AN ANALYSIS OF STABILITY OF THE FLUX RECONSTRUCTION FORMULATION WITH APPLICATIONS TO...

AN ANALYSIS OF STABILITY OF THE FLUX

RECONSTRUCTION FORMULATION WITH APPLICATIONS TO

SHOCK CAPTURING

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND

ASTRONAUTICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Abhishek Sheshadri

August 2016

c© Copyright by Abhishek Sheshadri 2016

All Rights Reserved

ii

Abhishek Sheshadri

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

(Antony Jameson) Principal Adviser




(Juan J. Alonso)




(Sanjiva K. Lele)

Approved for the Stanford University Committee on Graduate Studies

iii

Abstract

High-order methods in Computational Fluid Dynamics (CFD) have been growing in

popularity due to their promise of increased computational efficiency and fidelity to

flow physics. Amongst a plethora of methods proposed over the last few decades, Dis-

continuous Galerkin (DG) type (Finite Element Methods (FEM)) have drawn great

attention due to their attractive accuracy and stability properties, facility for per-

forming arbitrarily high order computations and the capability to handle complex

unstructured geometries, among other features. The Flux Reconstruction (FR) ap-

proach to high-order methods is a flexible, robust and simple to implement framework

that has proven to be a promising alternative to the traditional DG schemes on par-

allel architectures like Graphical Processing Units (GPUs) since it pairs exceptionally

well with explicit time-stepping methods.

While high order methods have already successfully displayed significant improve-

ments over low order methods on certain fronts, one of the major reasons limiting

their industry-wide adoption is their inferior robustness relative to low order methods.

These high order schemes are prone to developing instabilities while solving nonlinear

problems and the issue compounds with increasing order, thereby requiring a com-

promise between accuracy and stability. Instabilities due to discontinuous solutions

or shocks that develop in compressible flows and aliasing instabilties are two of the

most important ones.

This dissertation is divided into two major parts. In the first part, the stability of

the FR framework for solving linear advection and advection-diffusion equations on

iv

tensor product elements has been investigated and the approach has been proven to

be stable for both problems. In the second part, a robust and simple to implement

shock capturing method which can be adopted in any unstructured high order Finite

Element (FE)-type method has been developed. The proposed method does not

sabotage the accuracy of the solution in smooth regions and shows great promise in

our numerical simulations.

v

Acknowledgments

First and foremost, I would like to thank my advisor Prof. Antony Jameson for all

the support and inspiration he has provided me with during my doctoral studies.

His outstanding ability and dedication for solving practical problems motivated me

greatly to persevere through difficulties. The freedom he provided in choosing my own

problems helped me think independently and mature as a researcher. I thoroughly

enjoyed all his anecdotes and learnt a great deal from the discussions and chats in

the lab. I am proud to have worked with one of the pioneers of CFD.

I would like to express my sincere gratitude to my readers Prof. Sanjiva Lele and

Prof. Alonso for taking the time to review my thesis and for providing insightful

comments and advice. I would like to thank Prof. Matthias Ihme and Prof. George

Papanicolaou for serving on my oral exam committee. Prof. Matthias also reviewed

my thesis and provided great feedback and I really enjoyed the discussions I had with

him.

I am very thankful to Prof. David Williams, who was a great mentor through my

initial years. He has also helped me immensely by thoroughly reviewing my papers

and providing guidance and direction to my research. He set a great example with

his incredible work ethics and the depth to which he analyzed each problem. Next I

would like to thank my colleague Josh Romero. I developed a lot of my numerics on

in his code ZEFR and used it extensively for my numerical simulations. He has an

incredible passion and expertise in High Performance Computing and the performance

of his codes have always impressed me. I would also like to thank Dr. Manuel Lopez

vi

who was a great colleague and friend. I have worked with him on several projects and

coding competitions. I have learnt a lot from him and have always enjoyed working

with him.

I am extremely grateful to Robert and Audrey Hancock for supporting a major part of

studies at Stanford through the prestigious Stanford Graduate Fellowship. It provided

complete freedom in choosing my department, advisor and area of research and I am

extremely proud to be a SGF fellow. I also want to thank the ICME department in

this regard for supporting me through Teaching Assistantships for eight quarters. I

am very grateful to the Student Services Manager of ICME, Indira Choudhury, who

helped me procure a TA each quarter. I would like to thank Dr. Hung Le and Kapil

K Jain who I have TA’ed with for several quarters for providing me the opportunities

and I enjoyed working with them.

I would like to express my gratitude to Prof. Kay Giesecke for providing me the

opportunity to research interesting problems in Finance with him. He is an excellent

mentor and is very open to different ideas. I thoroughly enjoyed working with him

and wish I can work with him again.

I want to thank my colleagues Dr. Kartikey, Jacob, Jerry, David Manosalvas, Jonathan

Chiew and Zach for motivating me through peer pressure and inspiring me through

great research. I greatly enjoyed spending time with them and they played a major

role in molding me as a researcher. They have helped me focus when I lost direction

and I have learnt immensely from conversations with them. I would like to thank

our post-doctoral researchers Dr. Freddie and Dr. Jonathan Bull for bringing in in-

teresting ideas and insights into the lab. I have been working with Freddie on a few

projects over the last few months. The depth of his knowledge in High Performance

Computing is exceptional. I have learnt a great deal from him and he continues to

inspire me with his achievements.

I had 5 wonderful years with my roommates Sreenath and Muneeb who have been

amazing friends and mentors. All our cooking experiments, GoT nights and gaming

frenzies are unforgettable and I’ll miss them undoubtedly.

vii

I would like to sincerely thank all my friends who have defined my Stanford experience

and showered me with rich memories. Your company helped me get through the

toughest of times with a smile. All the crazy unplanned trips and poker all-nighters

are memorable and I wish I could spend more time with all of you.

I would like to thank my sister Amulya who has always been my best friend. She

has provided me with immense love and support and has been the one person I can

always open up to. Finally, I would like to express my deepest gratitude to my parents

Sheshadri and Hemalatha who have been the greatest support and motivation for me

throughout my life. They raised me with a love for science and made great sacrifices

for my education. I can never thank them both enough.

viii

Contents

Abstract iv

Acknowledgments vi

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Unstructured High Order Methods . . . . . . . . . . . . . . . . . . . 2

1.3 The Flux Reconstruction Formulation . . . . . . . . . . . . . . . . . . 3

1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Flux Reconstruction Formulation in One Dimension 8

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 FR Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Correction Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

ix

I Linear Stability Theory of the FR Formulation on Ten-

sor Product Elements 21

Preamble 22

3 Stability of the FR approach for Linear Advection Equation on Ten-

sor Product Elements 24

3.1 FR Approach on Quadrilaterals for a First Order PDE . . . . . . . . 24

3.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.2 FR Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.3 VCJH Correction Functions . . . . . . . . . . . . . . . . . . . 31

3.2 Key Difficulties in Extending the 1D Stability Analysis to Tensor Prod-

uct Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Necessity for a new norm . . . . . . . . . . . . . . . . . . . . . 32

3.2.2 Varying Jacobian of the Geometric Transformation . . . . . . 34

3.3 Proof of Stability of the FR Approach on Quadrilaterals . . . . . . . 35

3.4 Insights Gained from Stability Analysis . . . . . . . . . . . . . . . . . 54

3.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5.1 Upwind Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5.2 Central Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Stability of the FR Approach for Linear Advection-Diffusion Equa-

tion on Tensor Product Elements 60

x

4.1 FR Approach on Quadrilaterals for a Second Order PDE . . . . . . . 60

4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.2 FR Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Proof of Stability of the FR Approach on Quadrilaterals . . . . . . . 69

4.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

II Numerical Simulation of Compressible Flows 91

5 Shock Capturing 95

5.1 Comparison of shock capturing methods . . . . . . . . . . . . . . . . 95

5.1.1 Limiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1.2 Artificial Viscosity Methods . . . . . . . . . . . . . . . . . . . 96

5.1.3 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.1.4 Reconstruction Approaches . . . . . . . . . . . . . . . . . . . 101

5.2 Shock Capturing Strategy . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3 Modal Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3.2 Extension to multiple dimensions . . . . . . . . . . . . . . . . 105

5.3.3 Parameter Selection . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4 Positivity Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . 108

xi

5.4.1 Adaptation to the FR approach . . . . . . . . . . . . . . . . . 111

5.5 Time-Stepping scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6 Shock Detection 114

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2 Comparison of Current Shock Sensors . . . . . . . . . . . . . . . . . . 115

6.3 Concentration Method . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3.1 Concentration Kernels . . . . . . . . . . . . . . . . . . . . . . 118

6.3.2 Concentration kernel for spectral projection . . . . . . . . . . 120

6.3.3 Concentration Property for Jacobi Polynomials . . . . . . . . 120

6.3.4 Non-Linear Enhancement . . . . . . . . . . . . . . . . . . . . 125

6.4 Design and Implementation of the Shock Detector . . . . . . . . . . . 130

6.4.1 Parameter Selection . . . . . . . . . . . . . . . . . . . . . . . . 130

6.4.2 Shock Sensor in 1D . . . . . . . . . . . . . . . . . . . . . . . . 131

6.4.3 Extension to 2D and 3D Tensor Product Elements . . . . . . . 133

6.4.4 Extension to Triangles and Tetrahedra . . . . . . . . . . . . . 139

7 Numerical Experiments 142

7.1 Sod Shock Tube Problem . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.1.1 Testcase 1: Weak Filtering . . . . . . . . . . . . . . . . . . . . 146

7.1.2 Testcase 2: Strong Filtering . . . . . . . . . . . . . . . . . . . 147

7.2 Shock-Entropy Interaction . . . . . . . . . . . . . . . . . . . . . . . . 147

xii

7.3 Blast Wave Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.4 Inviscid Transonic Flow: Structured Quadrilateral Mesh . . . . . . . 156

7.4.1 Convergence to Steady State . . . . . . . . . . . . . . . . . . . 162

7.4.2 Convergence Acceleration . . . . . . . . . . . . . . . . . . . . 165

7.5 Inviscid Transonic Flow: Unstructured Triangle Mesh . . . . . . . . . 166

7.6 Viscous Supersonic Flow: Hybrid Mesh . . . . . . . . . . . . . . . . . 173

7.7 Mach 3 Wind Tunnel With a Step . . . . . . . . . . . . . . . . . . . . 180

7.8 Shock Wave-Laminar Boundary Layer Interaction . . . . . . . . . . . 188

7.9 Inviscid Transonic Flow over a Sphere . . . . . . . . . . . . . . . . . . 193

8 Conclusions 197

Appendices 201

A Partial Sobolev Norm 202

B Proof of Θadv ≤ 0 207

Bibliography 211

xiii

List of Tables

2.1 Table provides the values of the VCJH parameter c for recovering dif-

ferent existing schemes. c− is the smallest value of c which can provide

a stable scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.1 Norms of the difference between the numerical and analytical solutions

at t = 0.4 for the two testcases. . . . . . . . . . . . . . . . . . . . . . 145

xiv

List of Figures

2.1 Figure shows an illustration of a discretization of the domain into ele-

ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Figure shows the solution and flux point locations for a one-dimensional

element for a scheme with p = 4, i.e., a 5th order scheme . . . . . . . 11

2.3 Figure shows a generic discontinuous solution in the reference domain

(uD) represented by a 4th degree polynomial . . . . . . . . . . . . . . 12

2.4 Figure shows the discontinuous flux polynomial in the reference domain

(fD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Figure shows the physical solution polynomial in two neighboring ele-

ments in a generic scenario. The physical solution is in general discon-

tinuous giving rise to a Riemann problem. . . . . . . . . . . . . . . . 14

2.6 Figure shows a schematic of the flux correction process in the reference

element ΩS. The dashed black line represents the discontinuous flux

in the reference domain fD. The thick magenta line represents the

corrected total flux polynomial that would result in a continuous flux

in the physical domain . . . . . . . . . . . . . . . . . . . . . . . . . . 15

xv

2.7 Figure shows one possible choice for the left and right correction func-

tions employed for a p = 4 FR scheme. These correction functions are

polynomials of degree p + 1, i.e., degree 5. The correction functions

shown here correspond to a VCJH parameter c = 0 which recovers the

nodal DG method for linear problems . . . . . . . . . . . . . . . . . . 17

2.8 Figure shows the left correction function hL(ξ) for various values of c 19

3.1 Mapping between the physical domain (on the left) and the reference

element (on the right) . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Figure showing the solution and flux points in the reference element

for a p = 2 scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Plot of the evolution of the L2 norm of the solution from t = 0 to

t = 20 T for the FR approach using upwind interface fluxes for different

values of the VCJH parameter c . . . . . . . . . . . . . . . . . . . . . 56

3.4 Plots of the evolution of the L2 norm of the solution for the FR ap-

proach using central interface fluxes for different values of the VCJH

parameter c. The bottom two plots are zoomed in versions of the plot

at the top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1 Mapping between the physical domain (on the left) and the reference

element (on the right) . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Figure showing the solution and flux points in the reference element

for a p = 3 scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


proach using upwind interface fluxes for different values of the VCJH

parameter c. The plot on the right shows a zoomed version of the one

on the left . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xvi


proach using central interface fluxes for different values of the VCJH

parameter c. The plot on the right shows a zoomed version of the one

on the left . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5 Figure shows the problem arising from representing a step disconti-

nuity using a Legendre polynomial basis. (a) shows the polynomial

representation for various orders. (b) provides a plot of the pointwise

error for these approximate representations. It can be noticed that the

convergence rate is reduced to first order . . . . . . . . . . . . . . . . 93

5.1 Figure shows the spectral action of the exponential filter as the param-

eters are varied. (a) shows this action for varying filter orders while

(b) shows the same for varying filter strengths . . . . . . . . . . . . . 106

6.1 Figure shows the portraits of the enhanced kernel (before clipping it

below a threshold) for the step discontinuity. (a) and (b) show the

step and its kernel portrait when the step is located in the center of

the element. (c) and (d) show the same when the step is located near

the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2 Figure shows the portraits of the enhanced kernel (before clipping it

below a threshold). (a) and (b) show the function and its kernel por-

trait for a ramp (derivative discontinuity, or C0 function) while (c) and

(d) show the same for a smooth function. . . . . . . . . . . . . . . . . 126

6.3 Figure compares the portraits of the enhanced kernel for the step and

ramp functions for various polynomial orders. The sensor seems to be

able to consistently generate adequate separation of scales to distin-

guish between a jump and a ramp across all polynomial orders . . . . 127

xvii

6.4 Figure compares the portraits of the enhanced kernel for the step and

ramp functions for different values of the non-linear enhancement ex-

ponent q. It is evident that higher the nonlinear exponent, the better

the separation of scales . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.5 Figure compares the portraits of the enhanced kernel using two dif-

ferent concentration factors - polynomial and exponential, for the step

function case. (a) shows this comparison for a non-linear enhancement

exponent q = 2 while (b) shows the same for q = 2 . . . . . . . . . . . 129

6.6 Figure shows the performance of the sensor on a generalized step in

2D on a quadrilateral element. (a) shows a general two-dimensional

step. (b) shows the enhanced kernels for the x-slices and (c) shows the

enhanced kernel portraits for y-slices . . . . . . . . . . . . . . . . . . 134

6.7 Figure shows the performance of the sensor on a ramp in 2D on a

quadrilateral element (a) shows a two-dimensional ramp (b) shows the

enhanced kernels for the x-slices and (c) shows the enhanced kernel

portraits for y-slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.8 Figure shows the performance of the sensor on a cylindrical step in

2D on a quadrilateral element (a) shows a cylindrical step function in

2D (b) shows the enhanced kernels for the x-slices and (c) shows the

enhanced kernel portraits for y-slices . . . . . . . . . . . . . . . . . . 136

6.9 Figure shows the performance of the sensor on a generalized step in 2D

on a triangle element (a) shows a generic step function inside a trian-

gular element. (b) shows the tensor product Gauss-Legendre points of

the quadrilateral when collapsed onto the triangle by collapsing over

the top left point (c) and (d) show the enhanced kernels for the x and

y slices respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

xviii

6.10 Figure shows the performance of the sensor on a generalized step in 2D

on a triangle element. (a) shows a generic step function inside a trian-

gular element. (b) shows the tensor product Gauss-Legendre points of

the quadrilateral when collapsed onto the triangle by collapsing over

the bottom right point (c) and (d) show the enhanced kernels for the

x and y slices respectively . . . . . . . . . . . . . . . . . . . . . . . . 141

7.1 Figure shows the results from the simulation of the flow through a

shock tube results at t = 0.4 for testcase 1 with a weak filter α = 0.4 146

7.2 Figure shows the results from the simulation of the flow through a

shock tube results at t = 0.4 for testcase 2 with a strong filter α = 4 . 147

7.3 Plots of the density at t = 1.8 for different combinations of polynomial

degree P and number of elements N . . . . . . . . . . . . . . . . . . 148

7.4 (a) shows a comparison of the plots of the density at t = 1.8 from

a fourth order and an eighth order scheme with the same Number of

Degrees of Freedom (NDoF). (b) shows a magnified version of (a) . . 151

7.5 (a) shows a comparison of the plots of the density at t = 1.8 from

a fourth order and an eighth order scheme with the same NDoF. (b)

shows a magnified version of (a) . . . . . . . . . . . . . . . . . . . . . 152

7.6 Figure shows a comparison of our blastwave simulation results with

those from Woodward and Colella at t = 0.026. (a) shows the velocity

and (b) shows the density. (c) shows the same quantities from the

reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.7 Figure shows a comparison of our blastwave simulation results with

those from Woodward and Colella at t = 0.038. (a) shows the velocity

and (b) shows the density. (c) shows the same quantities from the

reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

xix

7.8 Figure shows the 64x64 structured quadrilateral mesh employed for

this testcase. (a) shows the farfield mesh while (b) provides a closer

view of the mesh near the airfoil . . . . . . . . . . . . . . . . . . . . . 158

7.9 Figure shows the results for the simulation of an inviscid flow over a

NACA 0012 airfoil at Ma = 0.8 and AoA = 1.25. (a) shows a flood

plot of the density with the mesh overlaid on top and (b) shows the

sensor distribution. The critical sensor threshold beyond which we

filter is 16.52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.10 (a) shows the distribution of the Coefficient of Pressure over the airfoil

wall for the converged solution resulting from a simulation of an inviscid

flow over a NACA 0012 airfoil at Ma = 0.8 and AoA = 1.25. (b) shows

the result obtained using FLO82 run with 4096× 4096 points . . . . 161

7.11 Figure shows the convergence history for the simulation of an inviscid

flow over a NACA 0012 airfoil at Ma = 0.8 and AoA = 1.25. (a)

shows the residual change with time. (b) shows the L∞ norm of the

change in the density from one time-step to the next, normalized by

the time-step value. (c) shows the convergence history of the coefficient

of lift CL and (d) shows the convergence history of the coefficient of

drag CD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.12 (a) shows the CL and CD for this case obtained using OVERFLOW.

