2

1,M1


(cid:107)2



 
1(
1,
2(
2))



1(
1,
2(
2)) 

1


 
1(
1,
2(
2)) + (cid:107)
 − 


1(
1,
2(
2)) 

1

(cid:107)
(cid:107)2


2

2,M2


(cid:107)2



 
1(
1,
2(
2))

(cid:107)
(cid:107)2



1(
1,

2(
2)) 

1 + M−2
2

2

(cid:107)
(cid:107)2



 
1(
1,

2(
2)).
(3.71)

Therefore, (3.68) can be derived immediately from (3.69) and (3.71).

77

Likewise, Lemma 3.5.5 can be easily extended to the 
-dimensional approximation operator as

(cid:107)
 − Ò



(cid:107)2
≤ M−2








 

 (

,
2('


))
ˆ 


















(

) M2



(cid:107)
(cid:107)2







 (

,
2('



)) 


 +






=1

ࣔ


M−2
 





(cid:107)
(cid:107)2



 

 (

,



 
 (
 
 ,
2('


, 



 )))



(

) M2






M−2
 





(cid:107)
(cid:107)2




 (

,



 
 (
 
 ,
2('


, 



 ))) 







ˆ 
















+ M−2










+


=1

ࣔ



=1

ࣔ
, 


M−2
 



 M−2











=1

ࣔ

(cid:107)
(cid:107)2



 

 (

,




 ,
 
 (

×
 
 ,
2('


, 
,




))))

+ · · · + M−2






ˆ 


















(

) M2











=1

ࣔ


M−
 





(cid:107)
(cid:107)2


 


 (

,

1,··· ,

 ('



))) 


,

.


 = 

· · · 

where Ò






 ,M



1

1,M1

Theorem 3.5.7. Let 1 ≤ 

, 
0 = (0, 
), 

 = (

, 

),  = 
0 ×(cid:16)



 =



 ∈(cid:16) 
∩

, '
 = 



))(cid:17) ∩ 
,


(
0, 

1,··· ,

('
)),



0(
0, 

 

(

, 

1,··· ,

−1,

+1,··· ,

('


≤ 1 for 
 = 1, · · · , 
. If

2 < 







 and 1

< 






=1 




=1

ࣔ 


' 


(cid:17)








=1

(3.72)

(3.73)


=1 

,

78

then,

(cid:107)
 − 



0,N Ò



(cid:107)2B
,
1,··· ,

 ()





=1













(cid:107)
(cid:107)2

M−2
 


ˆ 








(
) N2
(cid:107)
(cid:107)

0(
0,
2('
)) 


ˆ 





(
) N2


(
) N2
(cid:16) 


ˆ 






(

)(cid:110)M2

−2






(cid:17)(cid:107)
(cid:107)


 (

,

1,··· ,

 ('


(cid:107)
(cid:107)


 (

,
2('


M−2
 


M−2
 









=1










≤ N−2
0

+ N−2
0

+ N−2
0





+




ˆ 




(cid:16)







=1


+ M2

−2







=1

ࣔ
, 







0(
0,



 
 (
 
 ,
2('


))) 

 + · · ·
(cid:17)(cid:107)
(cid:107)

0(
0,

1,··· ,

 ('
))) 




×
0)) + · · ·

,
2(
0)))




(cid:111)




,

(3.74)

where Ò


 = 


1

1,M1

· · · 






 ,M


and 
 is a real positive constant.

Proof. Directly from (3.45) we conclude that

(cid:107)
(cid:107)2B
,
1,··· ,

 () ≤ (cid:107)
(cid:107)2
Next, it follows from Theorem 3.5.4 that


 

(
0,
2('
)) +






=1

(cid:107)
(cid:107)2


2(
0,

 

 (

,
2('



))).

(cid:107)
 − 




(cid:107)2

0,N Ò
ˆ 





≤ N−2
0


,


(
0,
2('
))
(cid:107)
(cid:107)2

(
) N2


(cid:34)







M−
 





(cid:17)(cid:107)
(cid:107)

0(
0,

1,··· ,

 ('
)))

(cid:107)
(cid:107)2


=1






0(
0,


(cid:35)



0(
0,
2('
)) +
+(cid:16) 



M−
 






=1


 
 (
 
 ,
2('
))) + · · ·



.

(3.75)

Therefore, (3.74) is obtained immediately from (3.72) and (3.75).

Remark 3.5.8. Since the inf-sup condition holds (see Theorem 3.5.1), by Lemma 3.3.7, the error
in the numerical scheme is less than or equal to a constant times the projection error. Hence the
results above imply the spectral accuracy of the scheme.

79

3.6 Numerical Tests

< 2











the measured 
∞ error, (cid:107)
(cid:107) 
∞ = (cid:107)

 − 



(cid:107) 
∞ as the maximum bound of (cid:107)
(cid:107) 
2.

We provide several numerical examples to investigate the performance of the proposed scheme.
We consider a (1 + 
)-dimensional fully distributed diffusion problem with left-sided derivative
≤ 2 in
by letting 


 = 


 = 


 = 0, 


 = 1, 0 < 2



 < 2



 < 1 and 1 < 2





=1(−1, 1). We report

(3.50) for 
 = 1, · · · , 
, where the computational domain is  = (0, 2) ×

=

vector, given an exact solution 



. Here, we assume 



 = 

 ×



In each of the following test cases, we use the method of fabricated solutions to construct the load

=1 


. We project the spatial
part in each dimension, 


, on the spatial bases, and then, construct the load vector by plugging the
projected exact solution into the weak form of problem. This helps us take the fractional derivative
of exact solution more efficiently, while by truncating the projection with a sufficient number of
terms, we make sure that the corresponding projection error does not dominantly propagate into
the convergence analysis of numerical scheme.

Case I: We consider a smooth solution in space with finite regularity in time as





 = 
 
1+
 ×(cid:0)(1 + 
1) 
2(1 − 
1) 
3(cid:1)

(3.76)

to investigate the spatial/temporal p-refinement. We allow the singularity to take order of 
 = 10−4,
while 
1, 
2, and 
3 take some integer values. We show the 
∞-error for different test cases in
Fig.3.1, where by tuning the fractional parameter of the temporal basis, we can accurately capture
the singularity of the exact solution, when the approximate solution converges as we increase the
expansion order. In each case of spatial/temporal p-refinement, we choose sufficient number of
bases in other directions to make sure their corresponding error is of machine precision order.

Considering 
 = 10−4, 
1 = 2, 
2 = 
3 = 2 in (3.76), and the temporal order of ex-
pansion being fixed (N = 4) in the spatial p-refinement, we get the rate of convergence as
a function of the minimum regularity in the spatial direction. From Theorem 3.5.7, the rate
(cid:107)
(cid:107) 
2() ≤ (cid:107)
(cid:107) 
∞() ≤
of convergence is bounded by the spatial approximation error, i.e.

80

´ 




1





1

1


1(
1) M2
1
1

M−2
1
larity of the exact solution in the spatial direction for 
 < 1
practical rate of convergence ¯
1 = 16.05 in (cid:107)
(cid:107) 
∞() is greater than 
1 ≈ 2.50.

(cid:107)
(cid:107)

1(
1,
2(
0)) 

1, where 
1 = 
2 + 1

2 − 
 is the minimum regu-
2. Conforming to Theorem 3.5.7, the

Figure 3.1: Temporal/Spatial p-refinement for case I with singularity of order 
 = 10−4. (Left):

1 = 3, 
2 = 
3 = 2, and expansion order of N ×9. (Middle): 
1 = 2, 
2 = 
3 = 2, and expansion
order of 3 × M. (Right): 
1 = 3, 
2 = 
3 = 2, and expansion order of 4 × M.
Case II: We consider 



 = 
 
1+
 sin(2

1), where 
1 = 3, and let 
 = 0.1 and 
 = 0.9. We
set the number of temporal basis functions, N = 4, and show the convergence of approximate
solution by increasing the number of spatial basis, M in Fig. 3.2. The main difficulty in this
case is the construction of load vector. To accurately compute the integrals in the load vector,
we project the spatial part of forcing function, sin(2

1), on the spatial bases. To make sure that
the corresponding error is of machine-precision order and thus, not dominant, we truncate the
projection at 25 terms, where there error is of order 10−16. Therefore, the quadrature rule over
derivative order should be performed for 25 terms rather than only a single sin(2

1) term. This
will increase the computational cost.
Case III: (High-dimensional p-refinement) We consider the exact solution of the form





 = 
 
1+
 × 3


(1 + 

) 
2
(1 − 

) 
2
+1

(3.77)


=1

with singularity of order 
 = 10−4, where 
1 = 3, and 
2
 = 
2
+1 = 1. Similar to previous cases,
we set the number of temporal bases, N = 4, and study convergence by uniformly increasing the
number of spatial bases in all dimensions. Fig. 3.3 shows the results for (1 + 2)-dimensional and
(1 + 3)-dimensional problems with expansion order of N × M1 × M2, and N × M1 × M2 × M3,

81

Figure 3.2: Spatial p-refinement for case II, 
1 = 3, 
 = 0.1, and 
 = 0.9.

respectively. Following Case I, the computed rate of convergence ¯
1 = ¯
2 = ¯
3 = 16.13 in (3.77)
for 
 = 10−4 is greater than the minimum regularity of the exact solution 
 ≈ 2.05, which is in
agreement with Theorem 3.5.7.

Figure 3.3: Spatial p-refinement for case III with singularity of order 
 = 10−4. (Left): (1 + 2)-
dimensional, 
1 = 3, 
2
 = 
2
+1 = 1, where the expansion order is N × M1 × M2.
(Left):
(1 + 3)-dimensional, 
1 = 3, 
2
 = 
2
+1 = 1, where the expansion order is N × M1 × M2 × M3.

In addition to the convergence study, we examine the efficiency of the developed method and
fast solver by comparing the CPU times for (1 + 1)-, (1 + 2)-, and (1 + 3)-dimensional space-
time hypercube domains in case III. The computed CPU times are obtained on an INTEL(XEON
E52670) processor of 2.5 GHZ, and reported in Table 3.1.

82

problems. 



 = 
 
1+
 ×
3

Table 3.1: CPU time, PG spectral method for fully distributed (1+d)-dimensional diffusion

=1(1 + 

) 
2
(1 − 

) 
2
+1, where 
 = 10−4, 
1 = 3, and the
expansion order is 4 × 11
.

CPU Time [Sec]

(cid:107)
(cid:107) 
∞()

d=1

1546.81
6.84 × 10−12


2
 = 
2
+1 = 2

d=2

1735.03
4.45 × 10−12


2
 = 
2
+1 = 3

d=2

1786.61
3.86 × 10−12

d=3

2407.22
2.71 × 10−12

d=3

2358.67
3.27 × 10−12

d=1

1596.16
6.27 × 10−12

83

CHAPTER 4

A FRACTIONAL SUBGRID-SCALE MODEL FOR TURBULENT FLOWS

4.1 Background

Due to the remarkable advancements in computational capabilities over the last decades, large
eddy simulations (LES) have been introduced as a powerful approach in the computation of turbulent
structures in [138, 139]. In modeling subgrid-scale (SGS) structures in LES of turbulent flows,
spatially-filtered representation of a turbulent field is required for a priori and a posteriori analyses.
As a key ingredient in the development of SGS models in turbulent flows, the statistical behavior
of small scale motions and their cumulative effects on the evolution of the large scales should be
incorporated. In comprehensive studies, including numerical and empirical approaches (see e.g.,
[31, 73, 74, 75, 76]), the intermittent statistical behavior of velocity gradients and the development
of anomalously intense fluctuations were investigated. These studies confirmed the non-Gaussian
statistics of SGS structures and the existence of intermittency in the inertial sub-range of turbulence.
By measuring the Lagrangian velocity of tracer particles in a turbulent flow, Mordant et al. in [140]
explored the intermittent statistics of probability distribution functions (PDF) of the velocity time
increments, which is even more highlighted than the corresponding Eulerian spatial increments.
Arnéodo et al. in [36] investigated the intermittency and universality properties of velocity temporal
fluctuations in highly turbulent flows by quantitatively comparing experimental and numerical data.
They described a stochastic phenomenological modelization in the entire range of scales, using a
multifractal description, which links Eulerian and Lagrangian statistics. Recently, Buzzicotti et al.
in [141] performed a priori analyses of statistical characteristics of resolved-to-subfilter scale (SFS)
energy transfer. They quantified the intermittent scaling of the SFS energy transfer as a function
of filtering type and described its non-trivial, anomalous deviations from the classical scaling as a
function of cutoff scale. In fact, the anomalous behavior of turbulent small scales monotonically
deviates from Gaussianity by enlarging the filter width (see e.g., [142, 143, 144, 145]). This means

84

that filtering a turbulent field incorporates nonlocal interactions of SGS motions into the resolved
scales, which is reflected in heavy-tailed distributions of velocity increments.

Before any discussion on the various strategies of SGS modeling, the reader is referred to
[146, 147, 148, 138, 81, 149, 150] for more history and background on LES of turbulent flows.
The SGS modeling strategies are categorized into (I) functional and (II) structural modelings (see
e.g., [147]). In the functional strategies, the closure problem can be expressed in the form of a
mathematical operator, which is acting on the mean velocity field. Such turbulence models seek
only to generate the net kinetic energy transfer from the resolved to small scales (see e.g., [151]).
However, structural modeling strategies would approximate the SGS stresses in terms of the filtered
velocity field, where the SGS structures and statistical properties are recovered from the resolved
scale information. Multifractal modelings were introduced in [152, 153] as a structural approach
to model the underlying intermittent cascading of energy. More specifically, in a study by Burton
and Dahem [154], a new approach was presented on the multifractal modeling of subgrid-scale
stresses in LES of turbulent flows motivated by a priori testing. Subsequently, Rasthofer and
Gravemier in [155] proposed a new method of SGS modeling from a multifractal description of the
vorticity field. Regarding the non-Gaussain statistics of small scale motions and nonlocal effects in
turbulent flows, Hamilington and Dahem in [156] obtained a nonlocal closure modeling from a new
derivation of the rapid pressure strain correlation. Recently, Maltba et al. [157] presented a new
semi-local formulation employing a modified large eddy diffusivity (LED) approach, which retains
the accuracy of a fully nonlocal approach. It turns out that in formulating SGS models, standard
integer-order operators have commonly been used to mathematically represent the anomalous
features of small scale motions.

In addition to the considerable progresses in developing nonlocal models using standard meth-
ods, fractional calculus appears to be a mathematical tractable tool to describe anomalous phenom-
ena, manifesting in nonlocal interactions, self-similar structures, sharp peaks, and memory effects
(see e.g., [42, 43]). It seamlessly generalizes the notion of standard integer-order calculus to its
fractional-order counterpart, which leads to a broader class of mathematical models. Cushman

85

and Moroni in [158] developed a theory for modeling anomalous dispersion, which relied on the
intermediate scattering function. In another experimental work in [159], they obtained the inter-
mediate scattering function using the Lagrangian trajectories for a conservative tracer in a porous
medium. Based on the anomalous characteristics of fluctuation processes in turbulence, several
studies were conducted to explore the nonlocal modeling of turbulent flows. Chen in [160] pro-
posed a fractional Laplacian stochastic equation to describe intermittent cascade of fully-developed
turbulence. Furthermore, Churbanov and Vabishchevich presented a new fractional model to de-
scribe turbulent fluid flows in a rectangular duct, [161]. Recently, Egolf and Hutter [162] proposed
nonlocal turbulent models in the form of fractional operators to generalize Reynolds shear stresses
in local zero-equation models. Epps and Cushman-Roisin in [163] derived the Navier-Stokes (NS)
equations with fractional Laplacian starting from the Boltzmann transport equation. In their study,
they modeled large displacements of fluid particles by Lèvy 
-stable distributions. Moreover, great
progresses have been made towards the theories and numerical solutions to fractional partial dif-
ferential equations (FPDEs). Samiee et al. [60, 61] developed a unified Petrov–Galerkin spectral
method for a class of FPDEs with two-sided derivatives employing the so-called Jacobi poly-
fractonomials. Zayernouri and Karniadakis in [56] introduced Jacobi poly-fractonomials as a new
family of basis/test functions, which are the explicit eigenfunctions of fractional Strum-Liouville
problems in bounded domains of the first and second kind. Zhou et al. developed two efficient first-
and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear
fractional differential equations with fractional order 
 ∈ (0, 1] in [164]. The reader is referred to
[165, 21, 166, 18, 167, 22] and the references given therein for more details on fractional modeling
of anomalous transport.

In comparison with the recent advances in SGS modeling of non-Gaussian features, the devel-
opment of fractional transport modeling of turbulent structures is still at its very early stage. In
the present work, we aim to open up a new perspective to functional modeling of the SGS stresses,
employing the fractional calculus. This approach implies that we never question the correctness of
Navier-Stokes (NS) subject to the Newtonian assumption. Starting from the Boltzmann transport

86

equation, we propose to approximate the filtered Maxwellian equilibrium distribution function of
velocity with a Lévy 
-stable distribution. Accordingly, we derive the filtered NS equations, in
which the divergence of SGS stresses is approximated by the fractional Laplacian of filtered velocity
field. From a physical point of view, (any generic) filtering the flow field in LES would further
contribute to the nonlocal effects, which are rigorously modeled to appear as a fractional Laplacian
term in the filtered NS equations. Certainly, by decreasing the filter width, the super-diffusive
fractional operator gradually vanishes in compliance with the induced nonlocality. Here, we briefly
highlight the main contributions of this work as follows:
• We develop a new functional approach to model the SGS stresses by employing the fractional
Laplacian of the filtered velocity within the Boltzmann transport framework. The fractional
exponent in the model arises from the heavy-tailed behavior of the SGS stresses.
• We show that the model is frame invariant and constrain it to a set of conditions to preserve the
second-law of thermodynamics.
• We perform the a priori studies to assess performance of the model primarily by the correlation
and regression coefficients utilizing the results of direct numerical simulation (DNS) for three-
dimensional forced and decaying homogeneous isotropic turbulence (HIT) problems. We also
investigate the nonlocality of proposed model, as a hallmark of fractional operators, in a range of
filter widths.

The chapter is organized as follows: in section 2, we introduce some preliminaries on fractional
calculus. We outline the problem and discuss the governing equations in section 3. In section 4, we
develop the fractional model from the Boltzmann transport equation and study its mathematical and
physical properties. In section 5, we provide the details of a priori analysis for three-dimensional
forced and decaying HIT and study performance of the proposed fractional model.

4.2 Preliminaries on Fractional Laplacian

For modeling SGS stresses in isotropic turbulent flows, the heavy-tailed behavior of Lévy

-stable distributions are highly in demand due to their success in capturing singularities and

87

modeling anomalous phenomena (see e.g., [31]). From the stochastic point of view, the dynamics
of the SGS features, which is modeled by an isotropic Lévy 
-stable process at the microscopic
level, can be upscaled by a fractional Laplacian operator. Such operator provides a rigorous tool
for the mathematical modeling of nonlocal phenomena [168]. We denote by (−Ɗ)
 the fractional
Laplacian with 0 < 
 ≤ 1,

(−Ɗ)

(
) =

ˆ

|
|2
(cid:0)
, 
−

·
(cid:1)
= F −1(cid:110)|
|2
F(cid:8)
(cid:9)(
)(cid:111)

1
(2
)


R


,


2


·
 



(4.1)
where F and F −1 represent the Fourier and inverse Fourier transforms for a real-valued vector

 = 
 
, 
 = 1, 2, 3, respectively, and 
 denotes the imaginary unit. Moreover, (·, ·)
2 denotes
the 
2-inner product on R
, 
 = 1, 2, 3. The Fourier transform of the fractional Laplacian is then
obtained as

F(cid:110)(−Ɗ)

(
)(cid:111)

= |
|2
F(cid:8)
(cid:9)(
).

It is worth noting that the integer-order Laplacian is recovered when 
 = 1. Considering the
definition of 
-Riesz potential as

I

(
) = 

,−


ˆ


(
) − 
(
)
|
 − 
|
−2




,

R


(4.2)

(4.3)

(4.4)

the fractional Laplacian can also be expressed in the integral form as

(
) − 
(
)
|
 − 
|2
+


(−Ɗ)

(
) = 

,


ˆ

R




,

where 

,
 = 22
Ɖ(
+
/2)


/2Ɖ(−
)

-Riesz potential is also formulated in [170] as

for 2
 ∈ (0, 
) and Ɖ(·) represents Gamma function (see [169]). The

I

(
) = (−Ɗ)−

(
) = F −1(cid:110)|
|−2
F(cid:8)
(cid:9)(
)(cid:111)
|
| F(cid:8)
(cid:9)(
)(cid:111)
R 
 
(
) = ∇ 
 I1
(
) = F −1(cid:110) − 

 


,

.

(4.5)

(4.6)

Considering (4.5), the Riesz transform is then given by

which is dealt with in formulating the SGS stresses in section 4.3.3.

88

4.3 Governing Equations

In the mathematical description of incompressible turbulent flows, the primitive variables,
including the velocity and the pressure fields are represented by 
(
, 
) = (
1, 
2, 
3) and 
(
, 
)
for 
 = 

 and 
 = 1, 2, 3, respectively. In the following, the flow field variables are governed by the
continuity and the Navier-Stokes (NS) equations, given as









+ 


 




 


= − 1




 






+ 1






 



 


,


, 
 = 1, 2, 3,

(4.7)

where 
 denotes the density and the viscous stress tensor 

 
 is defined as 

 
 = 
 ( 





 

which 
 represents the dynamic viscosity for a Newtonian fluid.

+ 

 






), in

In the LES of turbulent flows, the fluid motions are resolved down to some prescribed length
scale, filter width (L), which decomposes the velocity field, 
, into the filtered (resolved), ¯
, and
the residual, 
, components. The filtered velocity field is obtained by convolution, where ¯
 = 
∗

and 
 = 
(
) denotes the kernel of spatial filtering in the convolution (see [138, 139]). To produce
the filtered velocity field and the true values of SGS stresses, we can adopt any generic isotropic
filtering technique. Accordingly, the filtered NS equations in the index form are derived as


 ¯






+ 
 ¯

 ¯




 


= − 1




 ¯






+ 



 


(


) − 
T 



 



 


,


 ¯




 


(4.8)

where the kinematic viscosity is represented by 
 and the SGS stress tensor, T 


 
 = 



 − ¯

 ¯

,
which must be closed in terms of the filtered flow variables. As the most popular eddy-viscosity
closure, we exemplify the Smagorinsky model (SMG), introduced in [171]. The SGS stresses
, 



 = (

L)2 | ¯
|, and
in the SMG are modeled by T 

| ¯
| =


 
 = −2



 ¯

 
, where ¯

 
 =

2 ¯

 
 ¯

 
.

(cid:113)


 ¯




 


+ 
 ¯
 






4.3.1 Boltzmann Transport Equation (BTE)

The kinetic theory aims to describe the motion of particles in a gas from a microscopic point of
view. The state of the gas is obtained by a distribution function 
 (
, 
, 
) in the particle phase space

89

such that 
 (
, 
, 
) 



 is defined as the mass of gas particles in phase space within volume 




centered on 
, 
 at time 
, where 
 and 
 represent gas particle’s location and speed, respectively.
We note that 
, 
, and 
 are independent variables. Let 
 = 
 (
, 
, 
) : R+ × R
 × R
 → R (see
e.g., [163, 77]). Without loss of generality, we take 
 = 3. Then ensemble-averaged macroscopic
flow variables are given by:

ˆ

ˆ

R

1



R



 =



 =


 (
, 
, 
)

,


 
 (
, 
, 
)

,


 = 1, 2, 3,

(4.9)

(4.10)

where 
 and 

 denote the fluid density and the 
-th component of flow velocity field in the NS
equations, respectively. The accurate description of non-reacting ideal gas particles is governed by
the Boltzmann transport equation [77, 78], which is written as

+ 
 · ∇ 
 =(cid:0) 
 


(cid:1)






 









 ≡ − 
 − 
 






,

(4.11)

where 
 

 = 
 

(
, 
, 
) represents the equilibrium distribution function and 
 is the relaxation
time, which is the required time for fluid particles to reach equilibrium state. The left-hand side of
(4.11) represents the streaming of non-interacting particles and the right-hand side expresses the
collision term due to two-particle interactions [77]. Assuming that the system of gaseous particles
is in thermodynamic equilibrium, the equilibrium distribution is given by the Maxwell distribution
[79],


 

(Ɗ) =


(Ɗ),




3

(4.12)

where 
(Ɗ) = 
−Ɗ/2, Ɗ = |
−
|2

2

room-temperature 
 = 15◦
, we get 
 =(cid:112)3k

/
 = 502 
/
, in which k
, 
, and 
 represent

and 
 denotes the agitation speed. For instance, for air at the

the Boltzmann constant, room temperature, and the molecular weight of air [163], respectively.
It should be pointed out that 
(Ɗ) is an isotropic function with respect to the velocity variables.
Moreover, we define 
 as the macroscopic characteristic length, l as the microscopic characteristic
length associated with the Kolmogorov length scale, and 
 as the mean-free path, which is the
average distance, traveled by a particle between successive collisions. Let take 
(cid:48) the location of

90

particles before scattering, where 
 is the current location. Then, 
(cid:48) = 
 − 

 and 

 = (
 − 
(cid:48))
,
in which we assume that 
 remains constant during 
 − 
(cid:48). As discussed in [163], the analytical
solution for the mass probability distribution function is


 (
, 
, 
) =

ˆ ∞

0


−
 
 

(
 − 

, 
 − 


, 
) 

 =


−
 
 




,
 (Ɗ)

,

(4.13)

ˆ ∞

0


 and 
 



where 
 ≡ 
−
(cid:48)

,
 (Ɗ) = 
 

(
 − 

, 
 − 


, 
). To establish a mathematical framework for
deriving the NS equations from the Boltzman equation in (4.11), we restrain our attention to some
necessary assumptions, following [163].

Assumption 5. The underlying assumptions for deriving the NS equations from BTE are:
• The density, 
, and the thermal agitation, 
, speed are constant,
• 
 ∼ O(1),
• The mean flow velocity is less than the thermal agitation speed, i.e., |
| (cid:28) 
,
• 
 (cid:28) 
 (cid:28) 
 and 
 (cid:28) 
| ¯
|.

4.3.2 Filtered Boltzmann Transport Equation (FBTE)

To proceed for the LES of a turbulent flow, we can decompose 
 to the filtered (resolved) and the
residual (unresolved) values as 
 = ¯
 + 
 (cid:48). Recalling from section 3 that overbar represents the
¯
 = 
 ∗ 
 , where 
 is the kernel of spatial filtering with the filter
spatial isotropic filtering, i.e.
width, L. Then, we formulate the filtered BTE (FBTE) according to


 ¯





+ 
 · ∇ ¯
 = − ¯
 − 
 

(Ɗ)




.

(4.14)

. Following (4.13), we obtain the corresponding analytical solution to

We also define ¯Ɗ := |
− ¯
|2

2
(4.14) in terms of 
 

(Ɗ) as

¯
 (
, 
, 
) =


,
 (Ɗ) = 
 

(cid:0)Ɗ(
 − 

, 
 − 


, 
)(cid:1).

where 
 



ˆ ∞

0


−
 
 




,
 (Ɗ) 

,

91

(4.15)

It is well-known that the nonlinear term is responsible for the transfer of kinetic energy in the
cascade of turbulent kinetic energy from large to small scale turbulent motions. In principle, the
SGS stresses originate from the convection term in the NS equations. It accordingly appears natural
to recognize the advection term, 
 ·∇ 
 , in (4.11) as the resource of turbulent motions. Considering

,
 (Ɗ), which plays a
(4.15), the streaming and collision terms in (4.14) can be revised in terms of 
 


key role in the development of a model for the SGS stresses. More specifically, the effects of highly

,
 (Ɗ), is manifesting in the advection term
vortical flow field on the filtered shifted equilibrium, 
 



,
 (Ɗ) can lead us to
in (4.14), which gives rise to the SGS stresses. Clearly, the way we treat 
 


the development of a closure model in the LES of turbulent flows. It is very important to note that

 

(Ɗ) is not equal to the Gaussian distribution of ¯Ɗ, i.e.,


 

(Ɗ) =


−Ɗ/2 ࣔ




3




3


− ¯Ɗ/2 = 
 

( ¯Ɗ).

(4.16)

A common practice in dealing with 
 

(Ɗ) is to follow the eddy-viscosity approach (see
e.g., [172, 173, 174]). Generally, the residual scale motions can be modeled by approximating the
collision term through a modified relaxation time, 
ਭ. In an analogy with the standard Smagorinsky
SGS model, the Boltzmann equation with the modeled collision term [172, 175] can be proposed
as

+ 
 · ∇ ¯
 = − ¯
 − 
 

( ¯Ɗ)

,


 ¯





(4.17)
where 
ਭ accounts for the difference between 
 

(Ɗ) and 
 

( ¯Ɗ). Interestingly, 
ਭ is inherently
associated with the turbulent viscosity in the Smagorinsky model.


ਭ

Here, we outline a new framework to develop an LES modeling strategy from the BTE using a
non-Gaussian stochastic process. Without loss of generality, we consider 
(
) = 1L 
( 1
2L−|
|) as
the convolution kernel of box filtering, where 
(·) denotes a Heaviside step function. Therefore,


 

(Ɗ) = 
 ∗ 
 

(cid:0)Ɗ(
, 
, 
)(cid:1) =

ˆ


(
) 
 

(cid:0)Ɗ(
, 
, 
 − 
)(cid:1) 
 
,


 


(4.18)

where 
 
 = [−L
2 ]3. Technically, 
 

(Ɗ) represents a summation of exponential functions,
which leads to its multi-exponential characteristics especially when L gets increased. That is,

2 , L

92

by enlarging L, we are incorporating more information into 
 

(Ɗ) according to (4.18), which
essentially induces more heaviness to the statistical behavior of 
 

(Ɗ). Thus, 
 

(Ɗ) deviates
more and more from the Gaussianity of 
 

(Ɗ) (see e.g., [141, 176, 142]). It should be noted that
we are permitted to employ any generic type of filtering.

For the purpose of modeling 
 

(Ɗ), it is understood from [177, 178] that the multi-exponential
distributions can be fitted with a power-law model, in which the discrepancy between the model
and true values can be reduced by increasing the number of exponential functions. Accordingly,
we propose to model 
 

(Ɗ) − 
 

( ¯Ɗ) with a power-law distribution, which follows as


 

(Ɗ) − 
 

( ¯Ɗ) (cid:39) 
 




( ¯Ɗ) = Ô
 
 
( ¯Ɗ),

(4.19)

where 
 
( ¯Ɗ) = 

assume Ô
 is a real-valued constant number. Moreover, we consider 
 ∈ ( 

represents the dimension of physical domain.


3 
 
(Ɗ), in which 
 
(Ɗ) denotes an isotropic Lévy 
-stable distribution. We
2) and 
 = 3

2 , 1 + 


Remark 4.3.1. Unlike the fractional exponent in [163], 
 relies not only on the thermodynamic
properties and boundary conditions, but also it is a function of Taylor Reynolds number, 



(defined further in Table 4.1), and L. It is also worth mentioning that the power-law distribution
can be well-suited in the modeling of multi-exponential functions if the filter width is chosen large
enough to incorporate nonlocal interactions.

Therefore, we propose to model 
 

(Ɗ) in the collision term by using an isotropic Lévy 
-stable

distribution. Therefore, the FBTE is approximated by
+ 
 · ∇ ¯
 = − ¯
 − 
 

( ¯Ɗ) + 
 

( ¯Ɗ) − 
 

(Ɗ)


 ¯








(cid:39) − ¯
 − 
 

( ¯Ɗ) − 
 




( ¯Ɗ)




.

(4.20)

For the sake of simplicity, we take 
 ∗( ¯Ɗ) = 
 

( ¯Ɗ) + 
 




( ¯Ɗ). In comparison to the eddy-
viscosity models, we approximate the collision term by replacing 
 

(Ɗ) by 
 ∗( ¯Ɗ) rather than
modifying 
, which leverages incorporating nonlocal interactions in turbulent flows.

93

4.3.3 Derivation of the FSGS model

The macroscopic continuum variables, associated with (4.8), can be expressed in terms of filtered
distribution function in (4.20) as

ˆ

ˆ

R

1



R


¯
 =

¯

 =

¯
 (
, 
, 
)

,


 ¯
 (
, 
, 
)

,


 = 1, 2, 3,

(4.21)

(4.22)





ˆ

X(cid:0) 
 ¯

R
 X(cid:0) ¯
− 
∗( ¯Ɗ)
´
(cid:16) 
 ¯

+ ∇ · (
 ¯
)(cid:17)
+ ∇ · (
2 ¯
)(cid:17)





 ¯








where 
 = ¯
 for an incompressible flow. It follows from [163, 179] that by multiplying both sides
of (4.20) by a collisional invariant X = X(
) and then integrating over the kinetic momentum, we
attain

(4.23)
where the choices of X = 1, 
 lead to the conservation of mass and momentum equations,
respectively. As noted in [180], due to the microscopic reversibility of the particles (the collisions
are taken to be elastic),

+ 
 · ∇ ¯
(cid:1) 

 = −
(cid:1) 

 = 0. This allows (4.23) to be found as

X(cid:0) ¯
 − 
 ∗( ¯Ɗ)

(cid:1) 

,

ˆ

R


R







 = 0

=⇒ 
 

+ ∇ · (
 ¯
) = 0,




 ¯

+ ∇ ·
=⇒ 





ˆ




R




 = 0

(4.25)
in which 
 is independent of 
 and 
. Reminding that the filter convolution kernel, 
 = 
(
), is
independent of 
 and 
 and thereby, Assumption 5 still holds. In (4.25), by adding and subtracting
¯
 ¯
, the advection term, 
2, is evaluated as

R



2 ¯
 

 = 0,

ˆ

(cid:16)

R


ˆ

ˆ

(4.24)

(4.26)

(4.27)

(4.28)

Plugging (4.26) into (4.25), we obtain




where

ˆ


2 ¯
 

 =

R


=

(
 − ¯
)(
 − ¯
) ¯
 

 +
R

(
 − ¯
)(
 − ¯
) ¯
 

 + 
 ¯

2.

¯
 ¯
 ¯
 



R


R


ˆ
ˆ
(cid:16) 
 ¯

ˆ







 
 =

R


2(cid:17)

+ ∇ · ¯


= −∇ · 
,

(

 − ¯

)(
 
 − ¯

) ¯
 

.

94

It is worth mentioning that the Cauchy and filtered SGS stresses arise from 

 
. Considering (4.15),
we formulate 

 
 in (4.28) as



 
 =

:= 
 




(cid:0) ¯Ɗ(
 − 

, 
 − 


, 
)(cid:1), 
 




,
 := 
 

(cid:0) ¯Ɗ(
 − 

, 
 − 


, 
)(cid:1), thus, 
 ∗

 ∗(cid:0) ¯Ɗ(
 − 

, 
 − 


, 
)(cid:1). In Appendix C, we prove that the temporal shift can be dropped from


−
(

 − ¯

)(
 
 − ¯

) 
 ∗


,
( ¯Ɗ)

 

,

where 
 






(4.29)


,
 :=

R



,


0

ˆ

ˆ ∞

(4.29) following the derivations of fractional NS equations in [163]. Consequently, 

 
 in (4.29)
can be simplified to



 
 =

ˆ

ˆ

R

+

ˆ ∞

−
(

 − ¯

)(
 
 − ¯

) 
 


ˆ ∞

−
(

 − ¯

)(
 
 − ¯

) 
 






0







( ¯Ɗ)

 



( ¯Ɗ)

 

.

R


0

According to the kinetic definition of static pressure, 
 = 
 
2, we decouple 

 
 as



 
 = − ¯


 
 + T
 
 ,

where

ˆ

ˆ ∞

(

 − ¯

)(
 
 − ¯

) 
 ∗( ¯Ɗ)




−



− ¯


 
 =

 
 denotes the sum of shear stress tensor, T 
Ò




R


0

, and the SGS stress
It is worth noting that in (4.32) when 
 ࣔ 
, (

 − ¯

)(
 
 − ¯

) 
 ∗( ¯Ɗ) represents
´
R
(

 − ¯

)(
 
 − ¯

) 
 ∗( ¯Ɗ)

 = 0. Considering


 


(4.30)

(4.31)

(4.32)


 


ˆ

+ T 


ˆ ∞

and T
 
 = T 
Ò



tensor, T 


 
 .
an odd function of 

 and 
 
; consequently,

 ( ¯Ɗ) = 
 ∗( ¯Ɗ) + ( 
 ∗

 ( ¯Ɗ) − 
 ∗( ¯Ɗ)), T
 
 is then obtained as

 ∗
T
 
 =

(

 − ¯

)(
 
 − ¯

)( 
 ∗
By ascribing the Gaussian distribution 
 

( ¯Ɗ) to T 
Ò




 
 , T
 
 in (4.33) is decomposed to
tion, 
 




( ¯Ɗ), to T 

ˆ ∞
(

 − ¯

)(
 
 − ¯

)( 
 


ˆ ∞
(

 − ¯

)(
 
 − ¯

)( 
 





ˆ ∞
(

 − ¯

)(
 
 − ¯

)( 
 


T 
Ò



T 


ˆ
ˆ
ˆ

R


R


R


=

=

=


 



 



 








0

0

0

0

R


95


 ( ¯Ɗ) − 
 ∗( ¯Ɗ))
−


 

.

(4.33)

and the isotropic Lévy 
-stable distribu-

( ¯Ɗ) − 
 

( ¯Ɗ))
−


 

,

( ¯Ɗ) − 
 




( ¯Ɗ))
−


 




 ( ¯Ɗ) − 
 
( ¯Ɗ))
−


 

.

(4.34)

(4.35)

In Appendix C, we discuss the evaluations of T 
Ò




and T 



 


in terms of the macroscopic

quantities, including 
 and ¯
. Eventually, the shear stresses are given by


 


(cid:16) 
 ¯





 


+ 
 ¯







(cid:17)

T 
Ò





 


= 


,

(4.36)

where 
 = 

2
 denotes the kinematic viscosity. Furthermore, we formulate the divergence of
SGS stress tensor as

(∇ · T 
)
 =


(

)2


Ɖ(2
 + 1) 



(4.37)
where 
 = −
− 
/2. Regarding the definition of fractional Laplacian given in (4.4), we can rewrite
equation (4.37) as

R





¯

(
(cid:48)) − ¯

(
)
|
(cid:48) − 
|2
+




(cid:48),

ˆ

(4.38)

(∇ · T 
)
 = 

(−Ɗ)
 ¯

,
Ɖ(2
 + 1) 

 and 

 = 22
Ɖ(
+
/2)

in which 

 = 
(

)2



/2Ɖ(−
) Ô
 and Ô
 is a real-valued constant.
Therefore, the filtered NS equations, developed from the filtered kinetic transport equation, is
described by





 ¯






+ 
 ¯

 ¯




 


= − 1



+ 
Ɗ ¯

 − 

(−Ɗ)
 ¯

,


 ¯


(4.39)



where 
 ∈ (0, 1], 

 = 



 . With a proper choice of 
 = 
(


, L), in which 
|L=0 = 1, the
FSGS model is able to capture the heavy-tailed distribution of the SGS quantities and predict the
corresponding high-order statistical moments. By setting L = 0, we obtain 

=1 = 0, and hence


(−Ɗ)
 ¯

 = 0, which evidently recovers the exact NS equations, given in (4.7).

Remark 4.3.2. In [163], Epps and Cushman-Roisin evaluated the fractional NS equations from the
BTE by replacing 
 

(Ɗ) as a Gaussian distribution with a Lévy 
-stable distribution and splitting
the jumps of particles into small and large scales. From this perspective, the fractional exponent,

, in the fractional NS equations is introduced only as a function of fluid properties and boundary
conditions. Unlike that, we developed the proposed fractional SGS model from the FBTE by
approximating 
 

(Ɗ) − 
 

( ¯Ɗ) with a Lévy-
 stable distribution, in which 
 

(Ɗ) = 
 ∗ 
 

(Ɗ)
and 
 = 
(
). Besides, we found that the factional exponent depends on the flow properties, 


,
and also L. Therefore, by setting L = 0 we recover the standard NS equations at any 


.

96

From the Fourier definition of fractional Laplacian and the Riesz transform in Section 4.2, it is

straightforward to verify that

F(cid:110)(−Ɗ)
 ¯



(cid:111)

= 




(cid:16) − 




|
|

2F(cid:110) ¯


(cid:17) (|
|2)
−1

(cid:111)

,

which leads to

Therefore, we obtain

(−Ɗ)
 ¯
 = ∇ 
(R 
(−Ɗ)
−1

2 ¯
).

∇ · T 
 = ∇ · (R(−Ɗ)
−1

2 ¯
).

(4.40)

(4.41)

(4.42)

(4.43)

Using (4.42), we can find the equivalent form of the SGS stress tensor as
2 ¯

 + R
(−Ɗ)
−1

(R 
(−Ɗ)
−1

2 ¯

),

1
2

T∗

 
 = T 



 
 + 
 =
where 
 is a real-valued constant. T∗
correlation coefficients .

Remark 4.3.3. As described earlier in (4.2), F(cid:110)(−Ɗ)
 ¯


(cid:111)

= |
|2
F(cid:8) ¯
(cid:9). Similar to the eddy-


 
 is dealt with later in section 4.4 in the computation of the

viscosity models, ∇ · T 
 can be explicitly derived in the Fourier domain, hence maintaining the
high-order accuracy of scheme.

4.3.4 Physical Properties

In order to ensure that the developed FSGS model is physically and mathematically consistent with
the filtered NS equations, we introduce a mild condition for the model in accordance with the
second law of thermodynamics and also examine the frame invariant modeling as follows.

4.3.4.1 Second-law of Thermodynamics

The contribution of filtered momentum equation in the entropy production rate is formulated in
[181] as

(cid:0)T 
Ò


 : ∇ ¯
 + T 
 : ∇ ¯
(cid:1),

(cid:164)
 



 =

1



(4.44)

97

where 
 represents the temperature of flow and “:” denotes a double dot product operator.
In
thermodynamic analysis of the exact NS equations, discussed in [182], it is proven that 
 > 0, in
the description of T 
Ò



2 ¯
 in (4.42),
the underlying coefficient in the FSGS model, 

, should satisfy

. Regarding 
 > 0 and T 
 = R(−Ɗ)
−1

(cid:16) 
 ¯



+ 
 ¯
 







 


= 




 


(cid:17)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,



 ≤ 
 min

∇ ¯
 : ∇ ¯


(R(−Ɗ)
−1

2 ¯
) : ∇ ¯


(4.45)

to ensure the positivity of entropy generation rate.