It gives a table of CL and CD for OVERFLOW at various number of

degrees of freedom. (b) shows the values obtained by us for the case

described in this section. We get very close to their continuum values

with NDOF = 102,400 . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.13 Figure compares the convergence histories of the CL and CD with and

without employing p-multigrid. The p-multigrid simulation converges

about eight times faster than the simulation without it in terms of the

number of iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

xx

7.14 Figure compares the convergence histories of the shows the L∞ norm

of the change in the density from one time-step to the next, normalized

by the time-step value, for two cases: One with p-multigrid only and

the other with p-multigrid and a JST-style fourth order filter . . . . . 166

7.15 Figure shows the unstructured triangle mesh composed of 11,464 ele-

ments employed for this testcase. (a) shows the farfield mesh while (b)

provides a closer view of the mesh near the airfoil . . . . . . . . . . . 167

7.16 Figure shows the results for the simulation of an inviscid flow over a

NACA 0012 airfoil at Ma = 0.8 and AoA = 0. (a) shows a flood plot

of the density and (b) shows the sensor distribution. The sensor cutoff

for this case is 2.53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.17 (a) shows CL and CD for OVERFLOW at various number of degrees

of freedom. (b) shows the values obtained by us for the case described

in this section. We get very good results with a much smaller number

of degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.18 Figure shows the distribution of the Coefficient of Pressure over the

airfoil wall for the converged solution resulting from a simulation of an

inviscid flow over a NACA 0012 airfoil at Ma = 0.8 and AoA = 0 . . 172

7.19 Figure shows the hybrid structured and unstructured mesh composed

of 65,538 9-node second order quadrilateral elements and 7,252 6-node

second triangle elements employed for this testcase. (a) shows the

farfield unstructured triangle mesh while (b) provides a closer view of

the structured quadrilateral mesh near the airfoil . . . . . . . . . . . 174

7.20 Figure shows the flood plot of the density for the simulation of a su-

personic viscous flow over a NACA 0012 airfoil at Ma = 1.2, AoA = 2

and Re = 60,000 at t = 3.85s . . . . . . . . . . . . . . . . . . . . . . 175

xxi

7.21 Figure shows a plot of the sensor distribution obtained from the simula-

tion of a supersonic viscous flow over a NACA 0012 airfoil at Ma = 1.2,

AoA = 2 and Re = 60,000 at t = 3.85s . . . . . . . . . . . . . . . . . 176

7.22 Figure shows snapshots of evolution of the vortex trail shed from the

airfoil for the simulation of a supersonic viscous flow over a NACA

0012 airfoil at Ma = 1.2, AoA = 2 and Re = 60,000 . . . . . . . . . 177

7.23 Figure shows snapshots of evolution of the vortex trail shed from the

airfoil as well as the λ-shock for the simulation of a supersonic viscous

flow over a NACA 0012 airfoil at Ma = 1.2, AoA = 2 and Re = 60,000179

7.24 Figure shows the computational domain which is a rectangular wind

tunnel with a step. The flow is from the left at Ma = 3 . . . . . . . . 180

7.25 Figure shows a flood plot of the density along with 30 contour lines

plotted according to (7.14). (a) shows the density plot at t = 0.5s

while (b) shows the same for t = 1s . . . . . . . . . . . . . . . . . . . 181







7.28 Figure shows the results obtained from the simulation of the flow in

a Mach 3 wind tunnel with a step at t = 4s. (a) shows a flood plot

of the density along with 30 contour lines plotted according to (7.14).

(b) shows the sensor distribution. The cutoff value is 103.924 . . . . . 186

xxii

7.29 Figure shows the results obtained from the simulation of the flow in

a Mach 3 wind tunnel with a step at t = 4s. (a) shows a flood plot

of Pressure along with 30 contour lines plotted in a similar fashion

to (7.14). (b) shows the Entropy or the Adiabatic constant p/ργ . . . 187

7.30 Fig.(a) shows a Schlieren visualization of the flow field obtained in

Degrez et al. [107]. Fig.(b) shows a synopsis of the flow field . . . . . 188

7.31 Figure shows the computational domain along with the boundary con-

ditions imposed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.32 Figure shows the steady state results. (a) shows the non-dimensional

density and (b) shows the sensor. The density is normalized by its

freestream value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.33 Figure shows a comparison of the pressure distribution at the plate

and at y = 0.1 between our simulation and the results submitted to

the High Order Workshop by University of Bergamo . . . . . . . . . . 192

7.34 Figure shows the mesh at the walls of the sphere . . . . . . . . . . . . 193

7.35 Figure shows the Mach number distribution over the surface of the

sphere with the mesh overlaid on top . . . . . . . . . . . . . . . . . . 195

7.36 Figure shows the Mach number flood plot along the z = 0 slice . . . . 196

xxiii

Chapter 1

Introduction

1.1 Background

Computational Fluid Dynamics (CFD) has evolved and transformed significantly over

the last few decades and has enabled the simulation of a wide range of flow phenom-

ena. The numerical schemes developed in this regard can be broadly classified in

terms of spatial discretization of the PDEs into Finite Difference (FD), Finite Vol-

ume (FV) and FE methods. Irrespective of the class, a majority of the methods

developed and currently adopted in the industry are low order methods, i.e., those

which provide a maximum of second order accuracy in space. Low order methods pro-

vide intuitive, geometrically flexible and robust schemes which have been employed

to solve a plethora of flow problems and other nonlinear PDEs. However, the high

numerical dissipation associated with these methods makes them inadequate for prob-

lems that require the propagation of waves and other features like vortices for long

times periods. Examples of this include unsteady vortex-dominated flow simulations,

airframe noise computations, subsurface explorations, broad-band target illumination

and penetration etc. In the context of aerodynamics, such vortex dominated flows are

often encountered in high-lift systems, around helicopters and at off-design conditions

for aircrafts in general [1]. This need for higher fidelity has spurred the development

1

CHAPTER 1. INTRODUCTION 2

of high order methods over the last few decades.

High order methods offer not only lower numerical dissipation, but are also more

efficient than low order methods in terms of the order of accuracy achieved per com-

putational degree of freedom. High order extensions have been developed in each

of FD, FV and FE contexts. High-order extensions in FD methods can be achieved

in a rather straightforward fashion by widening the stencil. Compact Finite Difference

schemes [2] provide higher orders of accuracy with a compact stencil at the expense

of solving implicit difference equations. For regular structured domains, they provide

some of the best alternatives. However all FD methods suffer from a major drawback:

their inability to easily handle complex geometries.

FV methods are suitable for complex geometries and there have been a few extensions

to higher orders. Examples include the k-Exact methods [3, 4, 5], ENO [6, 7, 8] and

WENO [9, 10, 11] schemes. But these high-order extensions of FV methods are gen-

erally not compact and cannot be extended to arbitrarily higher orders easily. They

also grow in complexity as we move to higher dimensions and are computationally

expensive. See the article by Vincent and Jameson [1] for a broad discussion about

the potential of these high order FV methods and the textbook by Barth and Decon-

inck [4] for detailed descriptions. At the other end of the spectrum lies the class of

spectral methods which decompose the global solution into modes in frequency space

and can achieve arbitrarily high order accuracy easily. However, these methods are

highly inflexible to geometries and are quite limited in their utility to practical flow

problems.

1.2 Unstructured High Order Methods

Due to the limitations of high-order extensions of FD and FV methods, FE type

methods promising arbitrary order of accuracy on unstructured grids have attracted

great attention recently. These schemes are broadly referred to as Unstructured

High Order methods. The development of such methods was spurred by a new class


of FE schemes called Discontinuous Galerkin (DG) methods which are similar to the

classic Continuous Galerkin (CG) Finite Element methods but allow the solution

to be discontinuous along element boundaries; which makes them more suitable for

solving hyperbolic PDEs that arise in fluid dynamics.

DG methods were first introduced in the context of the neutron transport equation by

Reed and Hill [12]. Several variants of the DG method have been proposed. The Local

Discontinuous Galerkin (LDG) method was proposed by Cockburn and Shu [13] for

second order PDEs where the second order system is broken down into two first

order systems and the additional variable introduced is eliminated through judicious

choices for the interface fluxes. Compact versions of this method for second order

PDEs known as Compact Discontinuous Galerkin (CDG) were then developed by

Peraire and Persson [14]. Other notable schemes in this context include the Interior

Penalty method [15] and the Bassi-Rebay-2 method [16]. In order to alleviate the

large linear solves necessary for implicit implementations, the Hybrid Discontinuous

Galerkin (HDG) methods were proposed by Cockburn et al. [17]. These methods

reduce the number of global degrees of freedom to single counts on each element

interface. All DG methods share the common feature with CG methods of solving the

variational or the weak form of the PDE. These DG methods have been well studied

and detailed accounts of can be found in the textbooks by Cockburn et al. [18] and

Hesthaven and Warburton [19].

Another notable class of high order unstructured methods is the Spectral Difference

(SD) method originally proposed by Kopriva and Kolias [20] but later generalized to

triangular elements by [21]. These methods are similar to nodal DG methods but

solve the differential or the strong form of the PDE.

1.3 The Flux Reconstruction Formulation

Originally proposed by Huynh [22, 23], the FR formulation provides a unifying

framework for discontinuous FE Methods for utilization with explicit time-stepping


schemes. The FR framework is capable of recovering nodal DG and SD schemes as

special cases, at least for linear problems. The connections between FR and DG

methods have been examined in detail by Allaneau and Jameson [24], De Grazia et

al. [25] and Zwanenburg et al. [26]. More recently, the connections between the two

methods for the case of curvilinear meshes was established by Mengaldo et al. [27, 28].

The FR approach was studied further by Vincent et al. [29] who proposed new cor-

rection functions for the reconstruction process. These functions are now referred

to as Vincent Castonguay Jameson Huynh (VCJH) correction functions. A closely

related scheme called the Lifting Collocation Penalty (LCP) was developed by Wang

and Gao [30] in 2009. These methods are also sometimes referred to as Flux Recon-

struction, while some authors refer to them collectively as Correction Procedure via

Reconstruction (CPR). In this dissertation, we refer to the FR approach that uses

the VCJH correction functions whenever we use the terminology Flux Reconstruction

(FR).

The FR approach is very well suited for highly parallel architectures like GPUs. It is

one of the few high-order methods that is naturally adaptable to large GPU clusters.

For details of high-performance implementations of these methods, once can refer

to the articles by Manuel et al. [31], Vincent et al. [32] and Witherden et al. [33].

For a thorough comparison of the performance of FR with other low order standard

industry tools, see the article by Vermier et al. [34]. In addition to the flexibility

provided by other high order DG-type methods, like the ability to handle complex

geometries, the FR approach also provides a wide range of choices for time-stepping

methods, strategies for controlling dispersion and dissipation errors [35], multigrid

convergence acceleration techniques etc.

Recently, an alternative formulation where the flux correction is performed implicitly

through a simple Lagrange interpolation over the solution and flux points has been

proposed by Romero et al. [36, 37]. When the solution points are chosen to be the

Gauss Legendre points, this method is equivalent to the FR scheme which recovers

the nodal DG method. This method forfeits the flexibility provided by the correction

functions in favor of simplicity and improved efficiency and has been used in several


of our numerical experiments.

1.4 Motivation

While high order methods have shown great promise of improved accuracy and ef-

ficiency over their low order counterparts, they have hardly been adopted by the

industry primarily due to their lack of robustness and the lack of good shock captur-

ing techniques for these schemes [1]. This is not just due to increased complexity, but

rather arises out of a fundamental property of polynomials which are used in all the

unstructured high order methods discussed so far. When polynomials are employed

to represent a step discontinuity, the classic Gibbs phenomenon kicks in leading to

oscillations around the shock. These oscillations get worse with increasing order and

apart from destroying accuracy even away from the shock, they also render the sys-

tem unstable unless treated carefully. In addition to the issue of instability is the

problem of encountering negative solutions which can lead to a loss of hyperbolicity

of the PDE and lead to a numerical issue colloquially referred to as ‘NaNing’ caused

by fractional exponentiation of a negative quantity.

While the major advantage of high order methods is reduced dissipation relative to

low order methods, this lower dissipation acts as a disadvantage as it is directly

related to their lower robustness. Apart from shocks, aliasing instabilities are also

more prominent in high order methods. The focus of this dissertation therefore is on

investigating and improving the stability and robustness of high order methods while

working within the FR framework.

While our goal is to investigate non-linear instabilities relevant to practical problems,

linear stability of a numerical scheme is imperative for success in the nonlinear realm.

The linear stability of the FR approach has been previously studied in the one di-

mensional context and the scheme has been shown to be stable whenever the VCJH

parameter c is non-negative [29]. In fact, the VCJH correction functions were pro-

posed with stability in mind. This idea was later extended to triangles [38, 39] and


tetrahedra [40] and new correction functions and conditions on the parameters were

provided for energy stability. Parallel to these developments, the FR approach was

formulated on quadrilateral elements as a simple tensor-product like extension of the

one-dimensional approach. However, the extension of the proof of stability in 1D

to the tensor product case turned out to be much harder than expected and had

not been possible till now. Quadrilateral elements introduce several complications in

comparison to their simplex counterparts which we describe in detail in later chap-

ters. Considering that quadrilateral and hexahedral elements are widely employed

especially in boundary layer meshes, this investigation of linear stability of the FR

approach on quadrilateral elements is very important for the progress of the FR ap-

proach and its industry-wide adoption.

1.5 Contributions

Some of the major contributions in this dissertation are as follows:

1. The stability of the FR approach on quadrilateral meshes for the linear advec-

tion equation has been Investigated and the approach have been proven to be

stable in this context.

2. The contributions of a second order term towards stability have been examined

using the linear advection-diffusion equation and a proof of stability of the FR

approach on quadrilateral grids for this equation has been provided .

3. A novel approach for detecting shocks and other discontinuities which is appli-

cable to any unstructured high order method has been proposed. The method

is based on ideas used in image edge-detection using Fourier modes [41]. The

efficient adoption of such methods to polynomial based FE type methods has

been analyzed and discussed.

A clear strategy for eliminating fine-tuning of parameters has been discussed.

The shock sensor computation is completely element local and can be computed


using a single matrix pre-multiplication, thereby making it very efficient, which

is a necessity for explicit methods.

4. A robust and efficient sub-cell shock capturing approach using a two-step ap-

proach of sensing shocks using the detector and locally filtering the problematic

regions has been proposed. This operation is again applicable to any unstruc-

tured high order method, is element local and computationally efficient. This

framework can also be easily employed to treat aliasing instabilities without the

need for additional machinery.

5. These studies have also contributed to the development, Validation and Verifi-

cation (V&V), and maintenance of High-Fidelity Large Eddy Simulation (LES)

(HiFiLES), an open source CFD package capable of running on large GPU or

CPU clusters.

In Part I, the stability of the FR approach on quadrilateral meshes has been inves-

tigated and proven, first for the linear advection equation and then for the linear

advection-diffusion equation.