4.3.4.2 Frame Invariance

The SGS stresses and their divergence are separately proven to be frame invariant [183, 184],
which contribute to invariant characteristics of the NS equations. In order to reproduce all local
and nonlocal turbulent solutions in the LES of turbulence, SGS models should undergo certain
restrictions to follow such invariant properties; otherwise, the value of turbulent stresses may change
with any frame movement.

It is apparent that in the FSGS model, 

 is frame invariant. Additionally, as a generator of
Lévy-stable processes, the fractional Laplacian operator is proven to be rotationally and Galilean
invariant (see [185, 186]); therefore, we do not need to impose any additional constraint on the
FSGS model.

4.4 A Priori Analysis of the Fractional SGS Model

We perform a priori tests using DNS database to study the performance and capability of
the proposed model in capturing anomalous behavior of SGS quantities. To pursue the a priori
evaluations, we introduce two primary cases: three-dimensional forced and decaying homogeneous
isotropic turbulent flows with periodic boundary conditions as follows.

98

Table 4.1: Computational parameters and statistical features of a forced HIT problem, provided by
JHTDB [2]


(cid:48)











 =


(cid:48)



 =

437

Case (I): Forced HIT

(cid:113)2
3 



 



 =(cid:10)
(cid:48)

(cid:11)


 
(cid:48)
(
2/


2)




0.93

(
/


)
0.686


 = 2
(cid:10) ¯

 
 ¯

 
(cid:11)

(
2/


3)
9.28 × 10−2




(
2/


)
1.85 × 10−4

Forced HIT is a canonical benchmark in studying the performance of subgrid-scale models. This
test case has the obvious advantage of allowing the statistical features to be approximately stationary.
Here, the corresponding computational domain is specified as  = [0, 2
]3, which is uniformly
discretized on a Cartesian grid using 10243 grid points. The Johns Hopkins Turbulence Databases
(JHTDB)1 has provided public access to DNS database of a forced isotropic homogeneous turbulent
flow, which is characterized by the micro-scale statistical properties presented in Table 4.1. For
more information, the reader is referred to [2, 187].

In the a priori assessments of the FSGS model, the filtered velocity fields are obtained from
2
 = 2 
 for

the DNS data by using a three-dimensional box filtering, in which we set L
 = L

 = 0, · · · , 5, where L and 
 represent filter and grid widths, respectively.

Case (II): Decaying HIT

In terms of a priori tests, the DNS of decaying HIT set the ground to evaluate the modeling
capabilities of FSGS while 


 experiences a decaying process. Furthermore, the DNS dataset
of decaying HIT gives us the opportunity to conduct a series of a priori tests to evaluate the
performance of the proposed model for a wider range of Reynolds numbers.

Similar to Case (I), the computational domain is chosen to be the cube of  = [0, 2
]3 with
the periodic boundary conditions. We start from a three-dimensional fully-developed HIT as the
initial condition, which was previously obtained from the DNS of a forced HIT. The skewness and

1http://turbulence.pha.jhu.edu

99

Table 4.2: Micro-scale statistical characteristics of turbulence for the applied initial condition in
DNS of decaying HIT.

(cid:48)
(
/


)
0.186

(
2/


3)
4.17 × 10−3

(
2/


)
0.001



L
(


)
1.478

L
(
)
0.275










(
2/


2)

0.052











66

flatness of the velocity derivatives for the initial condition data are approximately −0.5 and 4.0,
respectively. Table 4.2 shows the micro-scale statistical properties of the initial condition, which
are described in Table 4.1. It should be mentioned that L and 

L represent the integral length
scale and the eddy turnover time, respectively.

Furthermore, we conduct the numerical simulation of decaying HIT using the incompressible
Navier-Stokes solver of NEKTAR++, which is an open-source spectral/Ò
 element framework [188,
189]. Using a 
0-continuous Galerkin projection, the discretized domain consists of 643 uniform
tetrahedral elements and the fifth-order modified polynomials, 
 = 5, as the basis functions within
each element. In other words, our computational domain would be a uniformly discretized cube
with 2563 grid points. The applied solver works based on the velocity-correction method and for
time integration we use the second-order IMEX scheme. Let 



 and 
 = (
3/
)1/4 denote the
maximum wave number of turbulence and the Kolmogorov length scale, respectively. As a measure
of accuracy, we evaluate 




 > 2.6, which ensures that Kolmogorov scale motions are well-
resolved. Figure 4.1 depicts the time evolution of normalized turbulent kinetic energy, 
(
(cid:48))/
0,
normalized dissipation rate, 
(
(cid:48))/
0, and 


(
(cid:48)), where 
(cid:48) = 
/

L is the dimensionless time and

0 = 
(
(cid:48) = 0), 
0 = 
(
(cid:48) = 0), and 

L are the values reported in Table 4.2. The kinetic energy
is monotonically decaying in Figure 4.1 while the dissipation rate first experiences an increase
up to approximately two large eddy turnover times, 
(cid:48) ≈ 2, and later monotonically decays. The
decay of dissipation rate occurs when the energy spectrum starts to completely decay at the entire
wavenumbers, which is consistent with the physics of decaying HIT problems [138]. To conduct
the a priori analysis of Case (II), we collect the velocity field data, starting from 
(cid:48) ≈ 2 , where



 ≈ 45, and we set L
 = 
 for 
 = 1, · · · , 15.

100

(a) Normalized kinetic energy and dissipation rate

(b) Taylor micro-scale Reynolds number

Figure 4.1: Time evolution of 
(
(cid:48))/
0, 
(
(cid:48))/
0, and 


(
(cid:48)) during the DNS of decaying HIT

4.4.1 Estimation of Fractional Exponent 


To achieve a high degree of accuracy and performance in the FSGS model, the model parameters
are considered to be a function of L
 and 


. By assuming Ô
 as a real-valued function of

 in (4.19), there is only one adjustable model parameter, 
, given in (4.39). Conventionally,
the correlation and regression coefficients are known as the primary tools in a priori tests for
tuning the parameters associated with an SGS model. Following [190], we denote by 

 ∈ [−1, 1]
and R
 for 
 = 1, 2, 3 the correlation and regression coefficients between [∇ · T 
] 

 

from
the filtered DNS data and [∇ · T 
] 




from the FSGS model, respectively. Moreover, the
correlation coefficient associated with a component of SGS stresses, T 


 
 , is indicated by 

 
 with
dual subscripts, where 
, 
 = 1, 2, 3. Since the FSGS model is strictly limited to access the straight
form of SGS stresses, we employ the equivalent SGS stresses, T∗

 
 , given in (4.43) to attain 

 
.
Therefore, 

 
 = 
 (T∗
), where T 

 

denotes the SGS stress tensor

 

obtained from the DNS data.


 
 , T 

 



 









 
 , T 

 


) = 
 (T 



 


Technically, the proper choice of 
 can be made by looking at a range of 
, in which we obtain
the relatively largest values of 

 while the corresponding R
 is around 1. As a rule of thumb, Ô

should be designed such that R
 ≈ 1 occurs, where the values of 

 are relatively maximum. With

101

Figure 4.2: 

 versus 
 for 
 = 1.85 × 10−4 in Case (I) and 
 = 10−3 in Case (II)

this in mind, we adopt Ô
 = ¯

2, where ¯
 = 1500. Figure 4.2 illustrates the variation of 

 versus

 ∈ [0, 1] for the specified properties of Case (I) and Case (II) in Tables 4.1 and 4.2 at room
temperature.

To estimate the optimal fractional exponent, 

 

, in the case of forced HIT problem (Case
(I)), we perform a comparative study of 

 and R
 versus L
 by carrying out several a priori tests.
In Figure 4.3a, we illustrate 

 and 

 for uniformly distributed 
 ∈ [0, 1] at the specific L
 = 8
for 
 = 1, 2, 3. It is important to note that the values of 

 

, obtained from the evaluations in
each direction, can be approximately represented by the same value. Accordingly, we reduce the
evaluation of 

 

 to only the first direction as presented in Figure 4.3b. After running enough
test cases, we show the variations of 

 

 versus L
 in Figure 4.4. It reveals that enlarging L

accelerates the reduction of 

 

 toward the smaller values. Recall that 


 remains approximately
unchanged over time in forced HIT problems, hence, 

 

 is primarily relying on L
.

In a similar fashion, we perform a priori tests of the FSGS model using the dataset of Case (II)
for the purpose of calibrating 

 

 to well-describe the non-Gaussian features of the SGS stresses.
Such analysis also provides a platform for studying the statistical behavior of the FSGS model
regarding a range of 


. Using the Kriging method [191, 192, 193] from 135 direct evaluations,
we approximate a high-resolution surrogate of 

 

, which is presented as a function of L
 and 




102

(a) L
 = 4

L
 = 2

L
 = 8
(b)

L
 = 32

Figure 4.3: Variation of the correlation coefficient, 

, and the regression coefficient, 

, in terms
of the fractional exponent, 
 ∈ (0, 1) using the DNS database of Case (I) for (a) 
 = 1, 2, 3 at
L
 = 4, and (b) 
 = 1 at L
 = 2, 8, 32

Figure 4.4: 

 

 versus L
 for the Case (I), of properties are given in Table 4.1

in Figure 4.5a. For three specific 


, we also show the curves of 

 

 versus L
 in Figure 4.5b.
Similar to the corresponding Figure in Case (I), 

 

 shows a substantial reduction by enlarging
L
. Additionally, Figure 4.5b confirms that, when 


 decreases in a decaying process, 

 


exhibits a sharp reduction in a more limited span of L
. In further discussions, we elaborate on the
results and the nonlocality effects induced by larger 


 on 

 

.

103

(a)

(b)

Figure 4.5: (a) The surface of 

 

, obtained by the Kriging method, versus L
 and 


 using the
data points, denoted by ਭ, which are estimated by the a priori tests of FSGS model for Case (II) and
(b) comparison between the curves of 

 

 versus L
, which are designated by 


 ≈ 26, 35, 45

4.4.2 Analysis of the Model Performance

Following the evaluation of 

 

, we perform a comparative study of performance for the FSGS
model employing the introduced correlation coefficients, i.e., 

 and 

 
. Using the high resolution
turbulent fields of Case (I), we compare the variation of 

 obtained from the FSGS and SMG
models in terms of L
 in Figure 4.6. Seemingly, the FSGS model shows acceptable correlations
with the true values obtained from the DNS data, which is the notion of adequate magnitude
and phase agreement. More significantly, by intensifying nonlocality in the filtered velocity field
through increasing L
, the FSGS model works relatively better in terms of capturing heavy-tailed
behavior of the SGS stresses in all directions. In Figure 4.7, we present the scatter plot analysis on
the values of (∇ · T 
)
 for 
 = 1, 2, 3, attained by the FSGS model and the filtered DNS data, for
L
 = 8, 64. The results confirm that, with a proper selection of 

 

 in the FSGS model, we can
achieve an approximate unit regression, which represents the same level of magnitudes in the scatter
plots. In order to study the influence of 


 on the performance, we reiterate the evaluation of
correlation coefficients at other instantaneous realization of DNS database for Case (I) by imposing
the same 

 

. The outcomes in Table 4.4 are seemingly in an acceptable agreement with the

104

Figure 4.6: Comparing the correlation coefficients, 

, from the FSGS model using the optimum
fractional exponent, with the corresponding ones implied by the Smagorinsky model

Table 4.3: A priori results for the correlation coefficients, 

 
, of SGS stresses obtained from the
FSGS and SMG models at two different Reynolds numbers, L


FSGS
SMG


11
0.17
0.16


12
0.29
0.29

L
 = 16

22

13
0.13
0.30
0.28
0.12


23
0.30
0.31


33
0.23
0.23


11
0.18
0.17


12
0.36
0.33

L
 = 32

22

13
0.13
0.33
0.32
0.11


23

33
0.33 0.27
0.32
0.27

results shown in Figure 4.6. Therefore, in the LES of forced HIT problems, 

 

 can be dealt with
as a constant parameter since L
 is considered as a constant value.

Despite the limitations of fractional approaches in approximating the SGS stresses, our findings
explicitly formulate 

 
 by evaluating T∗

 
 in (4.43) on the filtered velocity field. Consistent with
the results discussed previously, Table 4.3 reports the correlations of the components of SGS stress
tensor, 

 
, for L
 = 16 and 32. More clearly, the results support compatible behavior of the FSGS
model with the SMG model in the description of SGS stresses.

In the LES of decaying HIT problems, 

 

 retains 


 dependence since 


 as a macro-scale
property undergoes a temporal decay. Employing 

 

 in Figure 4.5, we study the accuracy of the
FSGS model in a broader framework, as shown in Figure 4.8. Similar to Case (I), the FSGS model
shows better correlations at a wide range of 


 by enlarging L
. Taken together, the FSGS model
seems to be in a relatively favorable agreement with the filtered DNS database yet not comparable
with structural models.

105

L
 = 4

L
 = 8

L
 = 16

L
 = 32

Figure 4.7: A priori results for the correlation between the true and model values for the components
of ∇ · T 
, where [∇ · T 
] 



 = 

 (−Ɗ)
 ¯V|
=

 

, yielding the correlation coefficients, as
shown
Table 4.4: Study of FSGS model in terms of L through a priori analysis at other time instants of
Case (I)

L


1

2

3




 = 427
16
4
0.15
0.20
0.16 0.21
0.16 0.20




 = 437
16
4
0.20
0.15
0.21
0.15
0.15
0.20




 = 421
16
4
0.20
0.15
0.22
0.15
0.15
0.21

Figure 4.8: Comparison of Kriging-constructed surfaces of 

 for 
 = 1, 2, 3, obtained from a
priori study of the FSGS and the SMG models, versus L
 and 


 on Case (II)

106

L
 = 4

L
 = 8

L
 = 16

L
 = 32

Figure 4.9: A priori results for the PDF of the true and modeled (∇ · T 
)1 regarding 
 variations
at each L


4.4.3 Towards Modeling Nonlocal Effects

As pointed out previously, performance of the proposed model relies strictly on the selection of


 

 as a function of 


 and L
. Regarding the connection between small scale turbulent motions
in the NS and BT equations in section 4.3.2, we explore the influence of nonlocal interactions on
the model’s performance at the microscopic level. Within the Boltzmann transport framework,

 

(Ɗ) demonstrates increasingly multi-exponential behavior by enlarging L. Practically, when
we increase L in the filtered NS equations, more nonlocalities are incorporated into 

 
 in (4.30)
through 
 

(Ɗ). From a physical point of view, vortices in turbulent flows tend to live longer
than their turnover time. During the formation of coherent structures, the mutual advection and
filamentation of vortices render nonlocal flow structures in isotropic turbulent flows. Filtering the
flow field variables integrates such nonlocalities in a single numerical grid point, which intensifies
the heavy-tailed characteristics of 
 

(Ɗ).

In the proposed framework, the multi-exponential behavior of 
 

(Ɗ) is readily modeled by

107

a Lévy 
-stable distribution described in (4.19), in which the tail heaviness is indicated directly
by L
. The multi-exponential pattern suggests that the heavy-tailed characteristics of 
 

(Ɗ) get
more intensified if we increase L
. Interestingly, as we decrease 
, the Lévy 
-stable distribution
exhibits more fat-tailed behavior, which is provably in demand for the best-description of 
 

(Ɗ).
Extending this argument to the macroscopic level, the FSGS model inherently moves from the
diffusion toward the advection to precisely represent the heavy-tailed behavior of SGS statistics by
choosing the smaller values of 
 in (4.14). This argument accounts for the abrupt reduction of 

 


versus L
, presented in Figures 4.4 and 4.5. For 
 < 

 

, the FSGS model is subject to overfit
the heavy-tailed behavior of 
 

(Ɗ), thereby losing the correlation.

On such a background, the FSGS model can be dealt with as a new framework to capture
anomalous features of SGS statistics at large values of L
. As shown in Figures 4.6 and 4.8, the
correlations associated with the FSGS model offer an improvement compared to the SMG model.
Moreover, we proceed to perform qualitative assessment of the FSGS model in predicting the PDF
of (∇ · T 
)
 depicted in Figure 4.9 for different values of L
 in Case (I). We should note that
the presented results are confined to 
 = 1 due to the similarities in other directions. It appears
that by employing the proper choice of 

 

, the PDF obtained by the FSGS model fits into the
heavy-tailed distribution of true SGS values while with 
 < 

 

 the PDF is overpredicted. This
argument emphasizes the reliability of the FSGS model on the selection 

 

 as a function of L

and 


.

Inevitably, due to the approximation we made in modeling 
 

(Ɗ) in the filtered Boltzmann
equation, there are some discrepancies between the results obtained from the fractional model and
the filtered DNS data. We believe that this new framework has the advantage of allowing us to
promote the accuracy of the model by involving more compatible options for approximating 
 

(Ɗ)
in (4.14).

108

A TEMPERED FRACTIONAL APPROACH TO LES OF TURBULENT FLOWS

CHAPTER 5

5.1 Background

With the recent notable developments in computer technologies and, by extension, in the
computational mechanics, there is a rapidly growing interest toward using large eddy simulations
(LES) in a wide range of applications. Over the past decade, LES modeling has received an
increasing attraction from scientific communities as a powerful and promising tool in connection
with turbulence phenomena [194, 195]. In LES, one resolves the large energy-containing eddies
by modeling the interplay between large and subgrid scale motions. Due to the tendency of small
scales to homogeneous and universal dynamics, LES offers more accurate predictions comparing
with the results of resolving the Reynolds-averaged Navier-Stokes (RANS) equations [196, 197].
Furthermore, it lightens the burden of computational costs imposed by accurately capturing the
dissipative scales, which renders LES more affordable than direct numerical simulations (DNS).

Concurrent with the recent computational advancements, a marked shift occurred toward using
artificial intelligence (AI) as an effective and tractable tool in turbulence modeling due to their
significant capabilities in discovering anomalous structures and reproducing nonlocal statistical
properties [198, 199]. In this paradigm, turbulence modeling comes into two divisions:
Machine learning based approaches: They introduce advancements in prediction capabilities
and reconstructing turbulence structures. Several assorted machine learning (ML) algorithms
were proposed for turbulence closure problems including kernel regression and a deep neural
network [200, 201, 202]. Essentially, pure machine learning-based approaches are limited by the
representativeness training dataset though they appear to be simpler for implementation. Moreover,
to pinpoint complex patterns, large volumes of data are required for the algorithms to learn physical
constraints (e.g., frame invariance) and statistical properties, which secondarily makes further
complications like optimizing of data compression [203]. This reveals the significance of physics

109

based models in mentoring the AI approaches and pushing hybrid models as a new direction,
which exploit ML algorithms with a significant reduction in the required input data [see e.g.,
204, 205, 206, 207, 208].
Physics based approaches: As a catch-all phrase for a spectrum of physics based techniques, they
introduce a mathematical representation of physical structures through a number of parameters with
a sufficient amount of information. Contrary to ML based approaches, physics based models do not
involve large volumes of data although they are inherently limited by the model incompleteness or
the complexity of parameterizing physical structures [203]. Accordingly, it is markedly essential
to entail the underlying statistical properties in formulating and inferring an optimum model in
a numerically rigorous framework, which links a variety of research disciplines like turbulence,
numerical and statistical analysis, and data science. This approach employs principles of physics
and borrows insights from the statistical analysis to form a model for real phenomena, which can
also be used to guide the ML algorithms [see e.g., 209, 210, 211].

Establishment of such a physically-consistent LES model ties strongly with characterization of
nonlocal turbulence mechanisms and a better understanding of anomalous structures. As a puzzling
feature, the non-Gaussian behavior of turbulent dynamics is linked to the spatial intermittency of
small-scale motions, which is embodied in the form of very thin and elongated vortices [212, 213].
Technically, the nonlocal closure of Navier-Stokes (NS) equations, originated from the Green’s
function of the Laplacian operator for solving Poisson pressure equation, induces long-range
interactions (nonlocal triadic structures) in spectral space of homogeneous turbulence [82].
In
a preliminary investigation of isotropic turbulence [214], the significant role of highly vortical
structures, typically tube-like, was disclosed on generating nonlocal dynamics and coherence of
turbulence. Supported by Laval et al.
[213], nonlocality as a crucial element in generating
intermittent structures has tendency to prevail the local interactions by orders of magnitude. She
and Leveque attempted to express self-similar structures in terms of a squence of moment ratios for
the energy dissipation field [215]. Recently, Mishra and Girimaji in [216] studied the role of pressure
on nonlocal mechanisms in incompressible turbulent flows and identified the intercomponent of

110

energy transfer by the rapid pressure strain correlation. For more information, the reader is referred
to [217, 218, 219, 220].

From this perspective, an ideal subgrid-scale (SGS) model represent correctly statistics of the
filtered real turbulence at the resolved levels. Given the dependence of an ideal model on an
infinite-dimensional set of multi-point statistics, it would be more practical to define a weaker set
of conditions in study of SGS parameterization [82]. As one of the earliest studies on statistical
analysis of LES, Meneveau in [76] derived a closed set of necessary, yet mild, conditions to fulfill
a priori consistency in SGS quantities. More generally, the Karman-Howarth (KH) theorem for
anisotropic turbulent flows were studied in [221] by eliminating pressure velocity correlations to
determine the two-point structure function equations. By proposing a hyper-eddy viscosity term
in [81], SGS dissipation spectrum were measured in locally isotropic turbulence to assess ability
of classical two-point closures in prediction of the mean energy transfer. Recently, some of the
prevailed challenges in developing an optimal LES model was reviewed succinctly by [80]. This
review presents a clear set of statistical characteristics in performing an a priori analysis and
providing adequate information for optimizing SGS models.

Statistical descriptions of an ideal closure model derive a desire for developing nonlocal ap-
proaches in terms of two-point high-order structure functions in a rigorous mathematical framework.
The eddy damped quasi-normal Markovian (EDQNM) approach, described in [222], undertakes
closing of SGS motions in spectral space by involving high-order statistical moments. As a func-
tional approach, direct interaction approximation pushes the non-Markovanized stochastic models
to the direction of turbulence closure problem, whose solutions are constructed in a fraction form
[223]. Furthermore, multifractal models [e.g., 152, 224] suggest a potential realizable strategy
to accurately capture anomalous scaling exponents, observed in turbulent velocity increments.
Recently, Charalampopoulos and Sapsis [225] developed a new data-driven approach in closure
modeling of temporal and spatial nonlocalities for a two-dimensional jet flow. In addressing sta-
tistical local and nonlocal interactions, this progress proceeds with modeling turbulent effects at
the kinetic level. Premnath in [173] developed a framework for applying dynamic procedure in

111

the lattice-Boltzmann method for LES of inhomogeneous and anisotropic turbulent flows. A new
collision approach were proposed by [226] for LES of weakly compressible flows using two forms
of the modified Bhatanagar-Gross-Krook (BGK) collision operators. For more comprehensive
review of the literature, we refer the reader to [227, 175].

Focusing on the key ideas of (i) describing of anomalous structures in turbulence and (ii)
nonlocal closure modeling, fractional calculus appears to be a tractable mathematical tool due to
their power-law or logarithmic types of kernel. As an alternative approach to standard methods, they
leverage their inherent potentials in representing long-range interactions, self-similar structures,
sharp peaks, and memory effects in a variety of application [see 228, 229]. Egolf and Hutter
in [162] generalized Reynolds shear stresses in local zero-equation to the fractional counterparts.
Furthermore, Epps and Cushmann-Roisin derived fractional NS equations from the Boltzmann
transport equation in [163], which supply profound understanding of turbulent nonlocal effects at
the kinetic level. For more information, Egolf and Hutter [230] provided a comprehensive overview
of fractional and nonlocal turbulence, spanning from coherent structures to state-of-the-art ideas
on closure modeling in canonical flows. Recently, Di Leoni et al. [231] contributed in fractional
LES modeling by developing a two-point correlation based model in a robust physically-meaning
framework.

In the class of nonlocal models, Samiee et al.

[232] laid out a mathematical framework
for developing fractional models, which starts treating turbulence effects at the kinetic level. In
a precise derivation, the proposed distribution function in the closed form of filtered collision
operator, turns into a fractional model in the LES equations. Throughout a data-driven approach,
Akhavan-safaei in [211] extended the fractional modeling to the LES of scalar turbulence using
two-point correlation functions between the SGS scalar flux and filtered scalar gradient.

In the specific case of isotropic turbulent flows, cascading of energy from large to small scales
expresses a self-similar behavior in the inertial range, and then it falls exponentially into the
dissipation range. Inspired by such real physics phenomena, we focus on developing a nonlocal
model by employing a tempered heavy-tailed distribution within the proposed fractional framework,

112

which contributes in tempered fractional SGS (TFSGS) modeling. Such a tractable fractional
operator offers a great flexibility in characterizing nonlocal structures in the turbulent inertial
and dissipation ranges through fractional and tempering parameters. To achieve the enhanced
performance of the proposed model, we also present an optimization algorithm, involving two-
point structure functions. Regarding the best approximation of an ideal physics based model, the
optimized TFSGS model restores many essential statistical properties of SGS stresses and presents
an a priori consistency in the dissipation spectrum.

This chapter is organized as follows.

In section 5.2, we introduce some preliminaries of
tempered fractional calculus. We outline a mathematical framework in section 5.3 to develop the
tempered fractional model from the Boltzmann transport equation and derive the corresponding
forms for SGS quantities. Within a statistical framework, we present a two-point structure based
algorithm to infer the optimal behavior of the tempered fractional model in section 5.4. Using the
DNS database of an stationary isotropic turbulent flow, we evaluate the statistical a priori analysis
and perform a comparative study on the two-point structure functions in section 5.5. Next, we carry
out an a posteriori investigation in section 5.6 by tracking turbulent kinetic energy.

5.2 Preliminaries on Tempered Fractional Calculus

Fractional calculus introduces well-established mathematical tools for an accurate description
of anomalous phenomena, ubiquitous in a wide range of applications from biotissues [166, 49] and
material science [233, 234, 235] to vibration [167], porous media [236, 237, 238] and turbulence
[211, 231, 163]. As alternative approaches to the standard nonlinear models, fractional models
offer a great potential in capturing heavy-tailed distributions, self-similar structures, nonlocal inter-
actions, and memory effects. This potential is substantially indicated by power-law or logarithmic
kernels of convolution type in the corresponding fractional operators. From the stochastic point of
view, fractional transport models arise from the heavy-tailed distribution functions in modeling the
underlying super- or sub-diffusive motions of particles in complex heterogeneous systems at the
microscopic level [60, 61, 72, 71]. Nevertheless, common patterns in nature follow finite variance

113

dynamics, which urges the role of tempered fractional calculus as a more sophisticated approach
in representing natural cut-offs in real applications and retaining their finite statistical properties.
Recalling from [239, 101], we begin by the definitions of the left- and right-sided tempered

fractional derivatives respectively as

D
,
±
 
(
) =




Ɖ(1 − 
)

ˆ ∞

0


(
) − 
(
 ∓ 
)



+1






,

(5.1)

where the fractional derivative order, 
 ∈ (0, 1), and the tempering parameter, 
 > 0. Also, Ɖ(·)
represents a Gamma function. For 
 ∈ (1, 2), the corresponding fractional derivatives are given
by:

D
,
±
 
(
) =


(
 − 1)
Ɖ(2 − 
)

ˆ ∞

0


(
 ∓ 
) − 
(
) ± 
 

(
)







+1






.

(5.2)

The link between the derivatives in (5.2) and (5.3) and their counterparts in the Riemann-Liouville
sense are described by



D
,
±
 
(
) = D
,
±
 
(
) + 


(
)


D
,
±
 
(
) = D
,
±
 
(
) + 


(
) ± 


−1 

(
)
In particular, for 
 ≥ 0 the tempered integer-order derivatives are reduced as





.



(cid:16)





(
)(cid:17)

,





which recover the classic integer-order derivatives as 
 → 0.



D
,
+
 
(
) = 
−



(5.3)

(5.4)

(5.5)

Let F [
] (
) denote the Fourier transform of 
, where 
 is the Fourier numbers. Then, we

= (
 ± 
 
)
 F [
] (
).

F(cid:104)

D
,
±
 
(
)(cid:105)
F(cid:104)

I
,
±
 
(
)(cid:105) (
) = (
 ± 
 
)−
 F [
] (
).

114

obtain

In this context, the corresponding Fourier transform of the left- and right-sided tempered fractional
integrals are given by

Evidently, tempered integrals and derivatives functions as inverse operators when 
 possesses
sufficient regularity [see 240, 239]. Moreover, tempered fractional operators preserve semi-group
property, which prepares a useful and rigorous framework for further numerical considerations.

5.2.1 Tempered fractional Laplacian
Denoted by (Ɗ + 
)
(·), we define the tempered fractional Laplacian of the integral form as

ˆ

(Ɗ + 
)

(
) = 

,
 P.V.



|
−
||
 − 
|2
+

2 ;1) for 
 ∈ (0, 1) and 
 ࣔ 1
where 

,
 =
(Ɗ + 
)
 is reduced to the so-called Riesz fractional form, described by

2
1(−
, 
+2
−1

−Ɖ( 

2)


2 Ɖ(−2
)

R


; 


2


1

2


(
) − 
(
)



,

(5.6)

2. In particular, for 
 = 1

(Ɗ + 
)

(
) = (−1)(cid:98)2
(cid:99)+1
ˆ

= 

 P.V.

R



D
,
+
 
(
) + 

D
,
−
 
(
)

2

(
) − 
(
)



|
−
||
 − 
|2
+1



,

2)
−Ɖ( 1
1
2 Ɖ(−2
)

1

2


where 

 =
transform of (Ɗ + 
)

(
), formulated as

F(cid:2)(Ɗ + 
)

(
)(cid:3)(
) = Ô
,
 ×
(cid:32)

cos(

) [see 240]. In Appendix D.1, we detail the derivation of the Fourier

2
1(−
,


 + 2
 − 1

2

;



2

;


2

2 + 
2)

(cid:33)

(5.7)

F [
] (
),


2
 − (
2 + 
2)


1

2;1) . It
in which Ô
,
 =
is worth noting that when 
 approaches 0, we recover the usual fractional Laplacian in both integral
or Fourier forms.

2 ;1) and 
 = |
|. For 
 = 3, we define Ô
 =

2
1(−
, 
+2
−1

2
1(−
,2+2

2 ;3

; 


2

1

5.3 Boltzmannian Framework

The kinetic Boltzmann transport (BT) is a formal framework for describing fluid particle motions
over a wide range of flow physics (e.g., rarefied gas flows and turbulence). This framework offers
a great potential for statistical description of turbulent small-scales towards a better understanding

115

of coherent structures in turbulence yet at the kinetic level. As an alternative approach in turbulent
closure modeling, reconciling SGS terms in the BT and the Navier-Stokes (NS) equations can
conceivably give rise to a rigorous physics-based model at the continuum level.

Within the BT framework proposed in chapter 4, we develop a SGS model, respecting the

statistical and physical properties of turbulent motions.

5.3.1 A review on the subgrid-scale modeling

In the description of incompressible turbulent flows, we consider large eddy simulation (LES)
equations [138], governing the dynamics of the resolved-scale flow variables,


 ¯





+ ¯
 · ∇ ¯
 = − 1



∇ ¯
 + 
∇ · ¯
 − ∇ · TR,

(5.8)

where in the index form ¯
(
, 
) = ¯

 and ¯
(
, 
) represent the velocity and the pressure fields for

 = 1, 2, 3 and 
 = 

. Moreover, 
 and 
 denote the kinematic viscosity and the density, respectively.
Considering L as the filter width, the filtered field is obtained in the form of ¯
 = 
 ∗ 
, where

 = 
(
) denotes the kernel of a spatial isotropic filtering type and ∗ is the convolution operator.
By implementing the filtering operation, we decomposes the velocity field, 
, into the filtered
(resolved), ¯
, and the residual, 
, components. In (5.8), the filtered strain rate, ¯
, and the SGS
stress tensor, TR, are defined by ¯S
 
 = 1

) and TR


 
 = 



 − ¯

 ¯

.

2( 






 


+ 

 






Since the filtering operator cannot commute with the nonlinear terms in the NS equations, SGS
stresses must be modeled in terms of the resolved velocity field. As a common yet reliable approach,
Smagorinsky [171] offered modeling the SGS stresses borrowing the Boussinessq approximation
from the kinetic theory such that TR = −2

 ¯
 and 

 is indicated by 

 = (

L)2 | ¯
|, where
| ¯
| =

2¯S
 
 ¯S
 
 and 

 is the Smagorinsky (SMG) constant.

(cid:113)

116

5.3.2 The BGK equation and the closure problem

Starting from the Boltzmann kinetic theory [77], the evolution of mass distribution function 
 is
governed by the Boltzmann transport (BT) equation as


 





+ 
 · ∇ 
 = ( 
),

(5.9)

in which 
 (
, 
, 
) 



 represent the probability of finding mass of particles, located within
volume 



 centered on a specific location, 
, and speed, 
, at time 
. It is worth noting that
in the particle phase space 
, 
, and 
 are independent variables. Technically, the left-hand side
of (5.9) concerns the streaming of non-reacting particles in absence of any body force and the
right-hand side represent the collision operator. The most common form of ( 
) with a single
collision is the so-called BGK approximation [77, 79], given by

( 
) = − 
 − 
 






,

(5.10)

where 
 represent the single relaxation time.
In the case of incompressible flows with roughly
constant temperature, 
 is assumed to be independent of macroscopic flow field velocity and
pressure. Moreover, under the circumstances of thermodynamic equilibrium of particles, 
 

(Ɗ)
serves as


 

(Ɗ) =


(Ɗ),




3

(5.11)

where 
(Ɗ) = 
−Ɗ/2, Ɗ = |
−
|2

2

agitation speed. More specifically, 
 =(cid:112)3k

/
, in which k
, 
, and 
 represent the Boltzmann

as an isotropic Maxwellian distribution and 
 denotes the

constant, room temperature, and the molecular weight of air.

By recalling the basics of BT equation from [163, 232], we introduce the following quantities:

 as the macroscopic length scale, 

 as the microscopic characteristic length associated with the
Kolmogorov length scale, 

 as the average distance, traveled by a particle between successive
collisions. Furthermore, we define 
(cid:48) as the location of particles before scattering, where 
 is the
current location. Thus, 
(cid:48) = 
 − (
 − 
(cid:48))
, where 
 is assumed to be constant during 
 − 
(cid:48). The

117

analytical solution of (5.9) and (5.10) in [163] is given by


 (
, 
, 
) =

=


−
 
 

(
 − 

, 
 − 


, 
) 



−
 
 




,
 (Ɗ)

,

(5.12)

ˆ ∞
ˆ ∞

0

0

where 
 ≡ 
−
(cid:48)


 and 
 




,
 (Ɗ) = 
 

(
 − 

, 
 − 


, 
).

Remark 5.3.1. In order to develop an LES model within the kinetic transport framework, we
constrain our attention to the BT equation with the BGK collision approximation, involving a single
relaxation time. Moreover, we follow Assumption 1 in [232, pp. 4] in the further derivations
to establish a physical connection between the collision operator and the convective terms at the
continuum level.

In description of turbulence effects at the kinetic level, we decompose 
 into the filtered,

¯
 ,
and residual values, 
 (cid:48), where ¯
 = 
 ∗ 
 . As defined previously, 
 represents the kernel of any
generic spatial isotropic filtering type. Then, the filtered kinetic transport for ¯
 suffices:


 ¯





+ 
 · ∇ ¯
 = − ¯
 − 
 

(Ɗ)




,

(5.13)

ˆ ∞

in which 
 is independent of 
 and 
. Ensuing (5.12), the analytical solution of (5.13) is described
by


,
 (Ɗ) = 
 

(cid:0)Ɗ(
 − 

, 
 − 


, 
)(cid:1). Let define ¯Ɗ := |
− ¯
|2

¯
 (
, 
, 
) =


,
 (Ɗ) 

,


−
 
 



0

where 
 


. Due to the nonlinear
character of the collision operator [172], the filtering operation does not commute with , which
yields the following inequality as


2

(5.14)


 

(Ɗ) =


−Ɗ/2 ࣔ




3




3


− ¯Ɗ/2 = 
 

( ¯Ɗ).

(5.15)

This inequality gives rise to the so-called turbulence closure problem at the kinetic level. From
the mathematical standpoint, the SGS motions stem from the convective nonlinear terms in the NS
equations, which resembles with the corresponding advective term of the BT equation. Therefore,

118

it seems natural to recognize 
 · ∇ 
 responsible for the unresolved turbulence effects in the BT
equation, they manifest implicitly via the filtered collision operator though. That is, the filtered
collision term in the right-side of (5.13) undertakes not only molecular collisions, but also the
embedded SGS motions. By emphasizing on the importance of modeling 
 

(Ɗ) in the filtered
collision operator, we review some different approaches in treating nonlinear effects.

Classical approaches: As a common practice in modeling the SGS closures, the attentions were
directed toward with eddy-viscosity approximations by employing a modified relaxation time, 
ਭ,
in the BT equation [e.g., 175]. Therefore, the proposed filtered BT equation reads as


 ¯





+ 
 · ∇ ¯
 = − ¯
 − 
 

( ¯Ɗ)


ਭ

.

(5.16)

In this approach, the inequality in (5.15) is disregarded through using 
ਭ, which renders the SGS
model inappropriate for reproducing many features of the SGS motions. Nevertheless, there some
non-eddy viscosity models within the lattice Boltzmann framework, which make use of (5.15) to
propose more consistent SGS model. For more details, the reader is referred to [241, 173, 242].

Fractional approach: In the proposed framework in [232], the modeling of turbulence nonlinear
effects begins with closing the filtered collision operator, where the multi-exponential behavior of

 

(Ɗ) is approximated properly by a heavy-tailed distribution function. Therefore, the 
 

(Ɗ) in
(5.13) is described by


 

(Ɗ) − 
 

( ¯Ɗ) (cid:39) 
 
( ¯Ɗ),

(5.17)
where 
 
( ¯Ɗ) = 


3 
 
(Ɗ) and 
 
(Ɗ) denotes an isotropic Lévy 
-stable distribution. By taking the
first moment of (5.13), one derives the corresponding fractional Laplacian operator, called FSGS
model, at the continuum level, where

(∇ · T
) = 

(−Ɗ)
 ¯
,

(5.18)
where 

 = 
(

)2
Ɖ(2
+1)
22
Ɖ(
+
/2)


/2Ɖ(−
) c
 for 
 ∈ (0, 1) and c
 is a real-valued constant. In
principle, the choice of distribution function in (5.17) gives rise to a nonlocal operator of the
resolved flow field in (5.8) as an SGS model.




119

Despite the notable potentials of the FSGS model in maintaining some important physical and
mathematical properties of the SGS stresses, it suffers from a shortcoming by lacking a finite
second-order statistical moment. To control this statistical barrier in the FSGS model and to
achieve more congruence between both sides of (5.17), we seek a finite variance alternative for the
Lévy 
-stable distribution by employing the tempered counterpart and thereby a more flexible and
predictive fractional operator in the LES equations in the following subsection.

5.3.3 Tempered fractional SGS modeling

In a more accurate study, multi-exponential functions express a power-law behavior in the moderate
range of distribution and eventually relaxes into an exponential decay [see 243]. By engaging more
exponential terms to a multi-exponential function, the corresponding power-law behavior extends
toward long ranges; however, it is bound to vanish exponentially at the tail of the distribution,
enforced by nature of the physical phenomenon. As a rich class of stochastic functions for fitting
into realistic phenomena, tempered stable distributions [239] resemble a sheer power-law at the
moderate range and then converge to an exponential decay.

Inspired by this argument, we propose to model 
 

(Ɗ) with a coefficient of tempered Lévy


-stable distribution, denoted by 
 
,
( ¯Ɗ), within the proposed fractional framework as


 

(Ɗ) − 
 

( ¯Ɗ) (cid:39) 
 




( ¯Ɗ) = c
,
 
 
,
( ¯Ɗ),

where c
,
 is a real-valued constant number. Moreover, we consider 
 ∈ (−1 − 


 = 3 represents dimension of the physical domain. Therefore the filtered BT equation reads as

2 , − 


(5.19)
2), 
 > 0 and


 ¯





+ 
 · ∇ ¯
 = − ¯
 − 
 

( ¯Ɗ) + 
 

( ¯Ɗ) − 
 

(Ɗ)




(cid:39) − ¯
 − 
 

( ¯Ɗ) − 
 




( ¯Ɗ)

.

(5.20)
For the sake of simplicity, we take 
 ∗( ¯Ɗ) = 
 

( ¯Ɗ) + 
 




( ¯Ɗ). The approximation in (5.20)
conceivably provides a good fit into the filtered collision operator by maintaining the significant
statistical features of 
 

(Ɗ) and sets a physically-richer starting point for developing a nonlocal
LES model at the continuum level.




120

In this regard, the macroscopic variables associated with the flow field can be reconstructed

according to

ˆ

¯
 =

¯

 =

¯
 (
, 
, 
)

,


 ¯
 (
, 
, 
)

,

ˆ

R

1



R



 = 1, 2, 3,

(5.21)

(5.22)

where ¯
 = 
 for an incompressible flow. To establish the connection between the kinetic descrip-
tion and the filtered NS equation, we proceed with deriving the macroscopic form of (5.20) by
multiplying it with 
, and integrating over the kinetic momentum, which yields

(cid:16)

ˆ


 ¯








R


+ ∇ · (
2 ¯
)(cid:17)



 = 0

ˆ

+ ∇ ·

=⇒ 

´


 ¯





(cid:16) ¯
− 
∗( ¯Ɗ)

(cid:17)


2 ¯
 

 = 0.

(5.23)

R


Recalling the assumptions in Remark 5.3.1 that


 = 0 due to microscopic re-
versibility of the particle collisions. Following the derivations in [232, pp. 5-6], we add and
subtract ¯
 ¯
 to the advection term and accordingly, (5.23) is found to be

R
 





(cid:16) 
 ¯









(cid:17)

+ ∇ · ¯
 ¯


= −∇ · 
,

where 
 in the index form is expressed as
ˆ



 
 =

R


(

 − ¯

)(
 
 − ¯

) ¯
 

.