In Part II, currently available shock capturing methods have been discussed and the

most effective strategy from our perspective for capturing shocks in unstructured

high order methods has been discussed. In this regard, a two-step strategy has been

proposed wherein a shock sensor is utilized to detect regions containing shocks which

is then followed by a filtering operation in those specific regions. Good general shock

sensors suited for the context of high-order unstructured methods were found to be

lacking and a novel and robust approach for the same has been proposed.

Chapter 2

Flux Reconstruction Formulation

in One Dimension

This chapter provides a brief review of the FR formulation in 1D for the nonlinear

advection equation. More detailed accounts of the FR Formulation for advection-

diffusion type problems in 1D can be found in [29, 38, 42, 43]. Detailed descriptions

of the FR formulation on tensor product elements for both linear advection and linear

advection-diffusion type problems are provided in chapters 3 and 4.

2.1 Preliminaries

Consider the 1D advection equation

∂u

∂t+∂f

∂x= 0; x ∈ Ω ≡ [xL, xR] and f = f(u) (2.1)

where u is a conserved scalar quantity or the solution and f is the flux. The FR

formulation belongs to the class of FEMs and similar to other methods in this class,

the procedure begins by partitioning the domain Ω into a finite number (N) of open

8

CHAPTER 2. FLUX RECONSTRUCTION FORMULATION IN 1D 9

Figure 2.1: Figure shows an illustration of a discretization of the domain into elements

non-empty, non-overlapping open sub-domains Ω′k as follows:

Ω′k = x|xk < x < xk+1 with x1 = xL and xN+1 = xR (2.2)

The boundaries of each such sub-domain, composed of their two end-points is denoted

by Γk, i.e., Γk = xk, xk+1. The closed region formed by the union of the open region

Ω′k and its boundary Γk is referred to as an element and is denoted by Ωk, i.e.,

Ωk = Ω′l ∪ Γk andN⋃k=1

Ωk = Ω (2.3)

Now the approximate solution to the conservation law (2.1) inside element Ωk is

denoted by uDk . This approximate solution is identically zero outside of Ωk. The

union of such elemental solutions is denoted by uD, i.e.,

uD(x, t) =N∑k=1

uDk (x, t) (2.4)

The superscript D is used to denote the fact that uD is in general discontinuous across

elements. This is a standard feature of all discontinuous FEMs including the DG

and SD methods. Similarly an elemental approximate flux fk which is identically

zero outside of Ωk and the overall approximate flux in the entire domain given by a

sum of such elemental fluxes can be defined:

f(x, t) =N∑k=1

fk(x, t) (2.5)

Note that specific notation to indicate that this flux is the approximate flux obtained

numerically is not introduced to maintain brevity of notation. Unless specified oth-

erwise, henceforth it can be assumed that the quantities being referred to are the


approximate quantities obtained numerically. The flux f has not been superscripted

by D since the flux is at least C0-continuous across elements.

From an implementation perspective, it is advantageous to transform each element

and the quantities inside it to a standard reference element so that the procedure

carried out can be identical. The reference element chosen for this purpose is ΩS =

−1 ≤ ξ ≤ 1. The transformation between Ωk and ΩS is given by the mapping

x = Θk(ξ) =

(1− ξ

2

)xk +

(1 + ξ

2

)xk+1 (2.6)

or its inverse

ξ = 2

(x− xk

xk+1 − xk− 1

)(2.7)

The mapping has a Jacobian denoted by Jk and is given by

Jk =dΘk

dξ=

1

2(xk+1 − xk) (2.8)

The physical quantities are then transformed in a suitable way to reference quan-

tities such that the equation in the reference domain in terms of the transformed

or reference quantities remains identical to the original conservation law (2.1). The

transformations for the quantities are given by

u(ξ, t) = JkuDk (Θk(ξ), t) and f(ξ, t) = fk(Θk(ξ), t) (2.9)

The transformed version of the conservation law (2.1) then becomes

∂u

∂t+∂f

∂ξ= 0; ξ ∈ ΩS (2.10)

Finally, the transformed solution u within the reference domain is represented using

polynomials. The degree of polynomials utilized for this purpose is directly related to

the order of accuracy of the scheme. In general, in order to obtain of scheme of (p+1)th

order accuracy, degree p polynomials are required for representing the solution within

each element. The polynomial solution is stored as values at a discrete set of p + 1


Figure 2.2: Figure shows the solution and flux point locations for a one-dimensionalelement for a scheme with p = 4, i.e., a 5th order scheme

points known as the solution points. A popular choice for this is the set of Gauss-

Legendre points. The two boundary points of the reference element, i.e., ξ = −1 and

ξ = 1 are known as the flux points. Figure 2.2 shows the location of the solution and

flux points when the solution is represented using a 4th degree polynomial.

Using the values of the solution at these solution points, the polynomial representation

of the solution can be constructed using Lagrange polynomials:

uD =

p∑i=0

uDi li(ξ) (2.11)

where

li(ξ) =

p∏j=0j 6=i

(ξ − ξjξi − ξj

)(2.12)

is the Lagrange polynomial corresponding to the ith solution point.


Figure 2.3: Figure shows a generic discontinuous solution in the reference domain(uD) represented by a 4th degree polynomial

2.2 FR Procedure

In order to solve the conservation law in the transformed or reference domain, we begin

by calculating the discontinuous flux in the reference domain from the discontinuous

flux. This is done by calculating the flux at the solution points from uD using the

flux function, i.e.,

fDi = f(uDi /Jk) (2.13)

where the subscript i refers to the value at the ith solution point. Using these values

of the transformed flux at the solution points, we can construct a degree p polynomial

for it in a similar fashion to that of the discontinuous solution.

fD =

p∑i=0

fDi li(ξ) (2.14)

The next stage of the FR process involves calculating transformed numerical fluxes

at the boundaries of the standard element ΩS. To this end, we first build the physical


Figure 2.4: Figure shows the discontinuous flux polynomial in the reference domain(fD)

solution polynomial similar to the transformed solution polynomial.

uD(ξ) =

p∑i=0

uDi li(ξ) (2.15)

where uDi = uDi /Jki . For a constant Jacobian, this just amounts to dividing the

reference solution polynomial uD by the Jacobian of the element. Now the values of

this physical solution at the ends of the reference element (ξ = ±1) are computed.

Let us denote the value at ξ = −1 as uDL and ξ = 1 as uDR .

Once this procedure has been performed in each element, we then move on to compute

the common numerical flux at all element interfaces or boundaries. Consider the

interface between elements Ωk and Ωk+1. Let us denote the right value in Ωk, i.e.,

uDRk as uD− and the left value in Ωk+1, i.e., uDLk+1as uD+ . Figure 2.5 shows the physical

solution polynomials (uD) in two neighboring elements Ωk and Ωk+1 along with uD− and

uD+ . Note that the discontinuity at the element boundary is exaggerated for purposes

of illustration and is much smaller in practice. This discontinuous solution gives rise


Figure 2.5: Figure shows the physical solution polynomial in two neighboring elementsin a generic scenario. The physical solution is in general discontinuous giving rise toa Riemann problem.

to a Riemann problem at each element interface. Using these values of uD− and uD+ ,

we can compute the common numerical flux f ∗ by using an approximate Riemann

solver like the Roe or Rusanov or through a Generalized Local Lax Friedrichs flux.

This common numerical flux f ∗ is then transformed back to the reference domain

using (2.9).

In order to obtain a conservative scheme, the flux needs to be continuous across

element boundaries. So the next stage involves correcting the flux polynomial inside

the element such that it attains the value of the common numerical flux at the element

interface, i.e., f ∗. This is performed in the reference domain. To this end, let us denote

the transformed common numerical flux values at the left and right boundaries of a

generic element Ωk as f ∗L and f ∗R respectively. Similarly, let fDL and fDR denote the

values of the discontinuous flux polynomial fD at the left and the right boundaries.

The correction operation is performed by adding a correction component fC to the

discontinuous part fD. This correction component can be generally written as follows:


Figure 2.6: Figure shows a schematic of the flux correction process in the referenceelement ΩS. The dashed black line represents the discontinuous flux in the referencedomain fD. The thick magenta line represents the corrected total flux polynomialthat would result in a continuous flux in the physical domain


fC = (f ∗L − fDL )hL(ξ) + (f ∗R − fDR )hR(ξ) (2.16)

where hL(ξ) and hR(ξ) are the left and right correction functions which are polynomi-

als of degree p+1, i.e., one degree higher than the solution polynomials. This ensures

that the order of accuracy of the scheme is preserved at (p+ 1). Using this correction

component, the approximate total transformed flux in the reference domain f can be

constructed by adding it to the discontinuous flux:

f = fD + fC (2.17)

This correction process is required to obtain a total flux that is continuous across

the elements in the physical domain. For this, we require fL = f ∗L and fR = f ∗R.

In order to achieve this, the correction functions are required to have the following

properties:

hL(−1) = 1 hL(1) = 0

hR(−1) = 0 hR(1) = 1 (2.18)

hL(−ξ) = hR(ξ)

where the last property ensures symmetry of the correction process. Figure 2.7 shows

correction functions that recover the nodal DG method.

Finally the derivative of the continuous total flux f is computed using

∂fi∂ξ

=

p∑j=0

fDjdljdξ

(ξi) + (f ∗L − fDL )dhLdξ

(ξ) + (f ∗R − fDR )dhRdξ

(ξ) (2.19)

This is then used in (2.10) to obtain an ODE which is then time-advanced using a

high-order time integration scheme like the RK4 method.


Figure 2.7: Figure shows one possible choice for the left and right correction functionsemployed for a p = 4 FR scheme. These correction functions are polynomials ofdegree p + 1, i.e., degree 5. The correction functions shown here correspond to aVCJH parameter c = 0 which recovers the nodal DG method for linear problems


2.3 Correction Functions

Apart from the properties (2.18), the correction functions need to satisfy a few extra

properties in order to guarantee stability for linear problems. Vincent et al. [29, 44]

developed a single-parameter family of correction functions which guarantee stability

of the FR formulation for 1D linear advection and advection diffusion problems. The

correction functions are required to satisfy the following properties:

1∫−1

hLdlidξdξ = c

dplidξp

dp+1hLdξp+1

(2.20)

1∫−1

hRdlidξdξ = c

dplidξp

dp+1hRdξp+1

(2.21)

where c is a scalar that must be in the following range

c− < c <∞ (2.22)

where

c− =−2

(2p+ 1)(app!)2where ap =

(2p)!

2p(p!)2(2.23)

In order to satisfy these properties as well as (2.18), the correction functions must be

defined as follows:

hL =(−1)p

2

[Lp −

(ηpLp−1 + Lp+1

1 + ηp

)]hLR =

(1

2

[Lp +

(ηpLp−1 + Lp+1

1 + ηp

)](2.24)

where

ηp =c(2p+ 1)(app!)

2

2(2.25)

and Lp is the Legendre polynomial of degree p.

The parameter c is called the VCJH parameter and it parametrizes this family of


Parameter type Value

c−−2

99225

cDG 0

cSD8

496125

cHU1

39690

Table 2.1: Table provides the values of the VCJH parameter c for recovering differentexisting schemes. c− is the smallest value of c which can provide a stable scheme

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

c−/2 c

DGc

SDc

HUc

large

Figure 2.8: Figure shows the left correction function hL(ξ) for various values of c


correction functions hL and hR. Varying c changes the stability, dissipation and dis-

persion properties of the schemes. Certain values of c enable the recovery of several

existing high-order schemes. The collocation-based nodal DG scheme can be recov-

ered when the left and right correction functions are set to the right and left Radau

polynomials respectively, i.e., by setting c = 0 (for any p). It has also been shown that

the SD method can be recovered (at least for a linear flux function) if the zeros of the

correction polynomials hL and hR are set to a set of p symmetrically located colloca-

tion points within ΩS. The different values of c recovering various known schemes for

p = 4 are shown in Table 2.1. c− refers to the smallest value of c for which stability

of the FR schemes is guaranteed in 1D. Figure 2.8 shows the correction functions for

several different values of c starting from c−/2. More details about the correction

functions can be found in papers by Huynh [22, 23] and Vincent et al. [29].

The FR schemes using these VCJH correction functions have been proven to be stable

for c > c− in 1D. Numerical experiments show that the schemes become less stable as

c is decreased and do become unstable at a point close to but not greater than c− [29].

While the 1D analysis of Vincent et al. [29] guarantees stability of these schemes for

c > c−, it does not provide intuition for the numerical observations of decreased

stability when c is decreased. The results obtained in Chapters 3 and 4 of this

dissertation provide intuition for these numerical observations along with investigating

the stability of the FR formulation on tensor product elements.

Part I

Linear Stability Theory of the FR

Formulation on Tensor Product

Elements

21

Preamble

Stability is an indispensable property for any numerical scheme. High order methods

in particular are known to develop instabilities in nonlinear problems. Several strate-

gies have been adopted to handle such instabilities in practice. However, the stability

of the numerical scheme for solving linear problems is imperative if there is to be any

hope in the nonlinear scenarios. Unlike nonlinear problems where a detailed theo-

retical investigation is often intractable, linear problems provide a great framework

to study various properties of the numerical scheme like stability, order of accuracy,

dispersion and dissipation properties etc.

Vincent et al. [29] showed the stability of the VCJH schemes in 1D for linear advection

based on a similar analysis by Jameson in [45]. Furthermore, Jameson et al. studied

the non-linear stability of the FR approach in 1D [46]. Castonguay et al. [38, 44]

extended the approach to triangular elements and proposed an energy stable family of

correction functions for triangles. Further extensions to advection diffusion problems

on triangles and tetrahedral elements along with proofs of stability on those elements

were provided by Williams et al. [40] [43].

However, the stability of these schemes on tensor product elements like quadrilaterals

and hexahedra has not been studied successfully. Even the simplest bilinear quadri-

lateral elements pose a challenge due to the variation of the Jacobian inside each

element unlike in 1D and on simplexes. In fact, direct extension of the 1D approach

to the proof of stability does not seem possible. In this dissertation, we see that,

even in the case of rectangular Cartesian meshes, investigating stability requires a

22

23

somewhat different approach from that used for 1D and simplex elements. We get

additional terms which affect stability, each of which is scaled by the VCJH parame-

ter, thereby giving us valuable insight into the behavior of these schemes on general

quadrilateral elements.

The aim of this part of the dissertation is to investigate the stability of the FR

approach on tensor product elements for linear advection as well as advection-diffusion

equations. For both cases, we prove the FR approach is stable on Cartesian meshes

whenever the VCJH parameter c is non-negative. In Chapter 3, we prove the stability

of the formulation for the advection equation and then extend it to the advection-

diffusion equation in Chapter 4.

The work for this part has been presented from the following publications:

• Sheshadri A, Jameson A. On the Stability of the Flux Reconstruction Schemes

on Quadrilateral Elements for the Linear Advection Equation. Journal of Sci-

entific Computing, 2016 [47]

• Sheshadri A, Jameson A. Erratum To: On the Stability of the Flux Reconstruc-

tion Schemes on Quadrilateral Elements for the Linear Advection Equation.

Journal of Scientific Computing, 2016 [48]

• Sheshadri A, Jameson A. Proof of stability of the Flux Reconstruction schemes

on quadrilateral elements for the linear advection-diffusion equation. (Under

review with the Journal of Scientific Computing)

Chapter 3

Stability of the FR approach for

Linear Advection Equation on

Tensor Product Elements

In this chapter, we investigate the stability of the FR formulation for the linear

advection equation. We begin by describing the 2D FR approach for quadrilateral

elements for a first order PDE with the linear advection equation as the example. We

then proceed to prove the stability of the approach on Cartesian meshes.

3.1 FR Approach on Quadrilaterals for a First

Order PDE

Before we go on to assess the stability of the VCJH-type Flux Reconstruction (FR)

approach on quadrilateral elements, let us first explain the approach for the linear

advection equation on general linear quadrilateral elements.

24

CHAPTER 3. STABILITY FOR LINEAR ADVECTION EQUATION 25

3.1.1 Preliminaries

Consider the 2D conservation law

∂u

∂t+ ∇ · f = 0 in Ω , (3.1)

where Ω is a bounded connected subset of R2 with boundary Γ composed of a finite

union of parts of hyperplanes. Further, f is a linear flux of the form

f = au with f =

(F

G

)and a =

(a

b

). (3.2)

Consider a partition TN of Ω into N non-empty, non-overlapping, conforming quadri-

lateral elements Ωk with boundaries Γk such that Γk =4⋃i=1

F ik where F ik are straight

lines representing the faces (or edges) of the element Ωk. Furthermore, we restrict

ourselves to non-mortar elements, i.e., if F ik ∩ Γk′ 6= ∅ for k′ 6= k, then F ik ∩ Γj =

∅, ∀j 6= k, k′ and F ik ∩ Γ = ∅.