Comparing (5.8) and (5.24), it turns out that pressure term, viscous and SGS stresses all trace back
to ∇ · 
, where 

 
 = − ¯


 
 + T
Ò




 
. By plugging (5.14) into the kinetic definitions of each
term in 

 
, we obtain

+ TR


 


ˆ
¯


 
 = −
ˆ ∞

ˆ

R


T
Ò




 


=

0

R


(

 − ¯

)(
 
 − ¯

) 
 ∗( ¯Ɗ)


(

 − ¯

)(
 
 − ¯

) × ( 
 



ˆ ∞

0


−


,


,
 ( ¯Ɗ) − 
 

( ¯Ɗ))
−


 

 = 2
 ¯S
 
 ,

where 
 = 

2
. Similarly, by employing 
 




 in (5.19), we attain

ˆ
ˆ ∞
ˆ ∞

0

ˆ

R


0

R


T


 
 = c
,


c
,


3

=


,
 ( ¯Ɗ) − 
 
,
( ¯Ɗ))
−


 



(

 − ¯

)(
 
 − ¯

)( 
 
,

(

 − ¯

)(
 
 − ¯

)(
 
,
( ¯Ɗ
,
) − 
 
,
( ¯Ɗ))
−


 

,

121

(5.24)

(5.25)

(5.26)

(5.27)

(5.28)

in which ¯Ɗ
,
 = ¯Ɗ(
 − 

, 
 − 


, 
). As discussed in Appendix C, the temporal shift can be
detached from 
 
,



,
 ( ¯Ɗ) and then ¯Ɗ
,
 is simplified to ¯Ɗ
 = ¯Ɗ(
, 
 − 


, 
). Therefore,
(

 − ¯

)(
 
 − ¯

)(
 
,
( ¯Ɗ
) − 
 
,
( ¯Ɗ))
−


 

.

ˆ ∞

(5.29)

ˆ

T


 
 =


c
,


3

0

R


 
 is to decouple the particle speed into time and displacement
and approximate the asymptotic behavior of 
 
,
( ¯Ɗ) with a tempered

The strategy to evaluate T






ˆ

K


T


 
 = c
,
 ¯



by employing 
 = 
(cid:48)−

power-law distribution. In a detailed discussion in Appendix D, we show that
(
 − 
(cid:48)) · ( ¯
 − ¯


) (
 
 − 
(cid:48)
|
 − 
(cid:48)|2
+5
 ¯

|
−
(cid:48)| 

(cid:48),

)
in which ¯

 = (2
+ 3)(



2
−1
2
) for 
 ∈ (0, 1
2 , 1) and recalling ¯

 = 

2) ∪ ( 1
¯
K

 (
) is indicated in (D.12). Eventually, we disclose the integral form of ∇ · TR as
ˆ
|
 − 
(cid:48)|2
+3
 ¯

|
−
(cid:48)| 

(cid:48),

(∇ · TR) 
 = c
,
 ¯



(2
 + ¯

)
(2
 + 3)

( ¯

 − ¯
(cid:48)

)

¯
K

 (
)

¯
K

 (
)

K


(

 − 
(cid:48)


=0

(cid:48))

R


R



=0

(5.30)



 
. Moreover,

(5.31)

where 

 = c
,
 ¯

. Reminding the integral representation of a tempered fractional Laplacian in
(5.6), we formulate the divergence of the SGS stresses as follows

(∇ · TR) 
 = 




K

 (
, 
)(Ɗ + ¯

)
 ¯

 ,

(5.32)

K



=0


 (
, 
) =

where 
K
new operator in (5.32) reduces to a fractional Laplacian, which recovers the FSGS model.

¯
K

 (
). Evidently, by setting K = 0, we find ¯
K


 (
) = Ɖ(2
) and the

(2
+ ¯

)
(2
+3)

Inferring from (5.32), our choice in the kinetic description of turbulent effects reflects in the
form of a tempered fractional operator through a rigorous connection between the filtered BT and
NS equations. More specifically, we adopt K = 1 and hence the tempered fractional SGS (TFSGS)
model can be formulated as

0(
) =

where 
1
governing LES equations read as

1(2
+3)

∇ · TR = 

(cid:2)
1
Ɖ(2
 + 1) − Ɖ(2
)(cid:17)
(cid:16)

1(
)(cid:0)Ɗ + ¯
1(cid:1) 
 ¯
(cid:3),
0(
)(cid:0) − (−Ɗ)
(cid:1) ¯
 + 
1
1(
, 
) = (2
+
)
1


and 
1

+ 
Ɗ ¯

 − 




 ¯






+ 
 ¯

 ¯




 


= − 1




 ¯






(5.33)
(2
+3) Ɖ(2
 − 1). Accordingly, the


(
, 
)(Ɗ + ¯

)
 ¯

,

1

(5.34)


=0

122

where 
 ∈ (0, 1

2) ∪ ( 1

2 , 1), 
 > 0, and 

 = 



 .

Remark 5.3.2. As a generator of tempered Lévy stable processes, the tempered fractional Laplacian
is proven to be rotationally and Galilean invariant [see 244, 245]. Therefore, by having 

 and 
1


as real-valued functions of 
 and 
, the TFSGS model adopts the frame invariance property in a
consistent fashion with the SGS stresses.

5.3.4 TFSGS formulations for the SGS stresses

To study the key role of tempering fractional operators in recovering turbulent statistical structures,
it is essential to establish a straightforward form of the modeled SGS stresses. Due to some
numerical complications in evaluating the integral in (5.31), we settle to proceed with the Fourier
representation of the TFSGS model. Employing the definition of I
 (
-Riesz potential) from
[170], it is possible to verify that

∇ · T
 = (Ɗ + 
)
 ¯
 = ∇ · ∇I
=1(cid:2)




(
, 
)(Ɗ + ¯

)
 ¯
(cid:3) .


1

(5.35)

1



=0

(·) = ∇ 
I
=1(Ɗ + 
)
(·) as a tempered fractional operator

Inspired by (5.35), we introduce R
,

such that





(
, 
)(cid:2)R
, ¯






1



=0

¯

(cid:3),





1




2

T


 
 =

where F(cid:2)

=1(cid:3) = 1
F(cid:2)R
,


¯

 + R
, ¯




2 and F(cid:2)∇ 
(cid:3)(
) = −

 
 and 
 denotes an imaginary unite. Following (5.7)
(cid:3)(
) = Ô


(cid:0)
2
 − (
2 + 
2)


into (5.36), we find the Fourier form of R
,


(5.36)

(5.37)

2
1(−
, 1 + 
;

as





2 + 
2)(cid:1).


2

3
2

;

−

 


2




5.4 Statistical Analysis

In pursuit of an ideal SGS model, nonlinearity induced by the convective terms and nonlocality
imparted by the pressure term in the NS equations contribute to a synthetic hierarchy of transport
equations and multi-point descriptions of SGS terms, as shown in [82]. The infinitely-extended

123

hierarchical triangle of nonlinearity and nonlocality brings up the idea of indicating a set of weaker,
and yet significant, statistical conditions and make the ideal LES model more attainable, as endorsed
by [80]. To identify such statistical features, Menevuea in [76] developed a rigorous framework via
a statistical a priori analysis and formulated some sufficient conditions for the assessment of LES
models. As one of the candidates for evaluating SGS models, we give a brief review of the argued
formulations in [76] and introduce an optimization strategy, which enables the TFSGS model to
correctly generate the requisite statistical conditions.

Hereafter, we consider the following notations in study of the SGS fields. Let T
,
 and T
,∗
denote the SGS stresses, implied by the true DNS data and the SGS model, respectively. We also
take 
 as the displacement vector between two points in the correlation functions and 
 denotes the
unit vectors of the axes in the Cartesian coordinates. Then, 
 = |
|. As discussed in [76], performing
an ensemble-average of the filtered NS equations offers a set of necessary conditions for an LES
simulation to ensure the equality of mean velocity profiles and the second-order moments, listed
as:

(a)

(b)

(c)

(cid:104)T
,



 


(cid:105) = (cid:104)T
,∗

 
 (cid:105),
(cid:105) = (cid:104) ¯

T
,∗

 
 (cid:105),
(cid:105) = (cid:104)¯S
 
T
,∗

 
 (cid:105),

(cid:104) ¯

T
,



 


(cid:104)¯S
 
T
,



 


in which conditions (b) and (c) are inferred from the ensemble-averaged SGS transport equation.
Focusing on the nonlocality axis of the closure triangle for a homogeneous isotropic turbulent

(HIT) flow, one obtains the so-called Karman-Howarth (KH) equation as

(cid:2) 






− 2
( 
2



2 + 4










)(cid:3) 


(
, 
) − ( 


+ 4



)



(
, 
) = ( 





+ 4



)
 


(
, 
),

(5.38)





for sufficiently large L (cid:29) 
, where 
 represents the longitudinal direction. Additionally, we denote
by 


(
, 
) = (cid:104) ¯

(
, 
) ¯

(
+ 
· 
, 
)(cid:105) and 



(
, 
) = (cid:104) ¯

(
, 
)2 ¯

(
+ 
· 
, 
)(cid:105) the second- and


(
, 
) ¯

(
+ 
 · 
, 
)(cid:105)
third-order velocity correlation functions, respectively and 
 


(
, 
) = (cid:104)T


124

refers to the stress-strain correlation function. Technically, the third-order correlation function in
(5.38) is subdivided into 



 stemming from the resolved velocity field and 
 


(
, 
) coming
from the SGS stresses.
It turns out from (5.38) that the SGS model should undergo a correct
prediction of 
 


(
, 
) to re-generate 


 and 



 accurately. Referring to [76, pp. 819-820],
we arrive at the following equation

(cid:104)[ ¯

(
 + 
 · 
, 
) − ¯

(
, 
)]3(cid:105) + 6 
 


(
, 
) = 6(cid:104)¯S

T




(cid:105)
,

(5.39)

which exhibits the only sufficient condition for modeling third-order structure in an HIT flow.
Therefore, by satisfying the equality of SGS dissipation via conditions (c), modeling 
 


 remains
the only requisite for capturing the third-order structure functions.

This finding reveals the significance of condition (c), which intrinsically ties with the stress-strain


(cid:105). Using the conversation in

correlation function, represented by 
 

(
, 
) = (cid:104)¯S

(
 + 
 · 
, 
)T

[81, pp. 317], 
 

 is derived in terms of 
 


 as


 

(
, 
) =



 


(
, 
)





7
2

+ 4
 


(
, 
)




+ 

2


2
 


(
, 
)



2

.

(5.40)

Emphasizing the role of tempering parameter in modulating the turbulent dissipation range, we
therefore adopt 
 

(
, 
) as a key quantity in optimizing the TFSGS model to address condition (c)
and capture the nonlocal structures in (5.39). It must be noted that in evaluating the aforementioned
conditions and high-order structures, T
 represents either T
,
, obtained by filtering the instant
DNS database, or T
,∗, implied by implementing the model to the true resolved velocity field.

5.4.1 Optimization strategy

Devising a robust optimization framework is an inevitable element in predictive fractional and
tempered fractional modeling [see 228, 218]. Regarding the given set of conditions for the closure
problem, we find conditions (a) and (c) practically crucial in developing an approach for estimating
the parameters and coefficient associated with the TFSGS model while condition (b) can be

 
(cid:105). As we learn from
substantially recovered by imposing (5.39), where 
 


(
, 
)|
=0 = (cid:104) ¯

T


125

the one-point correlation analysis in [232] and the further section, correlation between the SGS
stresses, obtained by the DNS data and the model, highly relies on 
 and 
 in the TFSGS model
rather plays a central role in capturing the SGS dissipation energy and nonlocal structure functions.
This approach provides the basis for an optimal estimation of the fractional exponents (
 and 
) by
employing the normalized 
 

 and 

, defined in Algorithm 5.1.
Algorithm 5.1 Estimation of the optimal model parameters for a specific L
[-1]INPUT:
OUTPUT:
PROCESS:1. Find 

 

 where the maximum of 


 =


 

(0,
)(cid:3) 
 for the inferred 

 

.3. Quantify the model constant such that

occurs.2. Find 

 

 where



 

, 

 

, c
,



 

(0,
)(cid:3)
 
 fits into(cid:2) 
 

(
,
)
(cid:2) 
 

(
,
)

, ¯

(
 + 
 · 
, 
)

¯

, 
 
,


, T
,
 


(cid:105)(cid:69)

(cid:68)




T
,





(cid:104)


 






c
,
 =

(cid:104)¯S
 
 T
,

(cid:104)¯S
 
 T
, 



 



 


(cid:105)
(cid:105), given 

 

 and 

 

.

VALIDATION ANALYSIS:
1. Evaluate 
 


(
) for the modeled and true SGS stresses to check if (5.39) is validated.
2. Perform a comparative study on the PDFs of (cid:104)¯S
 
T



 
(cid:105) and (cid:104)T



(cid:105).

By fixing the values of fractional exponents, it is possible to accurately quantify the model
coefficient and thereby re-producing the third-order structure in (5.39) via modeling 
 


.
In
Algorithm 5.1, we schematically present the proposed method for optimizing the parameters asso-
ciated with the TFSGS model at a given flow Reynolds number (Re) and a specific L. It must be
noted that in step 3, we define

T
, 


 


=

T
, 
 


 

c
,


=

¯


2


(
, 
)(cid:2)R
, ¯







1

1



=0



(cid:3) .



 + R
, ¯






(5.41)

Moreover, superscripts “
” and “
 
” represent the values obtained by filtering the true DNS data
and the SGS model, respectively.

5.5 A Priori Study

As discussed in previous sections, the TFSGS model inherits the potentials of a tempered
power-law distribution at the kinetic level, which account for the interplay between the resolved

126

and small-scale motions. To attain the optimal behavior of the TFSGS model, we follow the steps
in Algorithm 5.1 by performing an a priori analysis and evaluate the capabilities of the TFSGS
model in generating the statistical features of turbulent flows.

5.5.1 DNS Database and LES Platform

In terms of a priori tests, we conduct the numerical simulation of a forced HIT flow employing the
open-source pseudo-spectral NS solver for a triply periodic domain the code of which is presented
at [246]. It should be noted that in the next section, the LES solver is successfully prepared using
the DNS code and the presented stationary dataset as initial conditions for the final a posteriori
assessments.

√

(cid:113)

Using the NS solver, we performed DNS of a stationary HIT flow with 3203 resolution for a
periodic computation domain as  = [0, 2
]3 and the large scale forcing occurs at 0 < |
| ≤ 2
to maintain turbulence statistics stationary. Here, 
 represents the vector of Fourier wave numbers
2/3 is the maximum wave number solved numerically, where 
 = 320 is the
and 



 = 2 

number of grid points. In this case, 





 = 1.6 > 1 certifies that all the scales of motion are
well-resolved, where 

 refers to the Kolmogorov length scale. We detail the flow parameters
and some of the statistical properties at Table 5.1, in which 
 and 



 denote the expected
and
values of dissipation rate and turbulent kinetic energy, respectively. Moreover, 


 =



/
 represent the Taylor Reynolds number and micro-scale length, respectively,


 =
where 
(cid:48)
L , to
construct 40 sample snapshots as our database. Due to the incumbent homogeneity and isotropy
in the HIT flow, we find the database adequate for obtaining the required statistics in the further
analysis. The kurtosis and skewnewss values of the diagonal velocity gradients are also presented
in Table 5.1, supporting non-Gaussianity of turbulent structures.




 =(cid:112)2



/3. The simulation undergoes running for 30 eddy turn-over times, 


15

(cid:48)2


(cid:48)










For the purpose of crunching heavy DNS database and overcoming the burden of timely filtering
process in the two-point correlation analysis, we develop a scalable multi-threaded filtering code
using Numpy, threading, and astropy.convolution. The described platform is publicly acces-

127

Table 5.1: Flow parameters and statistical properties in the DNS of a forced HIT flow.






190







(cid:48)
(
/


)
0.67







(
2/


2)

0.68




(
2/


)
0.001




(
2/


3)

0.1



L
(


)
4.2

skewness

kurtosis

−0.5

6.5

sible through (https://github.com/FMATH-Group) and (https://github.com/samieeme/
Dissertation).

5.5.2 Optimal estimation of fractional parameters

In order to optimize the efficiency of the TFSGS model, we developed a multi-step strategy in
Algorithm 5.1. The proposed algorithm is equipped with verification and validation mechanisms
through conventional correlation coefficients and two-point structure functions. Recalling from
section 5.4.1 that T
,

denotes the true SGS stresses obtained by filtering the well-resolved DNS
data. Moreover, T
,∗
represents the general form of modeled SGS values, where ∗ can be replaced
by 
 
 or 

 in the TFSGS or SMG models, respectively.


 



 


and T
,
 










The first step in Algorithm 5.1 concerns detecting optimum value of fractional exponent, 

 

,
where the maximum of ensemble-averaged correlation between T
,

, denoted by 


,
occurs. Our premise is that the tempering parameter, 
, does not make any noticeable changes
in 


, namely less than 3 percents, which is endorsed by the results of Table 5.2. In the absence
of 
, we plot the variations of 


 versus 

 

 in Figure 5.1 for 
 = 1, 2, 3, in which each dashed
box specifies the interval of 
 yielding the maximum of 


. Without any loss of accuracy, we
adopt 

 

 = 0.76, 0.58, 0.51 as the corresponding minimum value in each specified interval for
L
 = L

2

 = 4, 8, 12, respectively, where 

 = 2



 represents the computational grid size.

From the kinetic perspective, by enlarging L
, 
 

(Ɗ) in (5.13) demonstrates an increasingly
multi-exponential pattern, which can be better described by a power-law distribution function.
This argument accounts for the prediction enhancement in Figure 5.1, achieved by the TFSGS
model and the abduct reduction of 

 

 versus L
, which is extensively discussed in [232, page

128

Figure 5.1: variation of the correlation coefficients (a) 
11, (b) 
22, and (c) 
33 versus 
 ∈ (0, 1)
for L
 = 4, 8, 12 by setting 
 (cid:39) 0 in (5.41). The maximum values lie in the dashed boxes.
Table 5.2: The ensemble-averaged correlation coefficients (


) between SGS stresses obtained by
the filtered DNS data (T
,

) for 
 = 1, 2, 3. In the fractional models,


 

 is set as 0.76, 0.58, and 0.51 for L
 = 4, 8, 12 respectively.

) and the TFSGS model (T
,
 










SMG
−
0.20
0.21
0.21

FSGS

0
0.21
0.21
0.22




11

22

33

L
 = 4
TFSGS
0.1
4
0.19
0.20
0.20
0.20
0.21
0.21

FSGS

0
0.29
0.29
0.30




11

22

33

L
 = 12
TFSGS
5
0.26
0.26
0.29

0.45
0.28
0.28
0.30

FSGS

0
0.23
0.24
0.25

L
 = 8
TFSGS
5
0.22
0.23
0.24

0.35
0.24
0.24
0.25

SMG
−
0.22
0.23
0.23

SMG
−
0.26
0.26
0.28

10]. Theoretically, the tempered power-law distribution can resemble a power-law or a Gaussian
distribution by letting 
 go to 0 or ∞, respectively. This grounds for the TFSGS model to span
the gap between the FSGS model, representing self-similar behavior of the inertial range, and the
SMG model, renowned for its dissipative characteristics. The results in Table 5.2 support this line
of reasoning by a row of correlation quantities for the given filter widths, particularly at L
 = 12.
On this background, we proceed with the second step in Algorithm 5.1 to indicate 

 

 through

 

(
,
)

 

(0,
) .
a comparative study of the normalized strain-stress correlation function, defined as 

With the knowledge of 
 

(
, 
), we extend the two-point correlation analysis to the spectral space

by evaluating the instantaneous radial dissipation spectrum, given by ˜D(
) = F(cid:2)
 

(
, 
)(cid:3)(
).

Ɗ =

129

0.00.20.40.60.81.0α0.150.200.250.30
11(a)Lδ=12Lδ=8Lδ=40.00.20.40.60.81.0α0.150.200.250.30
22(b)0.00.20.40.60.81.0α0.150.200.250.30
33(c)Figure 5.2: (a) Comparing results of the normalized two-point stress-strain rate correlation functions
(
Ɗ) and (b) error analysis of the longitudinal dissipation spectrum (E
) for fractional SGS and
Smagorinsky models, where L
 = 8 and 
 represents the radius of Fourier wave numbers. In both
plots, the arrows point to the dissipation range.

To evaluate the error between the dissipation spectrum, obtained by the true DNS data and the LES
models at high wave numbers, we define

(cid:12)(cid:12)(cid:12)(cid:2) ˜D(
)(cid:3)∗ −(cid:2) ˜D(
)(cid:3) 
(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) ˜D(
)(cid:3) 
(cid:12)(cid:12)(cid:12)(cid:12)

,

E
 =

where | · | represents norm of the vector. Figure 5.2 (a) displays 
Ɗ versus the spatial shift, 
,
for a logarithmic sequence of 
 spanning three orders of magnitude in the TFSGS model, where
L
 = 8. As stated earlier, the proposed model can take a journey from the FSGS to the SMG
models by tuning 
. Evidently, the true quantities of 
Ɗ, colored by black, are well-predicted by
the proposed model with 

 

 = 0.45, where 

 

 = 0.58 is fixed. Figure 5.2 (b) confirms our
findings quantitatively in a plot of E
 versus radius of wave numbers, 
, with log-scale axes. In
fact, this plot implies accuracy of the TFSGS model in capturing the two-point structure function
at the dissipation range, pointed by an arrow.

Employing the same analysis for L
 = 4, 12, we infer the optimal behavior of the TFSGS
model, evaluated for a logarithmic range of 
 with a fixed 

 

, in Figure 5.3. The inset plots show
E
 versus 
 using log-scale on both axes to magnify the dissipation range at high wave numbers.
Interestingly, at L
 = 4 the FSGS model is dissipative enough to outperform the tempered model

130

050100150200250300350400r/ηk0.00.20.40.60.81.0S∆(a)DNSSMGFSGSTFSGSλ=0.05TFSGSλ=0.35TFSGSλ=51020.000.250.500.751.001012×1013×1014×1016×101ξ10−510−410−310−210−1EH(b)Figure 5.3: Comparing results of the normalized two-point stress-strain rate correlation functions
for (a) L
 = 4 and (b) L
 = 12. The inset plots illustrate the scaled error of longitudinal dissipation
spectrum (E
) versus the radius of Fourier wave numbers (
). The arrows point to the dissipation
range.

in capturing the true 
Ɗ in Figure 5.3 (a). With all this in mind, these results certify the importance
of tempering in correct regeneration of two-point correlation functions particularly at larger filter
widths (L
 = 8, 12). Moreover, the SMG model, resembling the TFSGS with 
 ∼ 5, exhibits a
relatively steeper slope at the dissipation range, which is rooted in the diffusive form of its operator.
In this context, tempering plays a crucial role in characterizing dissipation structures by comparing
the widening gaps between the asymptotic cases (
 = 0.01 and 5) in Figure 5.3 (b). This brings up
the TFSGS model as superior physics based model regarding the counterparts, i.e., the SMG and
the FSGS models.

Given the values of 

 

 and 

 

, we proceed lastly with quantifying c
,
 as prescribed in
Algorithm 5.1. Under statistically stationary circumstances of the flow field, c
,
 remains fairly
unchanged for each L
 of interest, as reported in Table 5.3. It should be noted that c
,
 is only part
of the fractional coefficient, described in (5.41), in order to scale up the model in a constant 



and L
.

5.5.3

Interpretation of two-point structure functions

The third-order structure functions, arising from the KH equations, provide insights about the
statistics of unresolved scales and their strong interactions with large scale motions. As discussed

131

050100150200250300350400r/ηk0.00.20.40.60.81.0S∆(a)DNSSMGFSGSTFSGSλ=0.1TFSGSλ=1TFSGSλ=4101ξ10−410−310−210−1EH050100150200250300350400r/ηk0.00.20.40.60.81.0S∆(b)DNSSMGFSGSTFSGSλ=0.05TFSGSλ=0.45TFSGSλ=5101ξ10−310−210−1EHTable 5.3: Optimized parameters associated with the TFSGS model in terms of Algorithm 5.1 for
differing filter widths.
L
 = 4

L
 = 12

L
 = 8



0.76



(cid:39) 0

c
,

2.08



0.58



0.35

c
,

0.88



0.51



0.45

c
,

0.048


 



previously in section 5.4, 
Ɗ =

L , representing the scaled two-point velocity-stress correlation
function, is introduced as a sufficient condition for precise regeneration of third-order structure
functions and an a priori consistency in LES modeling. Following the derivation of the longitudinal
Taylor maicroscale in [138, chapter 6], 
 

(
) seems to be directly connected to the first-order
derivative of 
 


(
) at the dissipation range. This offers the capability of the optimum edition
of TFSGS model in capturing 
Ɗ and thereby fulfilling the essential conditions in (5.39).

In the first stage of the statistical analysis, we perform a comprehensive study on 
Ɗ in Figure
5.4 (a) for L
 = 8, in which the dissipation and inertial ranges are magnified in Figure 5.4 (b)
with semi-logarithmic scale on the x-axis and Figure 5.4 (c) with logarithmic scale on both axes,
respectively. The balance regions (BR), including extremum points, are thickened up in all the
graphs in Figure 5.4 (a). BR also indicates the transitional zone between dissipation and inertial
ranges. Aligned with the right side of the (5.39), the trend of 
Ɗ at small-scale interactions appear
to be a linear function of spatial displacement, 
, suggested by [76, Figure 2]. The results in Figure
5.4 (a) and more accurately in Figure 5.4 (b) offer that the optimum TFSGS model well-predict
the true DNS quantities at the left side of BR, not only the slope of 
Ɗ but also the maximum of

Ɗ occurring at a relatively close 
. This spotlights the importance of step three of Algorithm
5.1 in tuning slope of 
Ɗ at the dissipation range and effective role of the tempering parameter in
fitting the BR, associated with the filtered DNS data. In practice, increasing 
 pushes BR toward
the left side to preserve the increasing linear correlation as a notion of more dissipative behavior.
These findings are endorsed qualitatively for the other filter widths in Figure 5.5, considering 

 


in Table 5.3.

In analysis of 
Ɗ at the inertial range, the graph, associated with 

 

, shows a favorable match

132

Figure 5.4: Two-point velocity-stress correlation function in a stationary HIT flow for L
 = 8 using
box filtering. The segment of balance region (BR) has been thickened up in (a) for all the graphs.
The dissipation and the inertial ranges have been enlarged in plots (b) with semi-logarithmic scale
on the x-axis and (c) with logarithmic scale on the both axes, respectively.

Figure 5.5: Two-point velocity-stress correlation function in a stationary HIT flow for (a) L
 = 4
and (b) L
 = 12 using box filtering. The inertial range has been enlarged in the inset plots with
logarithmic scale on the both axes.

133

050100150200250300350400r/ηk0.020.040.060.080.100.12G∆(a)DNSFSGSTFSGSλ=0.05TFSGSλ=0.35TFSGSλ=5SMG1020.040.060.080.10(b)1022×1023×1024×10210−1(c)050100150200250300350400r/ηk0.000.010.020.030.040.05G∆(a)DNSSMGFSGSTFSGSλ=0.1TFSGSλ=1TFSGSλ=41021.5×1024×10210−20100200300400r/ηk0.020.040.060.080.10G∆(b)DNSSMGFSGSTFSGSλ=0.05TFSGSλ=0.45TFSGSλ=51021.5×1024×10200.104×10−26×10−210−1with true points, colored by black, in Figures 5.4 and 5.5. For the purpose of clarity, the inertial
ranges are magnified in log-log scale plots in Figure 5.4 (c) for L
 = 8 and the inset plots in Figure
5.5 for L
 = 4, 12, respectively. Motivated by these results, tempered fractional modeling seems
to be faithful in fitting structures at the dissipation and the inertial ranges and also estimating the
Inevitably, enlarging L
 accounts for
correct value of 
, associated with the extremum points.
inaccuracies in fitting the tail of graphs as observed in 5.5 (b). Notwithstanding, the mid-range
interactions are acceptably predicted by the optimized TFSGS model.

With an overview of the present results, the TFSGS model stands out as a structure-based
approach, which reasonably covers the gap between the FSGS and the SMG models. In L
 = 4,


 

 is found to be very close to zero, which renders tempering nonessential in capturing two-point
structures. As we increase L
, this gap starts widening up and tempering mechanism acts more
dynamically in finding the true BR and fitting the dissipation structures. This argument confirms
that the tempered fractional approach displays a great potential for parameterizing structure function
especially at larger filter widths while retaining fairly acceptable accuracy.

5.5.4 PDF of SGS stresses

Within the proposed statistical framework, the last step in Algorithm 5.1 focuses on the PDFs of
filtered DNS data. The key idea is to assess the performance of models and verify if the proposed
model maintains the true statistics. In this context, we present the scatter plots of T
, 

against
T
, 
 

in Figure 5.6 for three given filter widths and 
 = 3. We should note that the present results



are confined to 
 = 3 due to the similarities in other directions. The slope in each plot is indicated
by the corresponding correlation coefficients in Table 5.2. The most noticeable specific about these
results is that the data points are bounded within a same order of magnitude on both axes. As a
matter of fact, we achieve a roughly unit regression coefficient between T
, 

, where
our optimization strategy targets for correct estimation of the SGS dissipation. This analysis can
be extended to the PDF plots in Figure 5.7. With nearly the same correlation coefficients, the
SMG model fails to reproduce the true statistics, while the optimum TFSGS model offers a great

and T
, 
 














134

Figure 5.6: Scatter plots of the SGS stresses obtained by the filtered DNS data (T
, 

modeled stresses (T
, 
 

(c) L
 = 12.

33 ) versus the
) using optimized parameters in Table 5.3 for (a) L
 = 4, (b) L
 = 8, and

33

Figure 5.7: PDF of the ensemble-averaged(cid:0)∇ · T(cid:1)

the SMG models at (a) L
 = 4, (b) L
 = 8, and (c) L
 = 12.


=3 for the optimized (tempered) fractional and

match with the true graphs. As pointed out previously, in L
 = 4 the FSGS model represents the
equivalent form of the TFSGS with 

 

 = 0.76 and 

 

 (cid:39) 0.

From the understanding of energy cascading in turbulent flows, the SGS dissipation, 
, is
considered as an external parameter in two-point structure equations for describing small-scale
motions. In the statistical sense, we compare the PDFs of 
, implied by the models, with the true
PDFs, obtained by the filtered DNS data for L
 = 4, 8, 12. As shown in Figure 5.8, the fractional
models accurately predicts the forward scattering, associated with the positive dissipation, 
+,
while the SMG model appears to be too dissipative due to its positive eddy viscosity. Furthermore,
the TFSGS model presents an under-prediction of the backward scattering by producing a slim
amount of negative dissipation, 
−. On the side of numerical analysis, this limitation results in
preserving numerical stability by minimizing negative dissipation error. In next section, we provide

135

−0.10−0.050.000.050.10TR,TF33−0.10−0.050.000.050.10TR,D33
33=0.22(a)−0.10−0.050.000.050.10TR,TF33−0.10−0.050.000.050.10TR,D33
33=0.25(b)−0.10−0.050.000.050.10TR,TF33−0.10−0.050.000.050.10TR,D33
33=0.30(c)−2.0−1.5−1.0−0.50.00.51.01.52.0h(∇·TR)3i101102103PDF(a)FSGSTFSGSλ=1SMGDNS−2.0−1.5−1.0−0.50.00.51.01.52.0h(∇·TR)3i101102PDF(b)FSGSTFSGSλ=0.35SMGDNS−1.0−0.50.00.51.0h(∇·TR)3i101PDF(c)FSGSTFSGSλ=0.45SMGDNSFigure 5.8: PDF of the ensemble-averaged SGS dissipation for the optimized (tempered) fractional
and the SMG models at (a) L
 = 4, (b) L
 = 8, and (c) L
 = 12.

an extensive discussion in terms of an a posteriori analysis.

5.6 A Posteriori Analysis

With a focus on numerical stability, we extend the statistical a priori assessments to an a
posteriori analysis in order to evaluate the performance of proposed models in time. To outline
the a posteriori framework, we employ the flow solver, described in section 5.5.1, on 403 and
203 grids for the corresponding L
 = 4, 8, respectively. The simulations are initiated with an
instantaneous flow field, given from the filtered DNS of the stationary HIT flow. This analysis also
allows for structural comparisons between the fractional models with the filtered DNS data through
the resolved turbulent kinetic energy, 



(
) = (cid:104) ¯

 ¯


2 (cid:105)
, where
¯

(
, 

) denotes the 
th component of the instantaneous filtered vorticity field. It should be noted
that in this section (cid:104)·(cid:105)
 denotes the spatial averaging over the entire domain.

2 (cid:105)
, and the enstrophy, E(
) = (cid:104) ¯

 ¯



Figure 5.9 displays the decay of kinetic energy in the subgrid-scale level, which verifies the
computational stability for the fractional model. In most LES approaches, fidelity in representing
spatial structures is essentially compromised to preserve numerical stability by inducing the exces-
sive amount of energy dissipation. Nevertheless, the present results verify our findings in section
5.4 that the TFSGS model provides stable LES solutions while preserving high-order structure
functions in the a priori tests. Comparing the results in Figure 5.9, the fractional model also
exhibits an acceptable performance in predicting the time-evolution of kinetic energy after one or
two eddy turn-over times, 


L , for both resolutions.

136

−0.4−0.20.00.20.40.60.81.01.2h¯SijTRiji101102103104PDF(a)FSGSTFSGSλ=1SMGDNS−0.2−0.10.00.10.20.30.40.50.6h¯SijTRiji101102103PDF(b)FSGSTFSGSλ=0.35SMGDNS−0.10−0.050.000.050.100.150.200.250.30h¯SijTRiji101102PDF(c)FSGSTFSGSλ=0.45SMGDNSFigure 5.9: Decaying of the resolved turbulent kinetic energy, 



, for the optimized TFSGS and
the SMG models with (a) 403 and (b) 203grid points.

Figure 5.10: Time-evolution of the resolved enstrophy, E, for the optimized TFSGS and the SMG
models with (a) 403 and (b) 203grid points.

Given the analysis in section 5.4, third-order structure functions in (5.39) are connected to the
third-order moment of filtered velocity derivatives and thereby the enstrophy balance as described
in [76]. Interestingly, based on the analysis of two-point structures in subsection 5.5.1, the TFSGS
model presents fulfilling the essential conditions in Algorithm 5.1 and accordingly, maintaining
the equilibrium between the production and the dissipation of enstrophy at the resolved and the
subgrid-scale levels. Figure 5.10 demonstrates performance of the LES models in estimating this
equilibrium in time. It seems that enforcing a constant coefficient, c
,
, into the LES solver causes
the under-prediction of the enstrophy decaying rate in Figure 5.10. However, these results confirm
numerical stability of the proposed models for both resolutions.

137

01234567t/τL0.00.10.20.30.40.50.6Ktot(a)SMGTFSGSλ∼0.0FilteredDNS01234567t/τL0.00.10.20.30.4Ktot(b)SMGTFSGSλ∼0.35FilteredDNS01234567t/τL0246810E(a)SMGTFSGSλ∼0.0FilteredDNS01234567t/τL0.00.51.01.52.02.53.03.54.0E(b)SMGTFSGSλ∼0.35FilteredDNSCHAPTER 6

SUMMARY AND FUTURE WORKS

The main objective of this study was to develop a robust mathematical framework for fractional
modeling of anomalous phenomena, starting from the kinetic level. In the context of modeling
scalar dispersion in underground waters, we developed a fast and accurate numerical method for
the proposed fractional model and performed a comprehensive theoretical and numerical analysis,
as the essential components of the framework. Inspired by thr nonlocal triad interactions between
large and small scale motions, we also extended the framework to turbulent flows by developing
a SGS model using fractional operators. This framework offers the advantage of allowing us to
develop more physically-consistent SGS models, which suitably capture the anomalous features of
turbulent flows.

6.1 Concluding Remarks

• In chapter 2, we developed a new unified PG spectral method for a class of fractional partial
differential equations with constant coefficients in a (1+
)-dimensional space-time hypercube,

 = 1, 2, 3, etc, subject to homogeneous Dirichlet initial/boundary conditions. We employed
Jacobi poly-fractonomials, as temporal basis/test functions, and the Legendre polynomials as
spatial basis/test functions, yielding spatial mass matrices being independent of the spatial
fractional orders. Additionally, we formulated the novel unified fast linear solver for the
resulting high-dimensional linear system, which reduces the computational cost significantly.
In fact, the main idea of the study was to formulate a closed-form solution for the high-
dimensional Lyapunov equation in terms of the eigensolutions up to the precision accuracy
of computationally obtained eigensolutions. Moreover, we proved the corresponding well-
posedness and discrete stability analysis. We obtained the theoretical error estimates, proving
that the method converges spectrally fast under certain conditions. Finally, several numerical
test cases, including finite regularity and smooth solutions, have been performed to show the

138

spectral accuracy of the method.

• In chapter 3, we developed a unified PG spectral method for fully distributed-order PDEs with
constant coefficients on a (1+ 
)-dimensional space-time hypercube, subject to homogeneous
Dirichlet initial/boundary conditions. We obtained the weak formulation of the problem, and
proved the well-posedness by defining the proper underlying distributed Sobolev spaces and
the associated norms. We then formulated the numerical scheme, exploiting Jacobi poly-
fractonomials as temporal basis/test functions, and Legendre polynomials as spatial basis/test
functions. In order to improve efficiency of the proposed method in higher-dimensions, we
constructed a unified fast linear solver employing certain properties of the stiffness/mass
matrices, which significantly reduced the computation time. Moreover, we proved stability
of the developed scheme and carried out the error analysis. Finally, via several numerical
test cases, we examined the practical performance of proposed method and illustrated the
spectral accuracy.

• More broadly, this study presented a new framework in chapter 4 to the functional modeling
of SGS stresses in the LES of turbulent flows, starting from the kinetic theory. Within the
proposed framework, we began with modeling the filtered equilibrium distribution function
as a key term to consider the power-law scaling of SGS motions in the filtered BTE. Due to
the multi-exponential behavior of the filtered equilibrium distribution function, we proposed
to approximate it with a Lévy-stable distribution, where the associated fractional parameter
strictly relied on the filter width. Subsequently, we derived the filtered NS equations from
the approximated filtered BTE, in which the divergence of SGS stresses was modeled via
a fractional Laplacian operator, (−Ɗ)
(·) for 
 ∈ (0, 1].
In general, we established a
framework, which permitted us to treat the source of turbulent motions at the kinetic level
by employing a compatible choice of distribution function and derive the corresponding
fractional operator in the filtered NS equations as an SGS model. Therefore, the proposed
framework, termed “FSGS modeling”, could potentially recover the non-Gaussian statistics

139

of SGS motions precisely. Next, we studied the physical and mathematical properties of
the proposed model and introduced a set of mild conditions to preserve the second law of
thermodynamics. Eventually, we carried out a priori evaluations of the FSGS model based
on the DNS database of forced and decaying HIT problems. In light of the analysis, there
was a relatively great agreement between the modeled and true SGS values in terms of
the correlation and regression coefficients. The performance of the FSGS model depended
rigorously on the choice of fractional exponent, 
, as a function of filter width, L, and flow
Reynolds number, 


. We showed that, by enlarging L, the heavy-tailed characteristics of
the SGS motions could become more intensified, which were conceivably well-described by
the FSGS model with smaller values of 
. With all this in mind, FSGS model provided a
new perspective, which respected the non-Gaussian behavior of SGS stresses by exploiting
fractional calculus within the Boltzmann transport framework.

• Motivated by the semi-power-law behavior of energy cascading phenomenon in turbulent
flows, we focus on developing a nonlocal model by employing a tempered heavy-tailed
distribution within the described fractional framework in chapter 5. To approximate muti-
exponential behaviors of the filtered equilibrium distribution in the collision operator, we
employed a tempered Lévy-stable distribution function, which presents a power-law at a
moderate range and then converges to an exponential decay. By ensemble-averaging of the
approximated Boltzmann transport, we derived the LES equations, in which the divergence
of SGS stresses emerged as a summation of tempered fractional Laplacian, (Ɗ+
)
(·), where

 ∈ (0, 1) and 
 > 0. Interestingly, the FSGS is found to be a particular form of the TFSGS
model when 
 approaches 0. Moreover, we formulated the SGS stresses straightforwardly in
terms of a combination of integer and fractional operators, which gives the advantage of being
feasible and quite easy to implement in the Fourier space. The corollary on frame invariant
property of the FSGS model were also extended to the current model, showing its physical
and mathematical consistency. In a statistical framework, we constructed a structure based
algorithm for optimizing the fractional models, which involved the closed essential conditions

140

for a weaker sense of an ideal LES model. Following the optimization strategy, we inferred
the optimum tempering parameter through a comparative study of two-point strain-stress
correlation functions while the fractional exponent was fixed for maintaining reasonable
values of correlation coefficients. Regarding the KH equation, the optimum TFSGS model
presented a great match with the true values of two-point velocity-stress correlation functions,
which ensures the accurate prediction of third-order structure functions. The success of
tempering mechanism in capturing structure correlation functions, particularly at larger L
,
originated from the capabilities of our choice in fitting semi-heavy-tailed behavior of the
filtered equilibrium distribution at the kinetic level. The inspection of statistical results
also supported accuracy of the fractional model in keeping unit regression and capturing
the corresponding PDF tails. As a notion of numerical stability, we demonstrated that the
optimized TFSGS model well-predicted the true forward scattering in a statistical sense
without generating any significant negative dissipation. Lastly, the TFSGS model underwent
the ultimate a posteriori analysis, which verified numerically stable performance of the
fractional model through tracking turbulent kinetic energy

6.2 Future Works

Despite the theorertical and statistical achievements, we believe that this framework, as a viable
and promising direction toward nonlocal modelings, can be upgraded for the purpose of studying
more complex turbulent regimes. On the theoretical side, the current framework deserves a careful
mathematical attention to be extended to anisotropic and inhomogeneous flows employing proper
forms of distributions at the kinetic level. Moreover, further works should be undertaken to
generalize the fractional model to a data-driven representation of spatial and temporal structures in
more complex turbulent regimes.

141

APPENDICES

142

APPENDIX A

DERIVATIONS IN CHAPTER 2

Here, we provide the force function based on the exact solutions.