To facilitate a uniform implementation of the method, each element Ωk can be mapped

to a square reference domain defined by ΩS =

(ξ, η)| − 1 ≤ ξ, η ≤ 1

as follows:

xk = Θk(ξ, η) =4∑i=1

Ni(ξ, η)vik (3.3)

Here xk represents the physical co-ordinates (x, y) of an arbitrary point in the el-

ement Ωk, vik denote the physical co-ordinates of the 4 vertices of Ωk and Ni(ξ, η)

are bilinear shape functions defined on ΩS. Figure 4.1 shows an example of such a

mapping. Further, let the Jacobian matrix associated with Θk be denoted by Jk and

its determinant by Jk. Jk varies from point to point within an element for a general

linear quadrilateral, unlike linear simplex elements.

In addition, we also transform the physical quantities u and f to the reference domain

using the following equations:

uk = Jkuk (3.4)


(−1,−1) (1,−1)

(1, 1)(−1, 1)

v1k

v2k

v3k

v4k

x

y

ξ

η

Figure 3.1: Mapping between the physical domain (on the left) and the referenceelement (on the right)

fk = JkJ−1k fk (3.5)

∇ · fk = Jk∇ · fk (3.6)

This transformation is designed to obtain the same form of the conservation law in

the reference domain. Using these equations we can see that the conservation law,

i.e., (4.3) can be written in the reference domain as follows

∂uk∂t

+ ∇ · fk = 0 (3.7)

Since we restrict ourselves to rectangular Cartesian meshes while discussing the sta-

bility of the schemes, it is worthwhile to note that the Jacobian matrix is a constant

for each element in such a mesh. We could further introduce some additional notation

to simplify the algebra. For rectangular Cartesian meshes we have

∂xk∂η

=∂yk∂ξ

= 0 (3.8)


Let Jxk = ∂xk∂ξ

and Jyk = ∂yk∂η

. We then have

Fk = JykFk Gk = JxkGk uk = JxkJykuk = Jkuk (3.9)

3.1.2 FR Procedure

Here we briefly describe the FR procedure as applied to a 2D conservation law with

a linear flux on a rectangular Cartesian mesh with linear elements. Details of the

implementation of the FR approach on more general quadrilateral elements and fluxes

can be found in [44].

In order to build a scheme of (p + 1)th order accuracy, we start by selecting a set

of (p + 1)2 points on the reference domain as our solution points. A possible choice

for the solution points is the tensor product of the 1D Gauss-Legendre points on the

square domain. We then represent our transformed solution within each element, i.e.,

uk, using a tensor product of pth degree Lagrange polynomial basis defined on these

solution points.

uD =

p∑i=0

p∑j=0

li(ξ)lj(η)uDij , (3.10)

where li(ξ) and lj(η) are the 1D Lagrange polynomials associated with the solution

points ξi and ηj respectively and uDij is the value of the transformed solution at (ξi, ηj).

Note that we have dropped the subscript k in order to keep the notation from getting

clumsy. Since uD is a transformed quantity, it is understood to be associated with

a certain generic element Ωk. Also, similar to a Discontinuous Galerkin method, we

allow our solution u to be discontinuous across the elements. Therefore, we represent

such discontinuous quantities with a superscript D.

We also have p+1 flux points along each boundary edge of the quadrilateral element.

These flux points are chosen to align with the solution points in the reference domain,

i.e., we would choose them to be the 1D Gauss-Legendre points along each edge if we

are using such solution points. The total continuous flux fk can be written as a sum


Solution Points

Flux Points

Figure 3.2: Figure showing the solution and flux points in the reference element fora p = 2 scheme

of a discontinuous component and a correction component.

fk = fkD

+ fkC

(3.11)

The discontinuous component, fkD

is the transformed version of the flux computed

directly from the solution values at the solution points and is represented using the

same pth degree Lagrange polynomial basis we used for the solution points. Therefore,

in each element we have

FD =

p∑i=0

p∑j=0

li(ξ)lj(η)FDij and GD =

p∑i=0

p∑j=0

li(ξ)lj(η)GDij (3.12)

where

FDij = JykF (uij) and GD

ij = JxkG(uij) (3.13)

The correction component of the flux is computed along 1D lines in both the ξ and

η directions and can be concisely written as follows

FC = −hL(ξ)

p∑j=0

((f ∗ − fD) · n

)Ljlj(η) + hR(ξ)

p∑j=0

((f ∗ − fD) · n

)Rjlj(η) (3.14)


GC = −hL(η)

p∑j=0

((f ∗ − fD) · n

)Bjlj(ξ) + hR(η)

p∑j=0

((f ∗ − fD) · n

)Tjlj(ξ) (3.15)

where if one wants to recover the VCJH schemes, hL and hR denote the left and right

1D VCJH correction functions of degree p + 1 respectively. L,R,B, T represent the

left (ξ = −1), right (ξ = 1), bottom (η = −1) and top (η = 1) edges respectively.

(.)Lj denotes the value at the jth flux point on the left boundary. f ∗ represents the

transformed common interface flux value. fD on the boundaries is obtained through

an extrapolation operation. Finally, lj denotes the jth member of the 1D Lagrange

basis of degree p defined on the edge and the summation is over the flux points on

the corresponding edge.

Remark 3.1.1. Note that we have used hL and hR as the correction functions for

GC as well because the correction along the η direction is performed in the same 1D

sense as that in the ξ direction.

Remark 3.1.2. In the above equations, note that the corrections coming in from left

and bottom edges have a negative sign associated with them, unlike in 1D, because we

use f · n. Since the outward-facing normal vector n has a negative sign on the left

and bottom edges, we need to compensate for it with an additional negative sign.

Also, for brevity of notation, we let

∆Lj =((f ∗ − fD) · n

)Lj

∆Rj =((f ∗ − fD) · n

)Rj

∆Bj =((f ∗ − fD) · n

)Bj

∆Tj =((f ∗ − fD) · n

)Tj

(3.16)

In order to compute the transformed common interface flux f ∗, we first need to

extrapolate the solution values to the flux points on the boundary. For example,

the 1D-edge polynomial formed by the extrapolated transformed solution on the left

boundary is computed as follows

uDL =

p∑i=0

p∑j=0

li(−1)lj(η)uDij (3.17)


We then transform this uDL to the physical domain and compute the flux from the

physical solution values at the boundary flux points. Let fD− , uD− denote the boundary

fluxes and solution values calculated as above from the current element and fD+ , uD+

denote the corresponding values from a neighboring element along some edge. We

then compute a common numerical interface flux value f∗ at the flux points on the

edge using either an approximate Riemann solver or the local Lax-Friedrichs approach.

The latter has the benefit of providing stability at a lower computational cost. The

common numerical flux value given by the local Lax-Friedrichs approach is

f ∗ = fD+λ

2

(max

u∈[uD− ,uD+ ]

∣∣∣∣∂f∂u · n∣∣∣∣)[[uD]] (3.18)

where · and [[·]] are the average and jump operators respectively and λ is an

upwinding parameter with 0 ≤ λ ≤ 1. λ = 1 gives a fully upwinded scheme while

λ = 0 is essentially the central flux definition. We then have to transform the normal

common interface flux from the physical domain back to the reference domain. For

example, on the left boundary we can do this using

(f ∗ · n)Lj = JLj(f∗ · n)Lj (3.19)

where JLj is the edge-Jacobian at the jth flux point on the left boundary. The edge-

Jacobian is an edge-based scaling factor which is just equal to the edge length in the

Cartesian case. Therefore (4.26) can be rewritten for the case of Cartesian meshes as

(f ∗ · n)Lj = Jy(f∗ · n)Lj (3.20)

where Jy is the edge length of the left (and right) edge. We can then go on and

compute the correction component of the flux using (4.29) and (4.30). Once we

have both the discontinuous and the correction components of the flux, we can then

calculate the transformed solution at the next time step in the kth element using

∂uDk∂t

= −∇ · fkD− ∇ · fk

C(3.21)


Note that the divergence of the total continuous flux is of degree p due to the (p+1)th

degree VCJH correction functions in the correction component fkC

.

3.1.3 VCJH Correction Functions

The 1D VCJH correction functions have been described in detail earlier in 2 and also

in the article by Vincent et al. [49].

Here we just repeat the properties we will need for the stability proof for convenience.

hL(ξ) = hR(−ξ) (3.22)

hL(−1) = 1 hL(1) = 0 (3.23)

1∫−1

dlidξhLdξ = c

dplidξp

dp+1hLdξp+1

(3.24)

3.2 Key Difficulties in Extending the 1D Stability

Analysis to Tensor Product Elements

As we mentioned earlier, the FR approach utilizing the VCJH correction functions

have been shown to be stable in 1D [29]. In fact, the VCJH correction functions

were designed to obtain stable FR schemes in 1D. Similarly, correction functions

that provide energy stable schemes were built on triangles and tetrahedra. In the

case of quadrilateral and hexahedral elements however, a simple extension of the 1D

approach is possible as seen in the previous sections and therefore designing new

two-dimensional correction polynomials is not necessary.

However, till now, it was not clear whether this tensor product extension of the 1D


scheme is stable, or if the same criteria for stability obtained in 1D will transfer

directly to tensor product elements. This was due to a couple of major difficulties in

extending the stability analysis in 1D to tensor product elements:

3.2.1 Necessity for a new norm

In the 1D stability analysis, a broken Sobolev norm of the form

‖uD‖p,2 =

[ N∑k=1

∫Ωk

(uDk )2 +c

2(Jk)

2p(∂puDk∂xp

)pdx

]1/2

(3.25)

is used to show stability. This norm of the solution is shown to be non-increasing

under certain conditions and the equivalence of norms guarantees L2 energy of the

solution cannot blow up.

In the case of triangles (and tetrahedra), a similar norm is used where the sum of all

degree p-derivatives appears in the place of the second term on the RHS of (3.25),

i.e.,

‖uD‖p,2 =

[ N∑k=1

∫Ωk

(uDk )2

2+

1

2AS

p+1∑j=1

cj(D(j,p)uD

)2dΩk

]1/2

(3.26)

where AS is the area of the reference triangle element and the differentiation operator

D(j,p) is defined as

D(v,w) =∂w

∂r(w−v+1)∂s(v−1)(3.27)

where r and s are coordinates in the reference domain.

In the case of tensor product elements, such a norm with just the degree p derivatives

is not sufficient for algebraic tractability. One of the major reasons for this is the

difference between tensor product elements and 1D or triangles when we take the

degree p derivative of the conservation law to get the second term in the RHS of

either (3.25) or (3.26). We will encounter this difficulty when we present the proof in

the later sections, but to get an intuition for this, we show a snippet of the key issue.


Let us first consider the 1D case. The derivative of the total continuous flux in the

reference domain is given by (See (2.19))

∂fi∂ξ

=

p∑j=0

fDjdljdξ

(ξi) + (f ∗L − fDL )dhLdξ

(ξ) + (f ∗R − fDR )dhRdξ

(ξ) (3.28)

When we take the degree p derivative of the conservation law to obtain the second term

on the RHS of (3.25), we would have to take the degree p derivative of the derivative

of the total flux. So consider taking the pth derivative of (3.28) and focus on the

first term on the RHS, i.e., the term representing the discontinuous flux calculated

directly from the discontinuous solution:

∂p

∂ξp

(∂fD

∂ξ

)=

∂p

∂ξp

( p∑j=0

fDjdljdξ

(ξ)

)= 0 (3.29)

since lj(ξ) are Lagrange polynomials of degree p. Therefore the degree-p derivative

of the discontinuous flux derivative in the conservation law vanishes; leaving only the

degree-p derivative of the correction component. The correction functions are then

designed or chosen so as to obtain stable contributions from such terms, which gives

rise to properties or criteria required to be satisfied by the correction functions such

as those in (3.24). The scenario is similar for triangles and tetrahedra. However, for

tensor product elements, the corresponding degree-p derivatives of the discontinuous

flux gradient do not vanish.

To see this, let us take the degree-p derivative w.r.t ξ of the equivalent term in the

2D conservation law in the reference domain (3.21):

∂p

∂ξp

(∇ · fD

)=

∂p

∂ξp

(∂FD

∂ξ+∂GD

∂η

)(3.30)


Using (3.13), we can rewrite this as

∂p

∂ξp

(∇ · fD

)=

∂p

∂ξp

( p∑i=0

p∑j=0

dlidξ

(ξ)lj(η)FDij +

p∑i=0

p∑j=0

li(ξ)dljdξ

(η)GDij

)

= 0 +

p∑i=0

p∑j=0

dplidξp

(ξ)dljdξ

(η)GDij

(3.31)

It is evident that only one of the terms vanishes. This non-vanishing term makes the

stability analysis much harder since this discontinuous part of the flux is not under

our control like the correction component. This requires the inclusion of the degree

2p-derivative of the solution in our Sobolev norm and extra handling of the degree

p-derivatives of the discontinuous components of the flux. Since the tensor product

of degree p Lagrange polynomials is truly a degree 2p-polynomial, this inclusion of

the 2p-derivative of the solution is not surprising.

3.2.2 Varying Jacobian of the Geometric

Transformation

The geometric transformation between the physical and reference domains has a con-

stant Jacobian in 1D and for straight-sided triangle and tetrahedral elements. How-

ever, the Jacobian matrix for the transformation of a general straight-sided quadrilat-

eral element to the square reference element is not constant. This makes the stability

analysis significantly more tedious and renders the algebra intractable. Therefore we

circumvent this difficulty by considering Cartesian meshes where the Jacobian of the

geometric transformation is constant.


3.3 Proof of Stability of the FR Approach on

Quadrilaterals

In this section we discuss the stability of the FR scheme on quadrilateral elements. We

restrict ourselves to rectangular Cartesian meshes with linear quadrilateral elements.

This is mainly to avoid the variation of the Jacobian inside the element which makes

the algebra difficult to manage. We state our main result in Theorem 3.3.5. Our

aim is to investigate the growth of an appropriate Sobolev norm of the solution and

identify factors that can cause instabilities. Before stating the theorem however, we

show some intermediate important results through Lemmas.

Lemma 3.3.1.

1

2

d

dt

∫Ωk

Jk(uDk )2dΩk = −

∫ΩS

uD(∇ · fD)dΩS −∫ΓS

uD(fC · n)dΓS

− c1∫

−1

dp+1hL(ξ)

dξp+1

∂puD

∂ξp

( p∑j=0

∆Lj lj(η)

)dη

︸︷︷︸A1

+ c

1∫−1

dp+1hR(ξ)

dξp+1

∂puD

∂ξp

( p∑j=0

∆Rj lj(η)

)dη

︸︷︷︸A2

− c1∫

−1

dp+1hL(η)

dηp+1

∂puD

dηp

( p∑j=0

∆Bj lj(ξ)

)dξ

︸︷︷︸A3

+ c

1∫−1

dp+1hR(η)

dηp+1

∂puD

dηp

( p∑j=0

∆Tj lj(ξ)

)dξ

︸︷︷︸A4

(3.32)

Proof. Let us start by rewriting (4.3) in the kth element by observing that the total


continuous flux is composed of the discontinuous and the correction components, i.e.,

∂uDk∂t

= −∇ · fDk −∇ · fCk (3.33)

Let Jk be the determinant of the transformation Jacobian Jk. Multiply (3.33) by

JkuDk and integrate over Ωk to get

1

2

d

dt

∫Ωk

Jk(uDk )2dΩk = −

∫Ωk

JkuDk (∇ · fDk + ∇ · fCk )dΩk (3.34)

Transforming the RHS to the reference domain using (4.6) and (4.8), we get

1

2

d

dt

∫Ωk

Jk(uDk )2dΩk = −

∫ΩS

uD(∇ · fD + ∇ · fC)dΩS (3.35)

Now consider the second term in the RHS above:

−∫

ΩS

uD(∇ · fC)dΩS = −∫

ΩS

uD(∂FC

∂ξ+∂GC

∂η

)dΩS

=

1∫−1

1∫−1

uDdhL(ξ)

dξ

p∑j=0

((f · n)∗Lj − F

DLj

)lj(η)dξdη

−1∫

−1

1∫−1

uDdhR(ξ)

dξ

p∑j=0

((f · n)∗Rj − F

DRj

)lj(η)dξdη

+

1∫−1

1∫−1

uDdhL(η)

dη

p∑j=0

((f · n)∗Bj − G

DBj

)lj(ξ)dξdη

−1∫

−1

1∫−1

uDdhR(η)

dη

p∑j=0

((f · n)∗Tj − G

DTj

)lj(ξ)dξdη

(3.36)


Now the first term on the RHS of (3.36) can be rewritten as follows by using integra-

tion by parts

1∫−1

1∫−1

uDdhL(ξ)

dξ

p∑j=0

((f · n)∗Lj − F

DLj

)lj(η)dξdη

=

1∫−1

1∫−1

uDdhL(ξ)

dξdξ

︸︷︷︸I.B.P

p∑j=0

∆Lj lj(η)dη

=

1∫−1

((uDhL(ξ))

∣∣∣∣ξ=1

ξ=−1

−1∫

−1

hL(ξ)∂uD

∂ξdξ

)( p∑j=0

∆Lj lj(η)

)dη

=

1∫−1

(− uDL (η)

)( p∑j=0

∆Lj lj(η)

)dη −

1∫−1

1∫−1

∂uD

∂ξhL(ξ)

( p∑j=0

∆Lj lj(η)

)dξdη

(3.37)

where the last step was obtained using (3.23). Since our transformed solution uD is

represented by a tensor-product Lagrange basis, we can use property (3.24) to obtain

1∫−1

∂uD

∂ξhL(ξ)dξ = c

∂puD

∂ξpdp+1hL(ξ)

dξp+1(3.38)

where c is the VCJH parameter. Now we can rewrite (3.37) using this property to


get

1∫−1

1∫−1

uDdhL(ξ)

dξ

p∑j=0

((f · n)∗Lj − F

DLj

)lj(η)dξdη

=

1∫−1

−uDL (η)

( p∑j=0

∆Lj lj(η)

)dη − c

1∫−1

dp+1hL(ξ)

dξp+1

∂puD

∂ξp

( p∑j=0

∆Lj lj(η)

)dη

=

1∫−1

−uDL (η)(− fC · n

)Ldη − c

1∫−1

dp+1hL(ξ)

dξp+1

∂puD

∂ξp

( p∑j=0

∆Lj lj(η))dη

=

1∫−1

uDL (η)(fC · n

)Ldη − c

1∫−1

dp+1hL(ξ)

dξp+1

∂puD

∂ξp

( p∑j=0

∆Lj lj(η)

)dη

(3.39)

Remark 3.3.1. Note that the term1∫−1

uDL (η)(fC · n

)Ldη is integrating along the left

boundary from η = −1 to η = 1. If we were to include this as a part of the integral

along the boundaries of the element, we would have to integrate in the opposite direc-

tion since we assume the counter-clockwise direction as positive for element boundary

integrals.

Writing down similar expressions for the other three terms on the RHS of (3.36), we


get

−∫

ΩS

uD(∇ · fC)dΩS =−∫ΓS

uD(fC · n)dΓS

− c1∫

−1

dp+1hL(ξ)

dξp+1

∂puD

∂ξp

( p∑j=0

∆Lj lj(η)

)dη

+ c

1∫−1

dp+1hR(ξ)

dξp+1

∂puD

∂ξp

( p∑j=0

∆Rj lj(η)

)dη

− c1∫

−1

dp+1hL(η)

dηp+1

∂puD

∂ηp

( p∑j=0

∆Bj lj(ξ)

)dξ

+ c

1∫−1

dp+1hR(η)

dηp+1

∂puD

∂ηp

( p∑j=0

∆Tj lj(ξ)

)dξ

(3.40)

Substituting these results back into (3.36), we get Lemma 3.3.1.


Lemma 3.3.2.

1

2

(1

2

) ∂∂t

∫Ωk

Jk

(∂puDk∂ξp

)2

dΩk = −1∫

−1

∂puD

∂ξp∂p

∂ξp

(∂GD

∂η

)dη

+

1∫−1

∂puD

∂ξpdp+1hL(ξ)

dξp+1

( p∑j=0

∆Lj lj(η)

)dη

︸︷︷︸A1

−1∫

−1

∂puD

∂ξpdp+1hR(ξ)

dξp+1

( p∑j=0

∆Rj lj(η)

)dη

︸︷︷︸A2

+

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

)(− ∂puD

∂ξp

∣∣∣∣η=−1

− cdp+1hL(η)

dηp+1

∂2puD

∂ξp∂ηp︸︷︷︸B3

)

−( p∑

j=0

∆Tj

∂plj(ξ)

∂ξp

)(∂puD

∂ξp

∣∣∣∣η=1

− c dp+1hR(η)

dηp+1

∂2puD


)

(3.41)

Proof. Multiply (3.33) by Jk and apply the operator ∂p

∂ξpto the entire equation to get

∂

∂t(Jk

∂puDk∂ξp

) = − ∂p

∂ξp(Jk∇ · fDk )− ∂p

∂ξp(Jk∇ · fCk ) (3.42)

Note that the derivative of uDk with respect to ξ is well defined as it is indirectly a

function of ξ and η. Now observing that Jk is a constant in a Cartesian mesh, the

above equation can then be written as

Jk∂

∂t(∂puDk∂ξp

) = − ∂p

∂ξp(Jk∇ · fDk )− ∂p

∂ξp(Jk∇ · fCk ) (3.43)


Multiply both sides of (3.43) by∂puDk∂ξp

and integrate over Ωk to obtain

1

2

∂

∂t

∫Ωk

Jk(∂puDk∂ξp

)2dΩk =−∫Ωk

∂puDk∂ξp

∂p

∂ξp(Jk∇ · fDk )dΩk

−∫Ωk

∂puDk∂ξp

∂p

∂ξp(Jk∇ · fCk )dΩk

(3.44)

Transforming RHS to reference domain we get

1

2

∂

∂t

∫Ωk

Jk(∂puDk∂ξp

)2dΩk =−∫

ΩS

∂puD

∂ξp∂p

∂ξp(∇ · fD)dΩS

−∫

ΩS

∂puD

∂ξp∂p

∂ξp(∇ · fC)dΩS

(3.45)

Let us first consider the 1st term on the RHS of (3.45),

−∫

ΩS

∂puD

∂ξp∂p

∂ξp(∇ · fD)dΩS = −

∫ΩS

∂puD

∂ξp

(>

0∂p+1FD

∂ξp+1+

∂p

∂ξp

(∂GD

∂η

))dΩS

= −2

1∫−1

∂puD

∂ξp∂p

∂ξp

(∂GD

∂η

)dη

(3.46)

where the last step was obtained by observing that the integrand is a constant w.r.t

ξ and hence the integral just amounts to 2. Now consider the 2nd term of the RHS


of (3.45).

−∫

ΩS

∂puD

∂ξp∂p

∂ξp(∇ · fC)dΩS = −

∫ΩS

∂puD

∂ξp

(∂p+1FC

∂ξp+1+

∂p

∂ξp

(∂GC

∂η

))dΩS

= + 2

1∫−1

∂puD

∂ξpdp+1hL(ξ)

dξp+1

( p∑j=0

∆Lj lj(η)

)dη

− 2

1∫−1

∂puD

∂ξpdp+1hR(ξ)

dξp+1

( p∑j=0

∆Rj lj(η)

)dη

+ 2

1∫−1

∂puD

∂ξp∂hL(η)

∂η

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

)dη

− 2

1∫−1

∂puD

∂ξp∂hR(η)

∂η

( p∑j=0

∆Tj

∂plj(ξ)

∂ξp

)dη

(3.47)

Now consider the 3rd term on the RHS of (3.47). We can write it as follows:

2

1∫−1

∂puD

∂ξp∂hL(η)

∂η

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

)dη

= 2

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

) 1∫−1

∂p

∂ξp

(uD

∂hL(η)

∂η

)dη

= 2

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

)∂p

∂ξp

( 1∫−1

uD∂hL(η)

∂ηdη

)︸︷︷︸

I.B.P+VCJH property

= 2

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

)(− ∂puD

∂ξp

∣∣∣∣η=−1

− cdp+1hL(η)

dηp+1

∂2puD

∂ξp∂ηp

)(3.48)

Using the same approach on the last term of equation (3.47), we can rewrite (3.47)


as

−∫

ΩS

∂puD

∂ξp∂p

∂ξp(∇ · fC)dΩS =

+ 2

1∫−1

∂puD

∂ξpdp+1hL(ξ)

dξp+1

( p∑j=0

∆Lj lj(η)

)dη

− 2

1∫−1

∂puD

∂ξpdp+1hR(ξ)

dξp+1

( p∑j=0

∆Rj lj(η))dη

+ 2

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

)(− ∂puD

∂ξp

∣∣∣∣η=−1

− cdp+1hL(η)

dηp+1

∂2puD

∂ξp∂ηp

)

− 2

( p∑j=0

∆Tj

∂plj(ξ)

∂ξp

)(∂puD

∂ξp

∣∣∣∣η=1

− cdp+1hR(η)

dηp+1

∂2puD

∂ξp∂ηp

)

(3.49)

Substituting the above results back into (3.45), we get the desired result stated in

Lemma 3.3.2.


Lemma 3.3.3.

1

2(1

2)∂

∂t

∫Ωk

Jk

(∂puDk∂ηp

)2

dΩk =

−1∫

−1

∂puD

∂ηp∂p

∂ηp

(∂FD

∂ξ

)dξ

+

1∫−1

∂puD

∂ηpdp+1hL(η)

dηp+1

( p∑j=0

∆Bj lj(ξ)

)dξ

︸︷︷︸A3

−1∫

−1

∂puD

∂ηpdp+1hR(η)

dηp+1

( p∑j=0

∆Tj lj(ξ)

)dξ

︸︷︷︸A4

+

( p∑j=0

∆Lj

∂plj(η)

∂ηp

)(− ∂puD

∂ηp

∣∣∣∣ξ=−1

− cdp+1hL(ξ)

dξp+1

∂2puD


)

−( p∑

j=0

∆Rj

∂plj(η)

∂ηp

)(∂puD

∂ηp

∣∣∣∣ξ=1

− cdp+1hR(ξ)

dξp+1

∂2puD


)

(3.50)

Proof. We can obtain this by applying the operator ∂p

∂ηpto (3.33) and arguing as in

the proof of Lemma 3.3.2


Lemma 3.3.4.

1

2(1

4)∂

∂t

∫Ωk

Jk

(∂2puDk∂ξp∂ηp

)2

dΩk = +∂2puD

∂ξp∂ηpdp+1hL(ξ)

dξp+1

( p∑j=0

∆Lj

∂plj(η)

∂ηp

)︸︷︷︸

B1

− ∂2puD

∂ξp∂ηpdp+1hR(ξ)

dξp+1

( p∑j=0

∆Rj

∂plj(η)

∂ηp

)︸︷︷︸

B2

+∂2puD

∂ξp∂ηpdp+1hL(η)

dηp+1

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

)︸︷︷︸

B3

− ∂2puD

∂ξp∂ηpdp+1hR(η)

dηp+1

( p∑j=0

∆Tj

∂plj(ξ)

∂ξp

)︸︷︷︸

B4

(3.51)

Proof. Multiply (3.33) by Jk and apply ∂2p

∂ξp∂ηpto the equation to get

Jk∂

∂t


)= −

:0

∂2p

∂ξp∂ηp(Jk∇ · fDk )− ∂2p

∂ξp∂ηp(Jk∇ · fCk ) = − ∂2p

∂ξp∂ηp(Jk∇ · fCk )

(3.52)

Multiply the above equation by∂2puDk∂ξp∂ηp

and integrate over Ωk to get

1

2

∂

∂t

∫Ωk

Jk


)2

dΩk = −∫Ωk


)∂2p

∂ξp∂ηp(Jk∇ · fCk )dΩk (3.53)

Transforming the RHS to the reference domain, we get

Jk2

∂

∂t

∫Ωk


)2

dΩk = −∫

ΩS


)∂2p

∂ξp∂ηp(∇ · fC)dΩS (3.54)

Upon substituting the expression for fC and noting the integrands are essentially


constants, we get (3.51). We can now move on to state the main result of this

paper.


Theorem 3.3.5. If the FR scheme for a 2D conservation law with periodic

boundary conditions is used in conjunction with the Lax-Friedrichs formulation

for the common interface flux

f ∗ = fD+λ

2

(max

u∈[uD− ,uD+ ]

∣∣∣∣∂f∂u · n∣∣∣∣)[[uD]] (3.55)

with 0 ≤ λ ≤ 1, and if a non-negative value of the VCJH parameter c is used,

then it can be shown that for a linear advective flux and any Cartesian mesh,

the following holdsd

dt‖uD‖2

W 2p,2δ

≤ 0 (3.56)

for a partial Sobolev norm defined as follows

‖uD‖2W 2p,2δ

=N∑k=1

∫Ωk

[(uDk )2 +

c

2

((∂puDk∂ξp

)2

+

(∂puDk∂ηp

)2)+c2

4


)2]dΩk

(3.57)

Note: For brevity of proof, we discuss the properties of the above norm in

Appendix A.

Proof. Multiply (3.41) and (3.50), i.e., Lemmas 3.3.2 and 3.3.3 by c, (3.51), i.e.,

Lemma 3.3.4 by c2 and add them to (3.32). Note that terms that are marked


(A1, A2, A3, A4, B1, B2, B3, B4) cancel out and we get

1

2

d

dt

(∫Ωk

[Jk(u

Dk )2 +

c

2Jk

((∂puDk∂ξp

)2

+

(∂puDk∂ηp

)2)+c2

4Jk


)2]dΩk

)

=−∫

ΩS


uD(fC · n)dΓS

− c1∫

−1

∂puD

∂ηp∂p

∂ηp

(∂FD

∂ξ

)dξ − c

1∫−1

∂puD

∂ξp∂p

∂ξp

(∂GD

∂η

)dη

− c∂puD

∂ξp

∣∣∣∣η=−1

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

)− c∂

puD

∂ξp

∣∣∣∣η=1

( p∑j=0

∆Tj

∂plj(ξ)

∂ξp

)

− c∂puD

∂ηp

∣∣∣∣ξ=−1

( p∑j=0

∆Lj

∂plj(η)

∂ηp

)− c∂

puD

∂ηp

∣∣∣∣ξ=1

( p∑j=0

∆Rj

∂plj(η)

∂ηp

)(3.58)

Now let us consider the 3rd term of the RHS of (4.58) which can be written as follows

for a linear flux (note that we leave out the factor c in order to just focus on the


algebraic manipulations)

−1∫

−1

∂puD

∂ηp∂p

∂ηp(∂FD

∂ξ

)dξ = −

1∫−1

∂puD

∂ηp∂

∂ξ

(∂pFD

∂ηp

)dξ

= −a1∫

−1

∂puD

∂ηp∂

∂ξ

(∂puD

∂ηp

)dξ

= − a2

1∫−1

∂

∂ξ

(∂puD

∂ηp∂puD

∂ηp

)dξ

= −1

2

1∫−1

∂

∂ξ

(∂puD

∂ηp∂pFD

∂ηp

)dξ

= −1

2

(∂puD

∂ηp∂pFD

∂ηp

)∣∣∣∣ξ=1

ξ=−1

= −1

2

(∂puD

∂ηp∂pFD

∂ηp

)∣∣∣∣ξ=1

+1

2

(∂puD

∂ηp∂pFD

∂ηp

)∣∣∣∣ξ=−1

= −1

2

(∂puD

∂ηp∂p(fD · n)

∂ηp

)R

− 1

2

(∂puD

∂ηp∂p(fD · n)

∂ηp

)L

(3.59)

Remark 3.3.2. In the above argument we have used the fact that we have a linear

advective flux fDk = J−1k auD. If J−1k a is written as a, and a = [a b]T , then we have

FD = auD. In the final step, we use the fact that on the left boundary, nL = [−1 0]T ,

which implies FDL = −(fD · n)L


Similarly the 4th term of the RHS of (4.58) can be written as

−1∫

−1

∂puD

∂ξp∂p

∂ξp(∂GD

∂η

)dη = −1

2

1∫−1

∂

∂η

(∂puD

∂ξp∂pGD

∂ξp

)dη

= −1

2

(∂puD

∂ξp∂pGD

∂ξp

)∣∣∣∣η=1

η=−1

= −1

2

(∂puD

∂ξp∂p(fD · n)

∂ξp

)T

− 1

2

(∂puD

∂ξp∂p(fD · n)

∂ξp

)B

(3.60)

Now note that ∆L = ((f ∗− fD) · n)L. Therefore we can rewrite the 7th term of (4.58)

as follows

∂puD

∂ηp

∣∣∣∣ξ=−1

( p∑j=0

∆Lj

∂plj(η)

∂ηp

)=

(∂puD

∂ηp∂p

∂ηp((f ∗ − fD) · n

))L

(3.61)

We can write similar expressions for the other terms on the RHS of (4.58). Therefore

when we substitute (3.59), (3.60) and these above results into (4.58), we get

1

2Jkd

dt

(∫Ωk

(uDk )2dΩk +c

2

∫Ωk

((∂puDk∂ξp

)2

+

(∂puDk∂ηp

)2)dΩk +

c2

4

∫Ωk


)2

dΩk

)

= −∫

ΩS


uD(fC · n)dΓS

+ c

([1

2

∂puDR∂ηp

∂p(fD · n)R∂ηp

− ∂puDR∂ηp

∂p(f · n)∗R∂ηp

]+

[1

2

∂puDL∂ηp

∂p(fD · n)L∂ηp

− ∂puDL∂ηp

∂p(f · n)∗L∂ηp

]+

[1

2

∂puDT∂ξp

∂p(fD · n)T∂ξp

− ∂puDT∂ξp

∂p(f · n)∗T∂ξp

]+

[1

2

∂puDB∂ξp

∂p(fD · n)B∂ξp

− ∂puDB∂ξp

∂p(f · n)∗B∂ξp

])


Transforming the RHS of the above equation to the physical domain, we get,

1

2Jkd

dt

(∫Ωk

(uDk )2dΩk +c

2

∫Ωk

((∂puDk∂ξp

)2

+

(∂puDk∂ηp

)2)dΩk +

c2

4

∫Ωk


)2

dΩk

)

= −Jk∫Ωk

uD(∇ · fD)dΩk − Jk∫Γk

uD(fC · n)dΓk

+ c

(J2p+1yk

Jk

[1

2

∂puDR∂yp

∂pFDR

∂yp− ∂puDR

∂yp∂p(f · n)∗R

∂yp

]k

+ J2p+1yk

Jk

[− 1

2

∂puDL∂yp

∂pFDL

∂yp− ∂puDL

∂yp∂p(f · n)∗L

∂yp

]+ J2p+1

xkJk

[1

2

∂puDT∂xp

∂pGDT

∂xp− ∂puDT

∂xp∂p(f · n)∗T

∂xp

]+ J2p+1

xkJk

[− 1

2

∂puDB∂xp

∂pGDB

∂xp− ∂puDB

∂xp∂p(f · n)∗B

∂xp

])It is clear from the above equation that Jk cancels across all the terms. After cancel-

lation of Jk, we sum over all the elements to get

d

dt‖uD‖2 = Θadv + cΘextra

where

‖uD‖2 =N∑k=1

(∫Ωk

[(uDk )2 +

c

2

((∂puDk∂ξp

)2

+

(∂puDk∂ηp

)2)+c2

4


)2]dΩk

)

is a broken Sobolev norm of the solution in the entire domain,

Θadv =N∑k=1

(−∫Ωk

uD(∇ · fD)dΩk −∫Γk

uD(fC · n)dΓk

)


and

Θextra =N∑k=1

(J2p+1yk

[1

2

∂puDR∂yp

∂pFDR

∂yp− ∂puDR


∂yp

]k

+ J2p+1yk

[− 1

2

∂puDL∂yp

∂pFDL

∂yp− ∂puDL


∂yp

]+ J2p+1

xk

[1

2

∂puDT∂xp

∂pGDT

∂xp− ∂puDT


∂xp

]+ J2p+1

xk

[− 1

2

∂puDB∂xp

∂pGDB

∂xp− ∂puDB


∂xp

])Θadv represents the summation of the first two terms in (3.3) over all the elements.