A.1 Force term of test case (I)

Let take the exact solution as







 = 
 
1 ×(cid:16)(1 + 
) 
2 − 
(1 + 
) 
3(cid:17)

,

(A.1)

where 
 ∈ [0, 2] and 
 ∈ [−1, 1]. To obtain 
 in (2.14) based on (A.1), first we need to calculate all
fractional derivatives of 



. To satisfy the corresponding boundary conditions, 

 = 2
2
−
2
+1.
− 1 and 

 ∈ [−1 , 1].
Take 

 = 
 
1 and 
 

Considering (2.2),


 = (1 + 

) 
2
 − 

 (1 + 

) 
2
+1, where 

 = 2 

−




−



0D2

where 
(
) = 2( 


and


 


 =





2

Ɖ[ 
1+1−2
] 
 
1−2
 = ( 

(cid:17)−2

(cid:104)


 

 = Ɖ[ 
1+1]
(cid:16) 

−



 ) − 1. Similarly,
Ɖ[ 
2
+1+1−2

] (1 + 
2
(

)) 
2
+1−2

(cid:105)


D2





(cid:17)−2

−2(cid:104) Ɖ[ 
2
+1]
(cid:16) 

−


Ɖ[ 
2
+1+1−2

] (1 + 
2
(

)) 
2
+1−2

(cid:105)

2
Ɖ[ 
2
+1+1]

Ɖ[ 
2
+1+1]



D2







 


 =





2) 
1−2
 Ɖ[ 
1+1]

Ɖ[ 
1+1−2
] (1 + 
(
)) 
1−2
,

(A.2)

Ɖ[ 
2
+1]
Ɖ[ 
2
+1−2

] (1 + 
2
(

)) 
2
−2

 −

,

(A.3)

Ɖ[ 
2
+1−2

] (1 + 
2
(

)) 
2
−2

 −

.

(A.4)

143

Therefore,


 = ( 

2











2

2

) 
1−2



=1
− 



(cid:16) 

 − 



(cid:16) 

 − 



( 

2
Ɖ[ 
2
+1 + 1]




Ɖ[ 
1 + 1]
(cid:17)−2

(cid:104)
) 
1(1 + 
) 
1(cid:16)
Ɖ[ 
1 + 1 − 2
] (1 + 
) 
1−2

+ 

(1 + 

) 
2
 − 

 (1 + 

) 
2
+1

=1
Ɖ[ 
2
 + 1]
Ɖ[ 
2
 + 1 − 2

] (1 + 
2
) 
2
−2


Ɖ[ 
2
+1 + 1 − 2

] (1 + 
2
) 
2
+1−2

(cid:105)



(cid:17)−2

−2(cid:104)
) 
1(1 + 
) 
1(cid:16)
− 

Ɖ[ 
2
 + 1]
Ɖ[ 
2
 + 1 − 2

] (1 + 
2
) 
2
−2


Ɖ[ 
2
+1 + 1 − 2

] (1 + 
2
) 
2
+1−2

(cid:105)



Take 

 = 
 
1 and 
 
 = 


(cid:0)


1(cid:1). Here, we approximate 
 
 as

=1(−1)2 
−1 (


1)2 
−1
(2 
 − 1)!

[(1 + 
 
) 
2 
 − 
 
 (1 + 
 
) 
2 
+1](cid:17)
[(1 + 
 
) 
2 
 − 
 
 (1 + 
 
) 
2 
+1](cid:17)

( 

2
Ɖ[ 
2
+1 + 1]


=1
− 




 
 = 




=1, 
ࣔ



=1, 
ࣔ


A.2 Force term of test case (II)

. (A.5)

,

(A.6)

where 

 controls the level of approximation error. Taking the same steps of (A.5), we obtain

) 
1−2



 = ( 

2
+ ( 

2

) 
1(1 + 
) 
1(cid:2)(

1)(cid:16) 
1 − 
1
− (

1)(cid:16) 
1 − 
1

Ɖ[ 
1 + 1]
(cid:17)−2
1
Ɖ[ 
1 + 1 − 2
] (1 + 
) 
1−2



(cid:17)−2
1−2





=1(−1)2 
−1 (


1)2 
−1
(2 
 − 1)!

=1(−1)2 
−1 (


1)2 
−1
(2 
 − 1)!

=1(−1)2 
−1 (


1)2 
−1
Ɖ[2 
]
(2 
 − 1)!

Ɖ[2 
 − 2
1] 
2 
−2
1(cid:3) .





2

2

Ɖ[2 
]
Ɖ[2 
 − 2
1] 
2 
−2
1

(A.7)

A.3 Proof of Lemma 2.4.1

Proof. In Lemma 2.1 in [109] and also in [108], it is shown that (cid:107) · (cid:107)
 

(') and (cid:107) · (cid:107)
 

(') are
equivalent. Therefore, for 
 ∈ 

('), there exist positive constants 
1 and 
2 such that

(cid:107)
(cid:107)

(') ≤ 
1(cid:107)
(cid:107)
 

('),
(cid:107)
(cid:107)

(') ≤ 
2(cid:107)
(cid:107)
 

('),

144

(A.8)

which leads to

(cid:107)
(cid:107)2

1(cid:107)
(cid:107)2
1 (cid:107) 
D



 

(') + 
2


(') ≤ 
2

 (
)(cid:107)2
= 
2
≤ ˜
1 (cid:107)
(cid:107)2
 

('),

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

(')
2 (cid:107) 
D


2(') + 
2


 (
)(cid:107)2


2(') + (
2

1 + 
2

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


2(')

where ˜
1 is a positive constant. Similarly, we can show that

(cid:107)
(cid:107)2
 

(') ≤ ˜
2 (cid:107)
(cid:107)

('),

where ˜
2 is a positive constant. This equivalency and (2.19) conclude the proof.

(A.9)

(A.10)

A.4 Proof of Lemma 2.4.4

Proof. Let '
 = 





D







 = 

D





(

D2







=1(

, 

). According to [70], we have 

D2



). Let ¯
 = 

D





ˆ 



(

D






, 
)'
 = (

D



1






 = 

D





 (

D







) and

(cid:3)
 
'


¯
(
) 


(

 − 
)



(A.11)


'



 = 0 due to the homo-












 
'


 .

(A.12)

(cid:111)

(

 − 
)1−

 












(

 − 
)






. (A.13)









ˆ 






=

ˆ

ˆ

'

−


 =



1
´
where '

'



geneous boundary conditions. Therefore,


=1, 
ࣔ
. Then, we have
ˆ

'


(

D2







, 
)'
 = −

Moreover, we find that

ˆ 















(

 − 
)






 =






= − 





(cid:2) 

(cid:111)










=















'



'
,

(cid:111)





=



Ɖ(1 − 

)
¯




(

 − 
)


¯
(
) 


(

 − 
)


´ 







¯



(

−
)


ˆ 



Ɖ(1 − 

)





¯
(
) 


(

 − 
)



ˆ 





=
 − 1
}


1 − 


(

 − 
)1−

 


 =




(cid:110)


. Then,
ˆ


 ¯
, 
)'
 =
ˆ 


ˆ 



Ɖ(1 − 

)








'









Ɖ(1−

)

Ɖ(1 − 

)

(cid:110)
ˆ 


(cid:110){ 
 (

 − 
)1−


1 − 


ˆ 



'








1

1
1 − 














145

Therefore, we get

(

D





 ¯
, 
)'
 = −

ˆ

'


1

Ɖ(1 − 
)


¯
(
)(cid:0) 






ˆ 









(

 − 
)






(cid:1) 

 
'
 = ( ¯
, 

D








)'


.

A.5 Proof of Lemma 2.10.4

Proof. Let 
 = 1, 
 = 2, and 
 = 3. We have

−

(cid:107)
 − 

3

2

1



2,M2

1,M1

3,M3
= (cid:107)
 − 

 + 

1

1

1,M1

1,M1

3

2

1



3,M3

2,M2

1,M1

(cid:107)
 

1(
1,
2('1
≤ (cid:107)
 − 

1

1,M1

1

2

1


1,M1

2,M2

1,M1


(cid:107)
 

1(
1,
2('1
3))

 + 

 − 

2


2,M2

(cid:107)
 

1(
1,
2('1
3))

 − 
3)) + (cid:107)

1

1,M1

3

2



3,M3

2,M2


1

1,M1


 − 

+(cid:107)


1

1,M1


2


2,M2





(cid:107)
 

1(
1,
2('1

2

1


2,M2

1,M1
3))

(cid:107)
 

1(
1,
2('1
(A.14)
3)),

where by Theorem 2.10.1

(cid:107)
 − 


1

1,M1

Furthermore,


(cid:107)
 

1(
1,
2('1

3)) ≤ M
1−
1

1

(cid:107)
(cid:107)

1(
1,
2('1
3)).

(A.15)


 − 

(cid:107)

1

1,M1
≤ (cid:107)(

1

1,M1
≤ M
1−
1
≤ M
1−
1

(cid:107)
 − 
1 M−
2


2

1


2,M2

1,M1
− I)(
 − 

2

2,M2

(cid:107)

1(
1,
2('1


(cid:107)
 

1(
1,
2('1
3))

)(cid:107)
 

1(
1,
2('1
3)) + (cid:107)
 − 


2

2,M2
(cid:107)
(cid:107)

1,
2('2,
2(
3))) + M−
2

2

1

2


(cid:107)
 

1(
1,
2('1
3))

3)) + (cid:107)
 − 

2

2,M2


2

2,M2

(cid:107)
 

1(
1,
2('1
3))
(cid:107)
(cid:107)

1(
1,

2(
2,
2(
3))).

(A.16)

146

Similarly,





2

2,M2


3


3,M3


(cid:107)
 

1(
1,
2('1
3))

 − 

 + 

2

2,M2

(cid:107)

1

1,M1
= (cid:107)

1

1,M1
+

2

2,M2
≤ (cid:107)(

1

1,M1
+(cid:107)

2

2,M2
≤ M
1−
1
1
+(cid:107)(

2

2,M2
1 M−
2
≤ M
1−
1
2
+M−
2
(cid:107)
 − 
2
1 M−
2
≤ M
1−
1
2 M−
3
+M−
2


3

2



3,M3

2,M2

3

2



2,M2

3,M3

(cid:107)
 

1(
1,
2('1
3))

)(cid:107)
 

1(
1,
2('1

3

2


2,M2

3,M3
3))

(cid:107)
 

1(
1,
2('1
3))

3

3,M3

)(cid:107)
 

1(
1,
2('1
3)) + (cid:107)
 − 

(cid:107)

1,
2('2,
2(
3))) + M
1−
1


 − 

2

1


2,M2

1,M1

 − 

2

1


2,M2

1,M1

 − 

3

2


3,M3

2,M2

 − 
− I)(

2

2,M2

 − 

3

2


2,M2

3,M3
− I)(
 − 

2

2,M2
− I)(
 − 

3

3,M3

(cid:107)
 

1(
1,

2(
2,
2(
3))) + M−
3
1 M−
3
(cid:107)
(cid:107)

1,
2,
3('3))) + M
1−
1
(cid:107)
(cid:107)
 

1(
1,

3(
3,
2(
2))).

(cid:107)
 − 

3

3,M3
2 M−
3
(cid:107)
(cid:107)
 

1(
1,

2,
3('1


3

3,M3

(cid:107)
 

1(
1,
2('1
3))
(cid:107)
(cid:107)

1,
3('2

3)) + (cid:107)
 − 

3

3,M3
1 M−
3
(cid:107)
(cid:107)
 

1(
1,

3(
3,
2(
2)))
(cid:107)
(cid:107)

1,
3('2


)(cid:107)

1(
1,
2('1

3)) + M−
3

3,
2(
2)))


3

3,M3

((cid:107)(

3

3

3

3

3

3

3,
2(
2)))


(cid:107)

1(
1,
2('1
3)))

Therefore,


2


2,M2


(cid:107)
 

1(
1,
2('1

3


3,M3
3))
1 M−
2
3)) + M
1−
1
(cid:107)
(cid:107)

1(
1,
2('1
(cid:107)
(cid:107)

1,
2('2,
2(
3)))
(cid:107)
(cid:107)
 

1(
1,

2(
2,
2(
3))) + M
1−
1
3,
2(
2))) + M−
2

1 M−
2
2 M−
3

(cid:107)
(cid:107)

1,
3('2

2 M−
3

(cid:107)
(cid:107)

1,
2,
2('3)))
(cid:107)
(cid:107)
 

1(
1,

2,
3('1
3))

3

3

2

1 M−
3

3

1

(cid:107)
 − 

1

1,M1
≤ M
1−
1
+M−
2
2
+M
1−
1
+M−
3
3

(cid:107)
(cid:107)
 

1(
1,

3(
3,
2(
2))).

Following the same steps, we get


2


2,M2


(cid:107)
 

2(
2,
2('2

3


3,M3
3))
2 M−
1
3)) + M
2−
2
(cid:107)
(cid:107)

2(
2,
2('2
(cid:107)
(cid:107)

2,
1('2,
2(
3)))
(cid:107)
(cid:107)
 

2(
2,

1(
1,
2(
3))) + M
2−
2
3,
2(
1))) + M−
1

2 M−
1
2 M−
3

(cid:107)
(cid:107)

2,
3('1

1 M−
3

(cid:107)
(cid:107)

2,
1,
3('3)))
(cid:107)
(cid:107)
 

2(
2,

1,
3('2
3))

3

3

1

2 M−
3

3

2

(cid:107)
 − 

1

1,M1
≤ M
2−
2
+M−
1
1
+M
2−
2
+M−
3
3

(cid:107)
(cid:107)
 

2(
2,

3(
3,
2(
1)))
147

(A.17)

(A.18)

(A.19)

and


2


2,M2


3


3,M3


(cid:107)
 

3(
3,
2('2))
(cid:107)
(cid:107)

3(
3,
2('2)) + M
3−
3
3 M−
1
(cid:107)
(cid:107)
 

3(
3,

1(
1,
2(
2))) + M
3−
3
3,
2(
1))) + M−
1

(cid:107)
(cid:107)

3,
2('1

3 M−
1
1 M−
2

(cid:107)
 − 

1

1,M1
≤ M
3−
3
3
+M−
1
1
+M
3−
3
+M−
2
2

3 M−
2

2

(cid:107)
(cid:107)
 

3(
3,

2(
2,
2(
1))).

1

(cid:107)
(cid:107)

3,
1('2
1 M−
2

3,
2(
2)))
(cid:107)
(cid:107)

3,
1,
2('3)))
(cid:107)
(cid:107)
 

3(
3,

1,
2('2))

2

2

(A.20)

148

APPENDIX B

DERIVATIONS IN CHAPTER 3

B.1 Entries of Spatial Stiffness Matrix

Here, we provide the computation of entries of the spatial stiffness matrix by performing an

affine mapping 
 from the standard domain 








∈ [−1, 1] to 
 
 ∈ [








, 








].

Lemma B.1.1. The total spatial stiffness matrix 

 


computed as:




is symmetric and its entries can be exactly


 
 






= 

 
 × 



 + 

 
 × 


 



 − 

 
 × 


 



 − 

 
 × 


 



 




,

(B.1)

where 
 = 1, 2, · · · , 
.


 




 
 ,

= 
1







 



 













D 
 

1


 


{



 }
,
 =

 





D 




1

 


Proof. Regarding the definition of stiffness matrix, we have

(cid:16)


(
 
)(cid:17)
(cid:16)

(
 
)(cid:17)
ˆ 1
ˆ 





 
(
 
) −1D 
 



+1(
 
) − 

−1(
 
)(cid:17)
(cid:16)

 
(cid:0)
(




)(cid:1) −1D 




−1
ˆ 1
ˆ 1
(cid:16)


+1(
 
) − 

−1(
 
)(cid:17)
−1
−1
(cid:104)(cid:101)

× 
 


−1,
+1 +(cid:101)


+1,
−1 −(cid:101)


+1,
+1 −(cid:101)

(cid:17)
(cid:16) 




where 
1 =(cid:101)

 


−






(
 
)(cid:17)
(cid:16)
)(cid:1) −1D 





 
(cid:0)
(




(cid:101)

2
ˆ 1
ˆ 1
)(cid:1)

 
(cid:0)
(




ˆ 1
−1
−1
−1
ˆ 1

)−




−1




 

Ɖ(
 + 1)
Ɖ(
 − 




−







 




(cid:105)


(
 
)(cid:17)

Ɖ(
 + 1)
Ɖ(
 − 




,−















 + 1)


 
 







 + 1)
,





D 




1





 


−1,
−1

(1 − 
2











 


,
 =



 
 ,

,


 


= 
1

and




(cid:16)

.




=

×










(B.2)


 




 
 









149

(cid:101)



 


,
 can be computed accurately using Gauss-Legendre (GL) quadrature rules as

(cid:101)










,


=




ˆ 1
−1


=1

Ɖ(
 + 1)
Ɖ(
 − 








(1 − 
2


)−





Ɖ(
 + 1)
Ɖ(
 − 




|
,




−














|
 + 1) 
 
|
 

 ×
|

(
 
) 













|
 + 1)
|
 





|
,−







|


(
 
)

 
 ,

(B.3)

|
}Q

in which Q ≥ M 
 + 2 represents the minimum number of GL quadrature points {




exact quadrature, and {

}

of the Jacobi polynomials where 

,



,
 = (−1)(
+
)(cid:101)

Following chapter 2,(cid:101)

 and 

 are chosen such that (−1)(
+
) is canceled. Accordingly, {


(
 
), we have(cid:101)


(−
 
) = (−1)


,



=1 for

=1 are the corresponding quadrature weights. Exploiting the property







.

,


 }
,
 =

 


 }
,
 =

 


 


 

, 

,






 }
,
 due to the symmetry of 


 


 }
,
. Eventually, we conclude that the stiffness matrix 


 

}
,
 as the sum of symmetric matrices is symmetric.

. Similarly, we get {

, 



 }
,
 = {


 


 }
,
 = {


 


{


 }
,
 = {


 

{


 }
,
 = {


 

, and thereby {

 



 






and 











 





 





 
















150

APPENDIX C

DERIVATIONS IN CHAPTER 4

In this chapter, we follow the derivations of fractional NS equations in [163] to evaluate the shear
and SGS stresses in (4.36) and (4.37).

C.1 Temporal Shift

Recalling from the Assumption 5 that 
 ∼ O(1), we take the temporal Taylor expansion of 
 ∗

as follows:

where 
 ∗


(cid:0) ¯Ɗ(cid:1) + 
 
 ∗

(cid:0) ¯Ɗ(cid:1) + 
 
 ∗




 ¯Ɗ



 ¯Ɗ


 ∗

= 
 ∗


,
 = 
 ∗(cid:0) ¯Ɗ(
 − 

, 
 − 


, 
)(cid:1) = 
 ∗

(cid:0) ¯Ɗ(cid:1) = 
 ∗(cid:0) ¯Ɗ(
, 
 − 


, 
)(cid:1), 

 = −

 and
3


(cid:0) ¯Ɗ(cid:1) + 
 
 ∗
(

 − ¯

)(
 
 − ¯

)(cid:0) 
 ∗
(cid:20)

(cid:0) ¯Ɗ(cid:1) + 2

(

 − ¯

)(
 
 − ¯

)

ˆ ∞
ˆ ∞

−2

2


 ¯Ɗ




ˆ
ˆ




 ¯Ɗ


−



−



 ∗


=1

=

=

R


R



2

0

0

(

 − ¯

) 
 ¯



.







 
 ≈

Considering (C.1), we can approximate (4.29) according to


 ¯Ɗ




 ¯Ɗ






 + O(

2)
(−

) + O(
2),

(

)(cid:1) 

 

,
( 3



=1


 ¯Ɗ




 
 ∗



 ¯Ɗ

(

 − ¯

) 
 ¯







(cid:21)

) (

)

(C.1)

(C.2)

(C.3)



 

.

(C.4)

Since ¯Ɗ is an even function of (

 − ¯

) for 
 = 1, · · · , 3, 
 

( ¯Ɗ) and 
 




( ¯Ɗ) and also their
are even functions of (

 − ¯

). Subsequently,
corresponding first derivatives 
 

there is an odd power of either (

 − ¯

) or (
 
 − ¯

), which makes
(

 − ¯

) 
 ¯




 ¯Ɗ and 
 
 






 ¯Ɗ
(cid:20) 
 
 ∗

(

 − ¯

)(
 
 − ¯

)

( 3




 = 0.







ˆ




R





 ¯Ɗ


=1

Therefore,



 
 ≈

=

ˆ ∞
ˆ
0

ˆ

−

ˆ ∞

R


0

(

 − ¯

)(
 
 − ¯

) 
 ∗
R


−
(

 − ¯

)(
 
 − ¯

) 
 ∗

151





) (

)

(cid:21)

(cid:0) ¯Ɗ(cid:1) 

 



(cid:0) ¯Ɗ(cid:1) 

 

.