This summation over elements of the two terms can be rewritten as a summation over

all element boundaries as shown by Castonguay et al. [38]. These terms arise while

applying the FR procedure to the linear advection equation on triangles, and they

have shown this term to be non-positive, i.e., Θadv ≤ 0. Although they proved this

on triangles, since the summation over elements can be converted to one over element

boundaries, the argument remains exactly the same for the case of quadrilaterals and

we therefore omit this proof.

Remark 3.3.3. If c = 0 as in the case of the DG-recovering FR approach, then the

contribution of the Θextra term is zero, therefore guaranteeing stability.

Remark 3.3.4. Note that∂puDL∂ηp

=∂puDξ=−1

∂ηp

is the pth degree edge derivative of the 1D polynomial formed on the flux points using

the extrapolated values of uD on the left boundary of the reference domain. Since

uDξ=−1 is a pth degree polynomial in η, its pth derivative with respect to η is a constant.

Similar arguments can be made about the flux derivatives in the term Θextra as well.

This expression can be rewritten as a sum over all the edges instead of all the elements.

Consider one such summation along an interior vertical edge. Let − and + subscripts

denote the element on the left and right. For the element on the left, this edge is its


right boundary and for the right element, it is the left boundary. Also, note that for

a Cartesian mesh with no mortar elements, the Jy for these left and right elements

are the same, since it is the edge length of their common boundary. Adding the 2

terms coming from each element from this edge, we get

J2p+1y

[1

2

∂puD−∂yp

∂pFD−

∂yp−∂puD−∂yp

∂p(f · n)∗−∂yp

− 1

2

∂puD+∂yp

∂pFD+

∂yp−∂puD+∂yp

∂p(f · n)∗+∂yp

](3.62)

In the Cartesian case, these are just the areas of the left and right elements respec-

tively. Now, we use the fact that f is a linear advective flux, i.e. FD = auD and

GD = buD. Also, from the definition of the Lax-Friedrichs flux, we have,

(f · n)∗− =1

2a(uD− + uD+) +

λ

2|a|(uD− − uD+) (3.63)

(f · n)∗+ = −1

2a(uD− + uD+) +

λ

2|a|(uD+ − uD−) (3.64)

Substituting these results in (3.62)), we get

J2p+1y

[− λ

2|a|(∂puD−∂yp

∂puD−∂yp

−∂puD−∂yp

∂puD+∂yp

)− λ

2|a|(∂puD+∂yp

∂puD+∂yp

−∂puD−∂yp

∂puD+∂yp

)](3.65)

which can be further simplified as

J2p+1y

[− λ

2|a|(∂puD−∂yp

−∂puD+∂yp

)(∂puD−∂yp

−∂puD+∂yp

)](3.66)

This reduces to

J2p+1y

[− λ

2|a|(∂puD−∂yp

−∂puD+∂yp

)2](3.67)

From this equation, we can see that the terms contributing to Θextra are non-positive.

The same is true for Θadv and therefore, for any c > 0, we have

d

dt‖uD‖2 = Θadv + cΘextra ≤ 0 (3.68)

which proves Theorem 3.5.


3.4 Insights Gained from Stability Analysis

The stability analysis in 1D as well as on triangles and tetrahedra give a result of the

formd

dt‖uD‖2 = Θadv ≤ 0

for suitable norms of the solution, i.e., the VCJH parameter does not appear explicitly.

Therefore, although the conditions for obtaining stable scheme are established from

the analysis and correction functions are designed to satisfy such conditions, the

dependence of the stability or the dissipation properties of the schemes on the VCJH

parameter is not evident from the stability analysis. However, from (3.68), we can

see that there is an explicit dependence of the evolution of the broken Sobolev norm

of the solution on the VCJH parameter c. This allows us to gain some major insights

about the stability and dissipation properties of the schemes which have been studied

numerically in 1D:

1. Since the Θextra term in (3.68) is non-positive, increasing c increases the negative

contribution from this term and therefore causes a faster decay of energy in the

Sobolev norm. For the case of central fluxes, i.e., λ = 0, we have Θextra = 0

(since all terms of Θextra reduce to the form of (3.67)). Additionally, since the

Θadv term is similar to those obtained on triangles (and in 1D), using results from

Vincent et al. [29] and Castonguay et al. [38], we can conclude that Θadv = 0

when a central flux is used. Therefore, for the case of a central flux, the energy

in the Sobolev norm does not change with time.

2. The same argument also shows that the schemes become less stable when c is

decreased. In particular, when c < 0, the second term on the RHS of (3.68) will

provide an unstable (positive) contribution and there is a competing effect of

the two terms Θextra and Θadv and can result in instability. In 1D, the stability

analysis guarantees energy stability for c > c−, a small negative value above

which the broken Sobolev norm in 1D is guaranteed to be a norm. However,

it does not suggest that the schemes become less stable as c is decreased below


this value. But the 2D stability analysis in this chapter provides this additional

insight which is also supported by numerical experiments in the following sec-

tion. Here again, when central fluxes are used, the contribution from the Θextra

term vanishes.

3.5 Numerical Experiments

In this section, we use numerical simulations to support our analysis in the previous

sections. We show results for both upwind and central fluxes and provide intuition for

some similar numerical results obtained in 1D. In the following numerical experiments,

the computational domain is chosen as Ω =

(x, y)|−5 ≤ x, y ≤ 5

. A 20×20 uniform

Cartesian quadrilateral mesh is used for the computations. The advection velocity is

chosen as a = [1 1]T . The initial condition is a centered Gaussian bump, i.e.,

u(x, 0) = e−(x2+y2)

A periodic boundary condition is used and the time period is 10. A fourth order

Runge Kutta method (RK44) with a constant time-step (dt = 0.01) is used for time-

stepping.

3.5.1 Upwind Flux

A fully upwind formulation for the common numerical flux at the element interfaces is

obtained by setting λ = 1 in the Local Lax Friedrichs formulation (3.18). Figure 3.3

shows the evolution of the L2 energy of the solution with time is shown upto 20

periods. It is clear from the rate of decay of the energy that the amount of dissipation

increases with c, as concluded from the stability analysis (See Section 3.4).


0 5 10 15 201.242

1.244

1.246

1.248

1.25

1.252

1.254

1.256

t/T

L2energy

ofuD

c−/2

cDGcSDcHU

c = 0.01

Figure 3.3: Plot of the evolution of the L2 norm of the solution from t = 0 to t = 20 Tfor the FR approach using upwind interface fluxes for different values of the VCJHparameter c


3.5.2 Central Flux

0 5 10 15 201.25

1.251

1.252

1.253

1.254

1.255

1.256

1.257

1.258

t/T

L2energy

ofuD

c−/2

cDGcSDcHU

c = 0.01

0 0.2 0.4 0.6 0.8 11.2533

1.2533

1.2533

1.2533

1.2533

1.2533

t/T

L2energy

ofuD

0.5 0.52 0.54 0.56 0.58 0.6

1.2533

1.2533

1.2533

1.2533

1.2533

1.2533

1.2533

1.2533

1.2533

1.2533

1.2533

t/T

L2energy

ofuD

Figure 3.4: Plots of the evolution of the L2 norm of the solution for the FR approachusing central interface fluxes for different values of the VCJH parameter c. Thebottom two plots are zoomed in versions of the plot at the top

A central formulation for the common numerical flux at the element interfaces is

obtained by setting λ = 0 in the Local Lax Friedrichs formulation (3.18). Figure 3.4

shows the evolution of the L2 energy of the solution with time is shown upto 20

periods along with zoomed in versions to clearly illustrate the trend with the VCJH

parameter c. In this case, as mentioned earlier in Section 3.4, the energy in the


Sobolev norm stays constant with time. Here we actually see that the L2 energy of

the solution oscillates. The trend with the VCJH parameter is opposite to the one

obtained for the upwind case. Such an oscillatory behavior of the L2 energy does not

contradict the stability results of the previous sections. While the stability analysis

guarantees that the energy in the Sobolev norm is constant and this does guarantee

that the L2 energy of the solution does not blow up, an exchange of energy between

the solution and its derivatives is possible leading to such oscillations.

Finally, as c is decreased below zero, the scheme is rendered unstable. This depends on

the particular problem and for the particular set of parameters used in this problem,

the scheme becomes unstable at approximately c = c− − 10−4 = −0.0014.

3.6 Conclusions

An investigation of the linear stability of the FR approach on quadrilaterals has been

performed for the first time. In the case of triangles, a straightforward extension of

the one dimensional FR approach was not possible, and hence a family of energy

stable FR schemes (ESFR) were built particularly for that case by Castonguay et

al. [38]. In contrast, for quadrilaterals, the tensor product nature of the geometry

allows for a simple extension of the 1D FR process as discussed in this paper. However,

a similar extension of the 1D stability analysis was not possible due to certain key

difficulties which required the formulation of a new norm and handling of additional

terms compared to 1D or triangles.

The stability analysis shows that the FR approach based on a tensor product for-

mulation employing VCJH schemes are stable, atleast on Cartesian meshes whenever

the VCJH parameter is non-negative. For upwind fluxes, as c decreases below zero,

the scheme becomes less stable and eventually turns unstable. For central fluxes,

the energy in the Sobolev norm is always constant, irrespective of c (as long as c is

larger than c−, below which the modified Sobolev norm is not a norm). All these

observations and stability results do not depend on the exact location of the solution


points.

Finally, an extension of this analysis of stability to the three dimensional case of

hexahedral Cartesian meshes is fairly straightforward since the ideas used for the

modification of the 1D or triangle norm can be used directly for the extension to

three dimensional tensor product elements.

Chapter 4

Stability of the FR Approach for

Linear Advection-Diffusion

Equation on Tensor Product

Elements

4.1 FR Approach on Quadrilaterals for a Second

Order PDE

4.1.1 Preliminaries

Consider the 2D conservation law

∂u

∂t+ ∇ · f(u,∇u) = 0 in Ω , (4.1)

60

CHAPTER 4. FR STABILITY FOR ADVECTION-DIFFUSION EQUATION 61

where Ω is a bounded connected subset of R2 with boundary Γ composed of a finite

union of parts of hyperplanes. Further, f is a linear flux of the form

f = au− b∇u with f =

(F

G

), a =

(ax

ay

)and b > 0. (4.2)

Consider a partition TN of Ω into N non-empty, non-overlapping, conforming quadri-

lateral elements Ωk with boundaries Γk such that Γk =4⋃i=1

F ik where F ik are straight

lines representing the faces (or edges) of the element Ωk. Furthermore, we restrict

ourselves to non-mortar elements, i.e., if F ik ∩ Γk′ 6= ∅ for k′ 6= k, then F ik ∩ Γj =

∅, ∀j 6= k, k′ and F ik ∩ Γ = ∅.

In the Flux Reconstruction approach, this equation is solved as a system of two

equations

∂u

∂t+ ∇ · f(u, q) = 0 (4.3)

q −∇u = 0 (4.4)

where f = au− bq. The first term is the advective part of the flux, while the second

represents the diffusive flux. i.e. fadv = au and fdif = −bq.

To facilitate a uniform implementation of the method, each element Ωk can be mapped

to a square reference domain defined by ΩS =

(ξ, η)| − 1 ≤ ξ, η ≤ 1

as follows:

xk = Θk(ξ, η) =4∑i=1

Ni(ξ, η)vik (4.5)

Here xk represents the physical coordinates (x, y) of an arbitrary point in the el-

ement Ωk, vik denote the physical coordinates of the 4 vertices of Ωk and Ni(ξ, η)

are bilinear shape functions defined on ΩS. Figure 4.1 shows an example of such a

mapping. Further, let the Jacobian matrix associated with Θk be denoted by Jk and

its determinant by Jk. Jk varies from point to point within an element for a general

linear quadrilateral, unlike linear simplex elements.


(−1,−1) (1,−1)

(1, 1)(−1, 1)

v1k

v2k

v3k

v4k

x

y

ξ

η

Figure 4.1: Mapping between the physical domain (on the left) and the referenceelement (on the right)

In addition, we also transform the physical quantities u, q and f to the reference

domain. The transformation equations for these quantities as well as a few others

which will assist in our stability proof are as follows:

uk = Jkuk (4.6)

qk = ∇u = JkJTk qk (4.7)

fk = JkJ−1k fk (4.8)

∇ · fk = Jk∇ · fk (4.9)

fk · qk = J2kfk · qk (4.10)

∫ΩS

fk · qkdΩS = Jk

∫Ωk

fk · qkdΩk (4.11)


∫ΓS

uk(fk · n)dΓS =

∫Γk

Jkuk(fk · n)dΓk (4.12)

Since we restrict ourselves to rectangular Cartesian meshes while discussing the sta-

bility of the schemes, it is worthwhile to note that the Jacobian matrix is a constant

for each element in such a mesh. We will further introduce some additional notation

to simplify the algebra. For rectangular Cartesian meshes we have

∂xk∂η

=∂yk∂ξ

= 0 (4.13)

Let Jxk = ∂xk∂ξ

and Jyk = ∂yk∂η

. We then have

Fk = JykFk; Gk = JxkGk; uk = JxkJykuk = Jkuk (4.14)

Using these equations we can see that the conservation law can be written in the

reference domain as follows∂uD

∂t+ ∇ · f = 0 (4.15)

qD − ∇u = 0 (4.16)

4.1.2 FR Procedure

In this section, we provide the details of the FR procedure used to solve the 2D

linear advection-diffusion equation. For a detailed description of the procedure in

1D, see [42]. As mentioned in the previous section, we solve it as a system of two

equations, i.e., (4.15) and (4.16). For constructing a scheme of (p+1)th order accuracy,

we start with a set of (p+ 1)2 points on the reference domain referred to as solution

points. There are several choices for the location of the solution points in the reference

domain. A possible choice is the tensor product of p + 1 1D Gauss-Legendre points.


Figure 4.2: Figure showing the solution and flux points in the reference element fora p = 3 scheme

In the FR schemes employed by us, we always use a tensor product of a choice of

1D points as our solution points for the 2D quadrilateral case. Therefore, we only

consider such a tensor product choice, although the location of the 1D points used to

generate this does not affect our proof.

The transformed solution within each element, i.e., uk is then represented using a

tensor product of pth degree Lagrange polynomial bases defined on the 1D points,

i.e.,

uD =

p∑i=0

p∑j=0

uDij li(ξ)lj(η) , (4.17)

where li(ξ) and lj(η) are the 1D Lagrange polynomials associated with the solution

points ξi and ηj respectively and uDij is the value of the transformed solution at (ξi, ηj).

Note that we have dropped the subscript k in order to keep the notation from getting

clumsy. Since uD is a transformed quantity, it is understood to be associated with

a certain generic element Ωk. Also, similar to a Discontinuous Galerkin method, we

allow our solution u to be discontinuous across the elements. Therefore, we represent

such discontinuous quantities with a superscript D.

We also have p+1 flux points along each boundary edge of the quadrilateral element.


These flux points are chosen to align with the solution points in the reference domain,

i.e., we will choose them to be the 1D Gauss-Legendre points along each edge if we

are using such solution points. Figure 4.2 shows the solution and flux point locations

in the transformed domain for a scheme of third order accuracy (p = 3).

At each timestep, we start with the discontinuous solution in the element. We then

compute the gradient of the solution (∇uD) by taking the gradient of (4.17). However,

this gradient is of degree p − 1. In order to obtain a gradient of degree p, we then

correct this gradient by using the information from the neighboring elements. The

first step in this regard is to extrapolate the solution uD to the flux points on the edges.

This extrapolation is performed by evaluating the appropriate Lagrange polynomials

at the element boundaries. For example, the interface solution at the left interface is

obtained as

uDL =

p∑i=0

p∑j=0

uDij li(ξ = −1)lj(η) , (4.18)

where uDL is a 1D polynomial (in η) representing the solution on the flux points on the

left boundary. We simultaneously perform a similar extrapolation in the neighboring

elements. We then transform this solution from the reference domain to the physical

domain in each element. Finally the common numerical solution at an interface is

computed using an appropriate Riemann solver. Common choices for this include

Central Flux (CF), Local Discontinuous Galerkin (LDG), Compact Discontinuous

Galerkin (CDG), Internal Penalty,Bassi Rebay 1 (BR1), and Bassi Rebay 2 (BR2)

approach. The LDG scheme in particular is identical to CDG in 1D and can recover

CF and BR1 schemes in 1D, 2D and 3D and we use the LDG scheme in this paper

for our proof. In the LDG method the common solution u∗ at the jth flux point on

an interface is obtained as

u∗f,j = uDf,j − β · [[uDf,j]] (4.19)

where

u =uin + uout

2& [[u]] = uinnin + uinnout


and β is a directional parameter that allows upwinding or downwinding. Here the

subscript ‘in’ refers to the element we are currently computing in and ‘out’ refers to

the neighboring element. n refers to the normal at the interface pointing outward

from the (appropriate) element. Finally we transform u∗ back to the reference domain

to obtain u∗.

The next step involves computing the correction to the gradient which is added to

∇uD to obtain qD, i.e.,

qD = ∇uD + ∇uC (4.20)

This correction to the gradient denoted by ∇uC is given by

(∇uC)ξ =dhL(ξ)

dξ

p∑j=0

(u∗ − uD)Lj lj(η) +dhR(ξ)

dξ

p∑j=0

(u∗ − uD)Rj lj(η) (4.21)

(∇uC)η =dhL(η)

dη

p∑j=0

(u∗ − uD)Bj lj(ξ) +dhR(η)

dη

p∑j=0

(u∗ − uD)Tj lj(ξ) (4.22)

where if one wants to recover the VCJH schemes, hL and hR denote the left and right

1D VCJH correction functions of degree p + 1 respectively. L,R,B, T represent the

left (ξ = −1), right (ξ = 1), bottom (η = −1) and top (η = 1) edges respectively.

(.)Lj denotes the value at the jth flux point on the left boundary. Finally, lj denotes

the jth member of the 1D Lagrange basis of degree p defined on the edge and the

summation is over the flux points on the corresponding edge. Note that we have used

hL and hR as the correction functions for η-component ( (4.22)) as well because the

correction along the η direction is performed in the same 1D sense as that in the ξ

direction.

It can be seen that the correction in (4.21) and (4.22) is equivalent to performing the

correction along 1D lines in both the horizontal and vertical directions. Note that,

unlike in 1D, this correction is not directly applied to the solution u, i.e., we do not

make u continuous at all the boundaries. We only need to ensure that the normal

component of the corrected gradient qD is continuous at element interfaces. Therefore


we apply the correction to the gradient directly without ever changing uD.

Before we proceed, it is worthwhile mentioning that qD is a pth degree polynomial

and is represented in a similar fashion to uD, i.e.,

qD =

p∑i=0

p∑j=0

qDij li(ξ)lj(η) , (4.23)

We now repeat this correction procedure for the flux. Here, we first start by extrap-

olating the corrected gradient to the element boundaries. We then transform this

gradient to the physical domain using (4.7). Similarly, we transform the extrapolated

solution at the boundaries obtained in the previous step to the physical domain us-

ing (4.6). We then calculate the flux in the physical domain f(uD, qD) at all the flux

points. Using the calculated flux at the boundaries from the neighboring elements, we

compute a common numerical flux by employing an appropriate method. In general,

the advective and diffusive parts of the flux are handled separately through different

approximate Riemann solvers. For linear problems like the linear advection-diffusion

equation, a common choice for the advective flux is the Lax Friedrichs (LF) flux. The

common numerical flux using the LF method is given by

f ∗adv = fDadv+λ

2

(max

u∈[uDin,uDout]

∣∣∣∣∂fadv∂u· n∣∣∣∣)[[uD]] (4.24)

where . and [[.]] operators are similar to those of (4.19) and 0 ≤ λ ≤ 1. For the

diffusive flux, we can use one of the aforementioned methods used for computing a

common solution. If we elect to use the LDG method, then the common numerical

flux is given by

f ∗dif = fdif+ τ [[u]] + β[[fdif ]] (4.25)

where the jump operator for the flux is given by

[[f ]] = fin · nin + fout · nout

and τ is a penalty parameter that penalises the jump in the solution.


Note that we have dropped the subscript ‘dif’ above for brevity of notation. We

finally transform f ∗ back to the reference domain to obtain f ∗. For example, on the

left boundary we can do this using

(f ∗ · n)Lj = JLj(f∗ · n)Lj (4.26)

where JLj is the edge-Jacobian at the jth flux point on the left boundary. The edge-

Jacobian is an edge-based scaling factor which is just equal to the edge length in the

Cartesian case. Therefore (4.26) can be rewritten for the case of Cartesian meshes as

(f ∗ · n)Lj = Jy(f∗ · n)Lj (4.27)

where Jy is the edge length of the left (and right) edge.

Then we compute the discontinuous flux fD at all the solution points using uD and qD

and then extrapolate this flux to the boundaries. This fD is a pth degree polynomial

and is represented using a Lagrangian basis similar to (4.23). We then calculate a

correction to the flux fC and add it to fD to get the corrected flux which is continuous

in the normal direction at the element boundaries, i.e.,

f = fD + fC (4.28)

The correction components are computed using the following equations:

FC = −hL(ξ)

p∑j=0

((f ∗ − fD) · n

)Ljlj(η) + hR(ξ)

p∑j=0

((f ∗ − fD) · n

)Rjlj(η) (4.29)

GC = −hL(η)

p∑j=0

((f ∗ − fD) · n

)Bjlj(ξ) + hR(η)

p∑j=0

((f ∗ − fD) · n

)Tjlj(ξ) (4.30)


Also, for brevity of notation, we let

∆Lj =((f ∗ − fD) · n

)Lj

∆Rj =((f ∗ − fD) · n

)Rj

∆Bj =((f ∗ − fD) · n

)Bj

∆Tj =((f ∗ − fD) · n

)Tj

(4.31)

∆uLj

= (u∗ − uD)Lj ∆uRj

= (u∗ − uD)Rj

∆uBj

= (u∗ − uD)Bj ∆uTj

= (u∗ − uD)Tj(4.32)

Note that we can finally calculate the transformed solution at the next time step in

the kth element using∂uDk∂t

= −∇ · fk (4.33)

4.2 Proof of Stability of the FR Approach on

Quadrilaterals

In this section, we prove that the FR approach for the linear advection-diffusion

equation on Cartesian meshes is stable when the VCJH parameter c ≥ 0. We prove

this for particular but common choices for the common numerical solution and fluxes

at the interfaces. We begin by stating and proving several intermediate results in the

form of Lemmas and finally combining them to obtain our stability result.


Lemma 4.2.1.

Jk

∫Ωk

fDdif,k · qDk dΩk =

∫ΩS

fDdif · ∇uDdΩS +

∫ΓS

∆u(fDdif · n)dΓS

−∫

ΩS

∂FDdif

∂ξhL(ξ)

p∑j=0

∆uLjlj(η)dΩS

︸︷︷︸A1

−∫

ΩS

∂FDdif

∂ξhR(ξ)

p∑j=0

∆uRjlj(η)dΩS

︸︷︷︸A2

−∫

ΩS

∂GDdif

∂ηhL(η)

p∑j=0

∆uBjlj(ξ)dΩS

︸︷︷︸A3

−∫

ΩS

∂GDdif

∂ηhR(η)

p∑j=0

∆uTjlj(ξ)dΩS

︸︷︷︸A4

(4.34)

Proof. Using (4.11) and (4.20), we have

Jk

∫Ωk

fDdif,k · qDk dΩk =

∫ΩS


∫ΩS

fDdif · ∇uCdΩS (4.35)

Consider the second term on the RHS of of (4.35). Using (4.21) and (4.22), we get∫ΩS

fDdif · ∇uCdΩS =

∫ΩS

FDdif

dhL(ξ)

dξ

p∑j=0

∆uLjlj(η)dΩS +

∫ΩS

FDdif

dhR(ξ)

dξ

p∑j=0

∆uRjlj(η)dΩS

+

∫ΩS

GDdif

dhL(η)

dη

p∑j=0

∆uBjlj(ξ)dΩS +

∫ΩS

GDdif

dhR(η)

dη

p∑j=0

∆uTjlj(ξ)dΩS

(4.36)


Now using integration by parts on the first term in the RHS of (4.36) and utiliz-

ing (3.23) gives

1∫−1

1∫−1

FDdif

dhL(ξ)

dξ

p∑j=0

∆uLjlj(η)dξdη

=

1∫−1

[FDdifhL(ξ)

]ξ=1

ξ=−1

p∑j=0

∆uLjlj(η)dη −

∫ΩS

∂FDdif

∂ξhL(ξ)

p∑j=0

∆uLjlj(η)dΩS

= −1∫

−1

FDdifL

p∑j=0

∆uLjlj(η)dη −

∫ΩS

∂FDdif

∂ξhL(ξ)

p∑j=0

∆uLjlj(η)dΩS

=

∫ΓS,L

∆u(fDdif · n)dΓS −∫

ΩS

∂FDdif

∂ξhL(ξ)

p∑j=0

∆uLjlj(η)dΩS

(4.37)

We can do similar manipulations to the other three terms of the RHS of (4.36).

Combining them gives Lemma 4.2.1.


Lemma 4.2.2.

Jk

∫Ωk

∂pfDdif,k∂ξp

· ∂pqDk∂ξp

dΩS =

∫ΩS

∂pfDdif∂ξp

· ∂p(∇uD)

∂ξpdΩS

+2

c

∫ΩS

hL(ξ)∂FD

dif

∂ξ

p∑j=0

∆uLjlj(η)dΩS

︸︷︷︸A1

+2

c

∫ΩS

hR(ξ)∂FD

dif

∂ξ

p∑j=0

∆uRjlj(η)dΩS

︸︷︷︸A2

− 2∂pGD

difB

∂ξp

p∑j=0

∆uBj

∂plj(ξ)

∂ξp− 2

1∫−1

∂

∂η

(∂pGD

dif

∂ξp

)hL(η)

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdη

︸︷︷︸B1

+ 2∂pGD

difT

∂ξp

p∑j=0

∆uTj

∂plj(ξ)

∂ξp− 2

1∫−1

∂

∂η

(∂pGD

dif

∂ξp

)hR(η)

p∑j=0

∆uTj

∂plj(ξ)

∂ξpdη

︸︷︷︸B2

(4.38)

Proof. We begin by taking the pth derivative w.r.t. ξ of fDdif,k and qDk and taking their

dot product. Since the Jacobian matrix Jk is a constant in the element Ωk, similar

to (4.10) we get

J2k

∂pfDdif,k∂ξp

· ∂pqDk∂ξp

=∂pfDk∂ξp

· ∂pqDk∂ξp

(4.39)

Integrating this over ΩS gives

Jk

∫Ωk

∂pfDdif,k∂ξp

· ∂pqDk∂ξp

dΩk =

∫ΩS

∂pfDk∂ξp

· ∂pqDk∂ξp

dΩS (4.40)


Using (4.20), (4.21) and (4.22), we get

Jk

∫Ωk

∂pfDdif,k∂ξp

· ∂pqDk∂ξp

dΩk =

∫ΩS

∂pfDdif∂ξp

· ∂p(∇uD)

∂ξpdΩS

+

∫ΩS

∂pFDdif

∂ξpdp+1hL(ξ)

dξp+1

p∑j=0

∆uLjlj(η)dΩS

+

∫ΩS

∂pFDdif

∂ξpdp+1hR(ξ)

dξp+1

p∑j=0

∆uRjlj(η)dΩS

+

∫ΩS

∂pGDdif

∂ξp∂hL(η)

∂η

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdΩS

+

∫ΩS

∂pGDdif

∂ξp∂hR(η)

∂η

p∑j=0

∆uTj

∂plj(ξ)

∂ξpdΩS

(4.41)

Consider the second term on the RHS of (4.41). Note that the integrand is a constant

for the ξ-integral, i.e.,

∫ΩS

∂pFDdif

∂ξpdp+1hL(ξ)

dξp+1

p∑j=0

∆uLjlj(η)dΩS = 2

1∫−1

∂pFDdif

∂ξpdp+1hL(ξ)

dξp+1

p∑j=0

∆uLjlj(η)dη (4.42)

Since FDdif is a tensor product of pth degree Lagrange polynomials in ξ and η, we can

use the third property of the VCJH correction functions, i.e., (3.24) as

∂pFDdif

∂ξpdp+1hL(ξ)

dξp+1=

1

c

1∫−1

∂FDdif

∂ξhL(ξ)dξ (4.43)

Substituting this into (4.42), we get

∫ΩS

∂pFDdif

∂ξpdp+1hL(ξ)

dξp+1

p∑j=0

∆uLjlj(η)dΩS =

2

c

∫ΩS

∂FDdif

∂ξhL(ξ)

p∑j=0

∆uLjlj(η)dΩS (4.44)

The third term on the RHS of (4.41) can be manipulated similarly. Now consider the


fourth term on the RHS of (4.41). Performing integration by parts for the η-integral,

we get

∫ΩS

∂pGDdif

∂ξp∂hL(η)

∂η

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdΩS

=

1∫−1

[∂pGD

dif

∂ξphL(η)

]η=1

η=−1

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdξ

−1∫

−1

1∫−1

∂

∂η

(∂pGD

dif

∂ξp

)hL(η)

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdξdη

= −1∫

−1

∂pGDdif,B

∂ξp

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdξ

− 2

1∫−1

∂

∂η

(∂pGD

dif

∂ξp

)hL(η)

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdη

= −2∂pGD

dif,B

∂ξp

p∑j=0

∆uBj

∂plj(ξ)

∂ξp

− 2

1∫−1

∂

∂η

(∂pGD

dif

∂ξp

)hL(η)

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdη

(4.45)

The fifth term on the RHS of (4.41) can be manipulated similarly. Substituting these

results into (4.41), we get Lemma 4.2.2.


Lemma 4.2.3.

Jk

∫Ωk

∂pfDdif,k∂ηp

· ∂pqDk∂ηp

dΩk =

∫ΩS

∂pfDdif∂ηp

· ∂p(∇uD)

∂ηpdΩS

+2

c

∫ΩS

hL(η)∂GD

dif

∂η

p∑j=0

∆uBjlj(ξ)dΩS

︸︷︷︸A3

+2

c

∫ΩS

hR(η)∂GD

dif

∂η

p∑j=0

∆uTjlj(ξ)dΩS

︸︷︷︸A4

− 2∂pFD

difL

∂ηp

p∑j=0

∆uLj

∂plj(η)

∂ηp− 2

1∫−1

∂

∂ξ

(∂pFD

dif

∂ηp

)hL(ξ)

p∑j=0

∆uLj

∂plj(η)

∂ηpdξ

︸︷︷︸B3

+ 2∂pFD

difR

∂ηp

p∑j=0

∆uRj

∂plj(η)

∂ηp− 2

1∫−1

∂

∂ξ

(∂pFD

dif

∂ηp

)hR(ξ)

p∑j=0

∆uRj

∂plj(η)

∂ηpdξ

︸︷︷︸B4

(4.46)

Proof. The derivation of this Lemma is similar to that of Lemma 4.2.2. We begin by

taking the pth derivative w.r.t. η of fDdif,k and qDk and taking their dot product. Then

we proceed in a similar fashion to Lemma 4.2.2 to obtain the proof.


Lemma 4.2.4.