C.2 Shear Stresses

Regarding (4.34), the shear stress tensors are described according to

T 
Ò





 


=

(

 − ¯

)(
 
 − ¯

)( 
 






( ¯Ɗ) − 
 

( ¯Ɗ))
−


 

,

ˆ ∞

ˆ

0

R


( ¯Ɗ) = 
 

( ¯Ɗ(
, 
−


, 
). The spatial shift 

 = 

|
| can be decomposed into small
in which 
 






 ≤ 
 and large 

 > 
 displacements, which are associated with ¯Ɗ ≤ 1 and ¯Ɗ > 1, respectively.
Therefore,

T 
Ò





 


=

ˆ
ˆ ∞
ˆ ∞

0
+

0

ˆ


≤


(

 − ¯

)(
 
 − ¯

)( 
 


(

 − ¯

)(
 
 − ¯

)( 
 



( ¯Ɗ) − 
 

( ¯Ɗ))
−


 


( ¯Ɗ) − 
 

( ¯Ɗ))
−


 

.









>


(C.5)

Since 
 

( ¯Ɗ) belongs to 
∞, which denotes the space of infinitely differentiable functions, we can
perform the local linear approximation of 
 





( ¯Ɗ), which yields in

where 
 
 



 ¯Ɗ

 





( ¯Ɗ) − 
 

( ¯Ɗ), we obtain


 





( ¯Ɗ
 − ¯Ɗ),

( ¯Ɗ) ≈ 
 

( ¯Ɗ) + 
 
 



 ¯Ɗ

(C.6)
2
3 
− ¯Ɗ/2 and ¯Ɗ
 = ¯Ɗ(
, 
 − 


, 
). Due to the exponential behavior of
= − 

ˆ ∞
ˆ

(

 − ¯

)(
 
 − ¯

)( 
 






( ¯Ɗ) − 
 

( ¯Ɗ))
−


 

 ≈ 0

(C.7)

0



>


and thereby

T 
Ò





 


≈

ˆ ∞

ˆ

0



≤


(

 − ¯

)(
 
 − ¯

)( 
 






( ¯Ɗ) − 
 

( ¯Ɗ))
−


 

.

(C.8)

Moreover, it is permissible to use the Taylor expansion of ¯Ɗ
 for 

 ≤ 
, which is formulated as




 + O(cid:0)|

|2(cid:1),

¯Ɗ
 = ¯Ɗ + 
 ¯Ɗ





where

3



=1


 ¯Ɗ





−2

2

=

(

 − ¯

) 
 ¯







.

152

(C.9)

Therefore,


 




 ¯Ɗ


 = 
 

(cid:0) ¯Ɗ(
, 
 − 


, 
)(cid:1) = 
 

(cid:0) ¯Ɗ(cid:1) + 
 
 


= 
 

(cid:0) ¯Ɗ(cid:1) + 
 
 


(cid:18) 
 
 


(cid:18)


 ¯Ɗ

−Ɗ
(

 − ¯

)(
 
 − ¯

)

ˆ


≤

ˆ
ˆ ∞

(

 − ¯

)(
 
 − ¯

)

ˆ ∞


 ¯Ɗ





≈ −


 ¯Ɗ

0
= − 2


5

0



≤


T 
Ò





 


Plugging (C.10) and (C.9) into (4.34), we attain


 ¯Ɗ





 ¯Ɗ








 + O(cid:0)|

|2(cid:1)
(−



) + O(cid:0)(

|
|)2(cid:1).
(cid:19)

3



−


 


(



)
(

 − ¯

) 
 ¯








=1

(cid:19)

(



)
−


 

.

(C.10)

(cid:17)

(cid:16) 
 ¯


´ ∞
0 
6
−Ɗ

, where Ɗ = 
2

We should note that the limits of integral in (C.11) can be extended to R3 due to (C.7). Following
every steps in the derivation of shear stresses from (4.33) to (4.36) in [163], we can formulate
´ ∞
0 
0

−


 = 

2

T 
Ò



= 

and 
0 = 4

15
SGS Stresses
C.3

in (4.36) from (C.11), in which we obtain 
 =


2 and 
 = |
 − ¯
|.

+ 
 ¯
 






−2



5



 



 


The SGS stresses are given by

ˆ ∞

ˆ

0

R


T 


 
 =

(

 − ¯

)(
 
 − ¯

)( 
 



 ( ¯Ɗ) − 
 
( ¯Ɗ))
−


 



(C.11)


3 
 
( ¯Ɗ) and 
 
( ¯Ɗ) denotes the isotropic Lévy-
 stable distribution.
in (4.35), where 
 
( ¯Ɗ) = 

Therefore,
ˆ ∞

ˆ

(

 − ¯

)(
 
 − ¯

)(
 



 ( ¯Ɗ) − 
 
( ¯Ɗ))
−


 

.

(C.12)


 
 = − 

T 


3

0

R






Asymptotically, 
 
( ¯Ɗ) behaves like a power-law distribution when ¯Ɗ > 1, i.e., 
 
( ¯Ɗ) ∼ ˜

 ¯Ɗ
 =
¯Ɗ−
+
/2 , where 
 = −
 − 


 

(Ɗ)
demonstrates a heavy-tailed behavior at ¯Ɗ > 1, it keeps the exponential trait for ¯Ɗ < 1 though.
Regarding the exponential behavior of 
 

( ¯Ɗ), 
 

(Ɗ) − 
 

( ¯Ɗ) can be fitted by a heavy-tailed
distribution like 
 
( ¯Ɗ), in which 
 
( ¯Ɗ) reduces exponentially in a close proximity of ¯Ɗ = 0.

2 and 

 = 22
Ɖ(
+
/2)


/2Ɖ(−
) .

It is worth mentioning that

153

Therefore, we can simplify (C.12) to


 
 ≈ − 



T 


3

ˆ ∞

0

ˆ
R
−



(

 − ¯

)(
 
 − ¯

)( ¯Ɗ

−
+
/2



− ¯Ɗ−
+
/2)
−


 

,

(C.13)

where 
 = 3 and 

 = {
 ∈ R
 
.
. | ¯Ɗ| < 
}, which is associated with ¯Ɗ (cid:28) 1. Due to the fact that

 
( ¯Ɗ) is continuously differentiable for |
| ∈ R
 − 

, we perform the Taylor expansion of 
 


 ( ¯Ɗ)
as follows:


 ( ¯Ɗ) − 
 
( ¯Ɗ) ≈ 

 
( ¯Ɗ)

 


 ¯Ɗ

( ¯Ɗ
 − ¯Ɗ) = (
 + 3
2

) 




3

( ¯Ɗ
 − ¯Ɗ)
¯Ɗ
+5/2

.

Under Assumption 5, in which 
 ∼ O(1), we obtain



 = 
|
|
 > O(
) → |
| > O( 




) = O(




/


) > O(
)

at large 

 > 
, which yields in ¯Ɗ = |
− ¯
|2
[163], we also conclude that


2 (cid:29) 1 and 

 − ¯

 ≈ 

. In virtue of (4.50-51) in


2 ≈ |
|2
3



=1

¯Ɗ
 − ¯Ɗ ≈ −2



( ¯

(
) − ¯

(
(cid:48)))


2

.

(C.14)

Utilizing the definition of 
 = 
−
(cid:48)
ˆ ∞


 



 
 ≈ (
 + 3
T 

2

) 




3

0

in Section 4.3.1 and (C.14), we reformulate (C.13) as
ˆ
R
−


= (2
 + 3)(



2
−1
2
)

) (
 
 − 
(cid:48)

) ( 
 
 − 
(cid:48)
( 

 − 
(cid:48)
ˆ ∞




−



 ×

1−2


) (
 − 
(cid:48)) · ( ¯
(
) − ¯
(
(cid:48)))

( ¯Ɗ
 − ¯Ɗ)
( |
−
(cid:48)|



 )2
+5



(cid:48)
(

)3

(

 − 
(cid:48)


−


,



(cid:48),




)





0




ˆ
R
−



|
 − 
(cid:48)|2
+5

(C.15)

which corresponds to (4.58) in [163]. Therefore, we can proceed the same derivations as discussed
in (4.58) to (4.64) in [163] to obtain

(∇ · T 
)
 =


(

)2


Ɖ(2
 + 1) 






(

)2





= 
.
.

Ɖ(2
 + 1) 



¯

(
(cid:48)) − ¯

(
)
|
(cid:48) − 
|2
+

¯

(
(cid:48)) − ¯

(
)
|
(cid:48) − 
|2
+




(cid:48)



(cid:48)

(C.16)

R


ˆ
ˆ
R
−



in which "p.v." denotes the principal value of the integral.

154

APPENDIX D

DERIVATIONS IN CHAPTER 5

D.1 Fourier Transform of Tempered Fractional Laplacian

As noted in [247, 248], the tempered fractional Laplacian operator can be represented in various

equivalent forms, i.e.,

ˆ
(Ɗ + 
)

(
) = −

,
 P.V.
ˆ

where 
 = |
|, 
 ∈ (0, 1
get

2) ∪ ( 1

F(cid:2)(Ɗ + 
)

(
)(cid:3)(
) =


(
) − 
(
)

R








|
−
||
 − 
|2
+


(
 + 
) + 
(
 − 
) − 2
(
)



 
2
+




,

2

P.V.

(D.1)
=
2 , 1), and 
 > 0. By performing the Fourier transform of (D.1), we



,

R


ˆ
ˆ

R




,

2

= −

,




·
 + 
−
·
 − 2



 
2
+

1 − cos(
 · 
)



 

+2


R








 F(cid:2)
(
)(cid:3)(
),

(D.2)

in which 
 denotes the Fourier numbers. For the sake of simplicity, we define

ˆ


(
) = −

1 − cos(
 · 
)



 

+2




,

R


ˆ


(
) =

which appears to be rotationally invariant. Moreover, we introduce 
 = |
| and 

 = 
 cos(
).
Without loss of generality, 
 can be chosen such that 
 cos(
) is aligned with the first primary
direction. Therefore, 
(
) can be re-expressed by

cos(
 · 
) − 1



 

+2


cos(

) − 1



/
(
/
)
+2







 = 
2


cos(

) − 1



/
(
)
+2


R




 =

(D.3)
where 
 = 

, 
 = |
|, and 

 = 


. Due to the invariant properties of 
(
), we proceed the deriva-
tions with transforming (D.3) into the corresponding spherical coordinate, (r, 
1, · · · , 

−1).



,

R


R


In terms of the transformation, we let 
 = |
| = r and 

 = 
 cos(
) = r cos(
1). Then, in a

ˆ

ˆ

general case for 
 > 1, 

 follows



 = J(r, 
1, · · · , 

−1) 
r 

1 · · · 


−1,

155

where

J(r, 
1, · · · , 

−1) =

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




(cid:12)(cid:12)(cid:12)(cid:12) = r
−1 sin
−2(
1) sin
−3(
2) · · · sin(

−2)






(r
 
)

for 
 = 1, · · · , 
 and 
 = 1, · · · , 
 − 1 [see 249]. Therefore, we find the general form of 
(
) as

where ¯
 =

. It is shown by [247] that

0

0

ˆ ∞



r/
(r)
+2


cos (r cos(
1)) − 1

r
−1 sin
−2(
1) 

1 
r,


(
) = 
2
 ¯

´ 

0 sin
−3(
2) 

2 · · ·

−
r/

r
+1

ˆ 2

´ 

0 sin(

−1) 


−1 = 2
(
−1)/2
Ɖ( 
−1
(cid:2) cos(cid:0)r cos(
1)(cid:1) − 1(cid:3) sin
−2(
1) 

1 
r
2 )
ˆ 2

(cid:34)
0
¯
 Ɖ(−2
)
1/2Ɖ( 
−1
2 )


 + 2
 − 1

ˆ ∞

0


2 − (
2 + 
2)


=

2
1(−
,

;



2

;

2

2)
Ɖ( 



(
) = ¯
 
2


where 2
1 denotes a Gaussian hypergeometic function. Therefore,

(cid:34)

F(cid:2)(Ɗ + 
)

(
)(cid:3) (
) = Ô
,
 ×
where Ô
,
 = 

,
 ¯
 Ɖ(−2
) 
1/2Ɖ( 
−1
2 )
Ɖ(
/2)

2 ;1) .
; 

D.2 Tempered Fractional Modeling of SGS Stresses

2
1(−
, 
+2
−1

=

2

1


2 − (
2 + 
2)


2
1(−
,


 + 2
 − 1

2

;



2

;


2

2 + 
2)

,

(D.4)

(cid:35)

,

(D.5)


2

2 + 
2)
(cid:35)

As we discussed in subsection 5.3.3, the SGS stresses are described by

T


 
 =


c
,


3

ˆ ∞

0

ˆ
R3

(

 − ¯

)(
 
 − ¯

)(
 
,
( ¯Ɗ
) − 
 
,
( ¯Ɗ))
−


 



(D.6)

where 
 
,
( ¯Ɗ) represents a tempered Lévy 
-stable distribution. Let consider 
 = −
 − 3

 ∈ (0, 1

¯Ɗ > 1, we decompose the domain of kinetic momentum such that R3 = I
 ∪(cid:16)

2 for
2 , 1). Regarding the equivalent Pareto-like behavior of Lévy distributions [250] at
, where

2) ∪ ( 1

R3 \ I


(cid:17)

I
 = {
 ∈ R
 
.
. | ¯Ɗ| < 
} and 
 (cid:28) 1. This allows for the following approximation as



,
( ¯Ɗ) (cid:39) 



0,


−
 ¯Ɗ


 ∈ I


1
2 ¯Ɗ

−(
+3

2), 
 ∈ R3 \ I


156

(D.7)

−Ɖ( 3
2)
3
2 Ɖ(−2
)

1

2


2
1(−
,1+
;3

2;1) . It is worth mentioning that 

,
( ¯Ɗ) reduces exponentially
where 

 =
in a close proximity of ¯Ɗ = 0. With all this in mind, the approximated function of 

,
( ¯Ɗ) in
(D.7) can properly capture the heavy-tailed behavior of the filtered collision term. Evidently, by
replacing 
−
 ¯Ɗ

1
2 for ¯Ɗ > 1, we arrive at the following expression

1
2
 with 
−
 ¯Ɗ



,
( ¯Ɗ
) − 

,
( ¯Ɗ) = 



and accordingly,

T


 
 =


c
,


3





ˆ ∞

0

ˆ
R3\IƊ

(cid:16) 
−
 ¯Ɗ

1
2

(
+3
2)



¯Ɗ

1
− 
−
 ¯Ɗ
2
(
+3
2)
¯Ɗ

2(cid:16)

1

(cid:17) (cid:39) 


−
 ¯Ɗ
2(cid:16)

1

(

 − ¯

)(
 
 − ¯

)
−
 ¯Ɗ

1
(
+3
2)



¯Ɗ

−

1
(
+3
2)

¯Ɗ

(cid:17)

,

1
(
+3
2)



¯Ɗ

−

1
(
+3
2)

¯Ɗ

(cid:17)


−


 

,(D.8)

where c
,
 is a real-valued constant. As a continuous differentiable function for ¯Ɗ > 1, we proceed
with the Taylor expansion of ¯Ɗ

according to

−(
+3
2)



−(
+3
2)
¯Ɗ



− ¯Ɗ

−(
+3

2) ≈ 
 ¯Ɗ

2)
−(
+3

 ¯Ɗ

( ¯Ɗ
 − ¯Ɗ) = −(
 + 3
2

) 



( ¯Ɗ
 − ¯Ɗ)
¯Ɗ
+5/2

.

In terms of the assumptions in remark 5.3.1, we use the same argument, presented by [232,
Appendix], on approximating ¯Ɗ
 − ¯Ɗ for ¯Ɗ (cid:29) 1, which allows for 

 − ¯

 ≈ 

 and thus

¯Ɗ
 − ¯Ɗ ≈ 2



( ¯

(
(cid:48)) − ¯

(
))


2

.

(D.9)

3



=1

Reminding the definition of 
 = 
−
(cid:48)


 


from section 5.3.2, we plug (D.9) into (D.8) and obtain


 
 = (2
 + 3)(
 c
,
 


2
−1
2
) ×
T


ˆ ∞

0


−


1−2


ˆ
R3\IƊ

(

 − 
(cid:48)


)(
 
 − 
(cid:48)


) (
 − 
(cid:48)) · ( ¯
(
) − ¯
(
(cid:48)))
|
 − 
(cid:48)|2
+5

 |
−
(cid:48)|






(D.10)


(cid:48)

.

approach is to dissociate the temporal element by employing the bionomial series of 

 |
−
(cid:48)|

In order to evaluate the outer integral in D.10 and find the corresponding coefficient, our



 as

157

follows:


− 
|
−
(cid:48)|






(cid:18) 1

(cid:19)

∞



 =

= 1 + 1

= (1 − 1 + 
− 
|
−
(cid:48)|



 ) 1





=0

 − 1)

 ( 1
(
 ¯
|
−
(cid:48)| − 1) + 1
∞

2!


,∞(
)
− ¯

|
−
(cid:48)|
K



,K(
)
− ¯

|
−
(cid:48)|,


=0

(cid:39)

=




(
 ¯
|
−
(cid:48)| − 1)

(
 ¯
|
−
(cid:48)| − 1) + · · ·

(D.11)


=0



 and ¯

 = ¯
. Under the assumption of 
 > 0.01, we can approximate the bionomial

 for K = 1.

where ¯
 = 

series with the first two leading terms, which yields 
0,1 = 1 − 1
Accordingly, by defining ¯

 = (2
 + 3)(



2
−1
2
) and


 and 
1,1 = 1

¯
K

 (
) =

ˆ ∞

0


−


1−2




,K(
)

,

(D.12)

we obtain the closed form of T


 
 as
ˆ
R
−



T


 
 = c
,
 ¯



¯
K

 (
)

K



=0

of I
 
 term by term, which yields

ˆ
R
−




I
 






=

3


(cid:110)


=1

(

 − 
(cid:48)


) (
 
 − 
(cid:48)

)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

(cid:48))
(
 − 
(cid:48)) · ( ¯
 − ¯

|
 − 
(cid:48)|2
+5
 ¯

|
−
(cid:48)|
I
 




(cid:48).

(D.13)

To ensue the proper form of the SGS stresses in the filtered NS equations, we take the derivative

−(
 
 − 
(cid:48)

)

(cid:48))
(
 − 
(cid:48)) · ( ¯
 − ¯

|
 − 
(cid:48)|2
+5
 ¯

|
−
(cid:48)|
(cid:48))
(
 − 
(cid:48)) · ( ¯
 − ¯

|
 − 
(cid:48)|2
+5
 ¯

|
−
(cid:48)|

−(

 − 
(cid:48)
(
 
 − 
(cid:48)


)

 


)(
 
 − 
(cid:48)

−

−(

 − 
(cid:48)

)(
 
 − 
(cid:48)
(2
 + 5)(

 − 
(cid:48)


)( ¯

 − ¯
(cid:48)

 )
|
 − 
(cid:48)|2
+5
 ¯

|
−
(cid:48)|

− ¯

|
−
(cid:48)|

) 
 ¯



)(

 − 
(cid:48)
|
 − 
(cid:48)|2
+5




(cid:48))
(
 − 
(cid:48)) · ( ¯
 − ¯


)2(
 
 − 
(cid:48)

)
|
 − 
(cid:48)|2
+5
 ¯

|
−
(cid:48)|
(cid:48))
(
 − 
(cid:48)) · ( ¯
 − ¯



(cid:48),
|
 − 
(cid:48)|2
+5
 ¯

|
−
(cid:48)|


)(
 
 − 
(cid:48)

)

+ ¯

(

 − 
(cid:48)

(cid:111)

158

which is clearly simplified to


I
 






= ( ¯

 + 2
 + 5 − 3 − 1 − 1)

ˆ
R
−



(
 
 − 
(cid:48)

)

(cid:48))

(
 − 
(cid:48)) · ( ¯
 − ¯

|
 − 
(cid:48)|2
+5
 ¯

|
−
(cid:48)| 

(cid:48).
´

(D.14)

Following the derivations in [232, 163], (D.14) can be formulated in the form of a tempered
´

fractional Laplacian by performing the technique of integration-by-parts for (D.14) as


 −




. We consider




 =


 = (
 
 − 
(cid:48)


)( ¯

 − ¯
(cid:48)


)
− ¯

|
−
(cid:48)|,

which directly leads to 

|
∈R3 (cid:39) 0. Therefore, we get

´



 =

(

 − 
(cid:48)

)
|
 − 
(cid:48)|2
+5
´




 = −




, in which



(cid:48),

(D.15)

(2
 + 3)|
 − 
(cid:48)|2
+3


 =


 = 
 
 
( ¯

 − ¯
(cid:48)

−1

)
− ¯

|
−
(cid:48)| + (
 
 − 
(cid:48)

,


)( 
 ¯







)
− ¯

|
−
(cid:48)| − ¯

(
 
 − 
(cid:48)


)( ¯

 − ¯
(cid:48)


)
− ¯

|
−
(cid:48)|

(cid:48).

We can make even more simplifications by eliminating the second term of 

 due to the incom-
pressibility assumption, i.e., 
 ¯





 the last term vanishes since




it represents an odd function of 
(cid:48). Therefore, the ultimate form of the TFSGS model is found to be

= 0. Moreover, by evaluating

´

K


ˆ
R
−



K


(2
 + ¯

)
(2
 + 3)

¯
K

 (
)


=0

K

 (
, ¯

)(Ɗ + ¯

)
 ¯

 ,

( ¯

 − ¯
(cid:48)

)

|
 − 
(cid:48)|2
+3
 ¯

|
−
(cid:48)| 

(cid:48),

(D.16)

(∇ · TR) 
 = c
,
 ¯



= 



where 

 = c
,
 ¯

.


=0

159

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