Jk

∫Ωk

(∂2pfDdif,k∂ξp∂ηp

)·(∂2pqDk∂ξp∂ηp

)dΩk =

4

c

1∫−1

∂

∂ξ

(∂pFD

dif

∂ηp

)hL(ξ)

p∑j=0

∆uLj

∂plj(η)

∂ηpdξ

︸︷︷︸B3

+4

c

1∫−1

∂

∂ξ

(∂pFD

dif

∂ηp

)hR(ξ)

p∑j=0

∆uRj

∂plj(η)

∂ηpdξ

︸︷︷︸B4

+4

c

1∫−1

∂

∂η

(∂pGD

dif

∂ξp

)hL(η)

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdη

︸︷︷︸B1

+4

c

1∫−1

∂

∂η

(∂pGD

dif

∂ξp

)hR(η)

p∑j=0

∆uTj

∂plj(ξ)

∂ξpdη

︸︷︷︸B2

(4.47)

Proof. We begin by taking the ∂2p(.)∂ξp∂ηp

of fDdif,k and qDk and taking their dot product.

Since the Jacobian matrix Jk is a constant in the element Ωk, similar to (4.10) we get

J2k

∂2pfDdif,k∂ξp∂ηp

· ∂2pqDk

∂ξp∂ηp=

∂2pfDk∂ξp∂ηp

· ∂2pqDk

∂ξp∂ηp(4.48)

Integrating this over the domain ΩS we get

Jk

∫Ωk


· ∂2pqDk

∂ξp∂ηpdΩk =

∫ΩS

∂2pfDk∂ξp∂ηp

· ∂2pqDk

∂ξp∂ηpdΩS (4.49)


Using (4.20), (4.21) and (4.22), and noting that ∂2p(ˆ∇uD)

∂ξp∂ηp= ~0 since ∇uD is a poly-

nomial of degree p− 1, we get

Jk

∫Ωk


· ∂2pqDk

∂ξp∂ηpdΩk =

∫ΩS

∂2pFDdif


dξp+1

p∑j=0

∆uLj

∂plj(η)

∂ηpdΩS

+

∫ΩS

∂2pFDdif


dξp+1

p∑j=0

∆uRj

∂plj(η)

∂ηpdΩS

+

∫ΩS

∂2pGDdif


dηp+1

p∑j=0

∆uBj

∂plj(ξ)

∂ξpdΩS

+

∫ΩS

∂2pGDdif


dηp+1

p∑j=0

∆uTj

∂plj(ξ)

∂ξpdΩS

(4.50)

Note that all the integrands in the integrals in the RHS of (4.50) are constants.

Therefore we have

Jk

∫Ωk


· ∂2pqDk

∂ξp∂ηpdΩk =

4∂2pFD

dif


dξp+1

p∑j=0

∆uLj

∂plj(η)

∂ηp

+ 4∂2pFD

dif


dξp+1

p∑j=0

∆uRj

∂plj(η)

∂ηp

+ 4∂2pGD

dif


dηp+1

p∑j=0

∆uBj

∂plj(ξ)

∂ξp

+ 4∂2pGD

dif


dηp+1

p∑j=0

∆uTj

∂plj(ξ)

∂ξp

(4.51)


Consider the first term in the RHS of (4.51). We can use (3.24) to get

4∂2pFD

dif


dξp+1

p∑j=0

∆uLj

∂plj(η)

∂ηp=

4

c

1∫−1

∂

∂ξ

(∂pFD

dif

∂ηp

)hL(ξ)

p∑j=0

∆uLj

∂plj(η)

∂ηpdξ

(4.52)

Performing similar manipulations on the other three terms in the RHS of (4.51) gives

us Lemma 4.2.4.

Lemma 4.2.5.

−bJk‖qDk ‖2 =

∫ΩS


∫ΓS


+ c∑e

[∂p(fDdif · n

)∂φp

∂puD

∂φp

]e︸︷︷︸

C1

− c1∫

−1

∂p

∂ξp(∂GD

dif

∂η

)∂puD∂ξp

dη

︸︷︷︸C2

−c1∫

−1

∂p

∂ηp(∂FD

dif

∂ξ

)∂puD∂ηp

dξ

︸︷︷︸C3

+ c∑e

[∂p(fDdif · n

)∂φp

∂p∆u

∂φp

]e

(4.53)

where the summation over e represents the summation over the 4 edges of the

square domain ΩS. φ = ξ on the top and bottom edges and φ = η on the left

and right edges.

Proof. Multiplying Lemma 4.2.1 by b, Lemmas 4.2.2 and 4.2.3 by bc2

and Lemma 4.2.4


by bc2

4and adding them, and noting that fDdif,k = −bqDk gives

−bJk‖qDk ‖2 =

∫ΩS


∫ΓS


+c

2

∫ΩS

∂pfDdif∂ξp

· ∂p(∇uD)

∂ξpdΩS +

c

2

∫ΩS

∂pfDdif∂ηp

· ∂p(∇uD)

∂ηpdΩS

− c∂pGD

difB

∂ξp

p∑j=0

∆uBj

∂plj(ξ)

∂ξp+ c

∂pGDdifT

∂ξp

p∑j=0

∆uTj

∂plj(ξ)

∂ξp

− c∂pFD

difL

∂ηp

p∑j=0

∆uLj

∂plj(η)

∂ηp+ c

∂pFDdifR

∂ηp

p∑j=0

∆uRj

∂plj(η)

∂ηp

(4.54)

where

‖qDk ‖2 =

∫Ωk

qDk · qDk dΩk

+c

2

∫Ωk

∂pqDk∂ξp

· ∂pqDk∂ξp

dΩk +c

2

∫Ωk

∂pqDk∂ηp

· ∂pqDk∂ηp

dΩk

+c2

4

∫Ωk

(∂2pqDk∂ξp∂ηp

)·(∂2pqDk∂ξp∂ηp

)dΩk

is a Sobolev norm of qD in the element Ωk. We do not formalize this because we

only use the fact that ‖qDk ‖2 ≥ 0 whenever c > 0. The norm definition here has been

used just for brevity of notation. Note that, while writing (4.54), the terms marked

A1, A2, A3, A4, B1, B2, B3, B4 get cancelled out.

Now let us consider the 3rd term on the RHS of (4.54).

c

2

∫ΩS

∂pfDdif∂ξp

· ∂p(∇uD)

∂ξpdΩS =

c

2

∫ΩS

∂pGDdif

∂ξp∂p

∂ξp(∂uD∂η

)dΩS

= c

1∫−1

∂pGDdif

∂ξp∂p

∂ξp(∂uD∂η

)dη

(4.55)


The first step in (4.55) is due to the fact that ∂p

∂ξp

(∂uD

∂ξ

)= 0, since uD is a polynomial

of degree p. The second step is because the integrand is a constant w.r.t ξ. Applying

integration by parts to the RHS of (4.55) yields

c

2

∫ΩS

∂pfDdif∂ξp

·∂p(∇uD)

∂ξpdΩS

= c

[∂pGD

dif

∂ξp∂puD

∂ξp

]η=1

η=−1

− c1∫

−1

∂p

∂ξp(∂GD

dif

∂η

)∂puD∂ξp

dη

= c

[∂p(fDdif · n

)∂ξp

∂puD

∂ξp

]T

+ c

[∂p(fDdif · n

)∂ξp

∂puD

∂ξp

]B

− c1∫

−1

∂p

∂ξp(∂GD

dif

∂η

)∂puD∂ξp

dη

(4.56)

We can do a similar manipulation on the 4th term on the RHS of (4.54). Now consider

the 5th term on the RHS of (4.54).

−c∂pGD

difB

∂ξp

p∑j=0

∆uBj

∂plj(ξ)

∂ξp= −c

∂pGDdifB

∂ξp∂p∆u

B

∂ξp

= c

[∂p(fDdif · n)

∂ξp∂p∆u

∂ξp

]B

(4.57)

Substituting these results into (4.54), we get Lemma 4.2.5.

Now we quote a result from [47] (see also the Erratum to that paper [48]). Lemmas

3.1,3.2,3.3 and 3.4 in [47] were obtained for the linear advection equation. However,

note that the conservation law (4.3) is the same, with the only difference being that

f is now given by f = fadv + fdif , whereas in [47], f = fadv. This discrepancy does

not make any difference for the arguments used to obtain Lemmas 3.1,3.2,3.3 and 3.4


in [47]. Therefore, quoting Equation (3.28) of [47], we have

1

2Jkd

dt

(‖uD‖2

k

)= −

∫ΩS


uD(fC · n)dΓS

− c1∫

−1

∂puD

∂ηp∂p

∂ηp

(∂FD

∂ξ

)dξ − c

1∫−1

∂puD

∂ξp∂p

∂ξp

(∂GD

∂η

)dη

− c∂puD

∂ξp

∣∣∣∣η=−1

( p∑j=0

∆Bj

∂plj(ξ)

∂ξp

)

− c∂puD

∂ξp

∣∣∣∣η=1

( p∑j=0

∆Tj

∂plj(ξ)

∂ξp

)

− c∂puD

∂ηp

∣∣∣∣ξ=−1

( p∑j=0

∆Lj

∂plj(η)

∂ηp

)

− c∂puD

∂ηp

∣∣∣∣ξ=1

( p∑j=0

∆Rj

∂plj(η)

∂ηp

)

(4.58)

where

‖uD‖2k =

∫Ωk

(uDk )2dΩk +c

2

∫Ωk

((∂puDk∂ξp

)2

+

(∂puDk∂ηp

)2)dΩk +

c2

4

∫Ωk


)2

dΩk

(4.59)

is the Sobolev norm of the solution over element k, i.e., in Ωk. Here again, we use

this for brevity of notation. Since f = fadv + fdif , we can separate out each term in

the above equation into the advective and diffuse parts.

In [47] and its Erratum [48], we have already shown that the advection terms lead to

non-positive contributions ensuring stability. Therefore, let us focus our attention on

the diffusion terms. For the diffusion terms alone, the following lemma holds.


Lemma 4.2.6.

1

2Jkd

dt

(‖uD‖2

k

)= Advection terms

−∫

ΩS

uD(∇ · fDdif )dΩS −∫ΓS

uD(fCdif · n)dΓS

− c1∫

−1

∂puD

∂ηp∂p

∂ηp

(∂FD

dif

∂ξ

)dξ

︸︷︷︸C3

−c1∫

−1

∂puD

∂ξp∂p

∂ξp

(∂GD

dif

∂η

)dη

︸︷︷︸C2

− c∑e

[∂puD

∂φp∂p

∂φp(f ∗dif · n

)]e

+ c∑e

[∂puD

∂φp∂p

∂φp(fDdif · n

)]e︸︷︷︸

C1

(4.60)

Proof. We begin by rewriting (4.58) by focusing only on the diffusion terms as

1

2Jkd

dt

(‖uD‖2

k

)= Advection terms

−∫

ΩS


uD(fCdif · n)dΓS

− c1∫

−1

∂puD

∂ηp∂p

∂ηp

(∂FD

dif

∂ξ

)dξ − c

1∫−1

∂puD

∂ξp∂p

∂ξp

(∂GD

dif

∂η

)dη

− c∂puD

∂ξp

∣∣∣∣η=−1

( p∑j=0

((f ∗dif − fDdif ) · n

)Bj

∂plj(ξ)

∂ξp

)

− c∂puD

∂ξp

∣∣∣∣η=1

( p∑j=0


)Tj

∂plj(ξ)

∂ξp

)

− c∂puD

∂ηp

∣∣∣∣ξ=−1

( p∑j=0


)Lj

∂plj(η)

∂ηp

)

− c∂puD

∂ηp

∣∣∣∣ξ=1

( p∑j=0


)Rj

∂plj(η)

∂ηp

)

(4.61)


Let us look at the 5th diffusion term on the RHS of (4.61):

−c∂puD

∂ξp

∣∣∣∣η=−1

( p∑j=0


)Bj

∂plj(ξ)

∂ξp

)= −c

[∂puD

∂ξp∂p

∂ξp


)]B

= −c[∂puD

∂ξp∂p

∂ξp(f ∗dif · n

)]B

+ c

[∂puD

∂ξp∂p

∂ξp(fDdif · n

)]B

(4.62)

Similar manipulations can be made to the last 3 diffusion terms on the RHS of (4.61).

Substituting these results into (4.61) gives us Lemma 4.2.6.

Theorem 4.2.7. If the FR approach employing VCJH correction functions

is used for solving the 2D linear Advection-Diffusion equation 4.1 with periodic

boundary conditions on a Cartesian mesh, and if the following choices are made

for computing common values at the element interfaces:

1. Local Lax Friedrichs flux (4.24) for the common numerical advective flux

f ∗adv with 0 ≤ λ ≤ 1

2. LDG formulation for the common solution u∗ (See (4.19)) and the com-

mon diffusive flux f ∗dif (See (4.25)) with τ ≥ 0,

then the following holds

d

dt‖uD‖2

W 2p,2δ

≤ 0 when c ≥ 0 (4.63)

for a partial Sobolev norm defined as

‖uD‖2W 2p,2δ

=N∑k=1

∫Ωk

[(uDk )2 +

c

2

((∂puDk∂ξp

)2

+

(∂puDk∂ηp

)2)+c2

4


)2]dΩk

(4.64)

Proof. Subtracting Lemma 4.2.5 from 4.2.6 and noting that the terms marked C1, C2


and C3 get cancelled out, we get

1

2Jkd

dt

(‖uD‖2

k

)+ bJk‖qDk ‖2 = Advection terms

−∫

ΩS


uD(fCdif · n)dΓS

−∫

ΩS

fDdif · ∇uDdΩS −∫ΓS


− c∑e

[∂puD

∂φp∂p


)]e

− c∑e

[∂p(fDdif · n

)∂φp

∂p∆u

∂φp

]e

(4.65)

Now consider the 1st and 3rd diffusion terms on the RHS of the (4.65)

−∫

ΩS

uD(∇ · fDdif )dΩS −∫

ΩS

fDdif · ∇uDdΩS

= −∫

ΩS

∇ · (uDfDdif )dΩS =

−∫ΓS

uD(fDdif · n)dΓS

(4.66)

Now consider the second diffusion term on the RHS of (4.65)

−∫ΓS

uD(fCdif · n)dΓS = −∫ΓS

uD((f ∗dif − fDdif ) · n

)dΓS

= −∫ΓS

uD(f ∗dif · n)dΓS +

∫ΓS

uD(fDdif · n)dΓS

(4.67)


Substituting (4.66) and (4.67) into (4.65), we get

1

2Jkd

dt

(‖uD‖2

k

)+ bJk‖qDk ‖2 = Advection terms

−∫ΓS

uD(f ∗dif · n)dΓS −∫ΓS


− c∑e

[∂puD

∂φp∂p


)]e

− c∑e

[∂p(fDdif · n

)∂φp

∂p∆u

∂φp

]e

(4.68)

Now let us transform the RHS of (4.68) to the physical domain.

1

2Jkd

dt

(‖uD‖2

k

)+ bJk‖qDk ‖2 = Jk(Advection termsphy)

− Jk∫Γk

uDk (f ∗dif,k · n)dΓk − Jk∫Γk

(u∗k − uDk )(fDdif,k · n)dΓk

− cJk∑e

[J2p+1ψk

∂puDk∂ψp

∂p

∂ψp(f ∗dif,k · n

)]e

− cJk∑e

[J2p+1ψk

∂p(fDdif,k · n

)∂ψp

∂p(u∗k − uDk )

∂ψp

]e

(4.69)

where ψ = x on the top and bottom edges and ψ = y on the left and right edges

of the element, and Advection termsphy refers to the version of advection terms after

transformation to the physical domain. Note that Jk cancels throughout the equation.

After cancellation of Jk, we can sum the expression on the LHS of (4.69) over all


elements in the domain Ω to obtain

1

2

d

dt

(‖uD‖2

)=− b‖qD‖2 +

N∑k=1

Advection termsphy

−N∑k=1

∫Γk

uDk (f ∗dif,k · n)dΓk

−N∑k=1

∫Γk

(u∗k − uDk )(fDdif,k · n)dΓk

− cN∑k=1

∑e

[J2p+1ψk

∂puDk∂ψp

∂p

∂ψp(f ∗dif,k · n

)]e

− cN∑k=1

∑e

[J2p+1ψk

∂p(fDdif,k · n

)∂ψp

∂p(u∗k − uDk )

∂ψp

]e

(4.70)

In [47], the contribution of Advection termsphy was shown to be non-positive for any

cartesian mesh and for c ≥ 0. For the rest of the terms, we can see that the summation

over elements can be expressed as a summation over all the edges of the mesh. Let us

consider one generic vertical edge in the mesh and accumulate all contributions from

these additional terms.

We now use the expression for common flux from (4.25) and for the common solution

from (4.19). Let us denote the terms coming from the element on the left side of

the edge by the subscript by − and the ones from the right by +. Consider the

contributions of the first diffusion term in the RHS of (4.70) to a vertical edge:

−∫e

uD−

[(FDdif,− + FD

dif,+

2

)+ τ(uD− − uD+) + βx(F

Ddif,− − FD

dif,+)

]dy

−∫e

uD+

[−(FDdif,− + FD

dif,+

2

)− τ(uD− − uD+)− βx(FD

dif,− − FDdif,+)

]dy

= −∫e

(uD− − uD+)

[(FDdif,− + FD

dif,+

2

)+ τ(uD− − uD+) + βx(F

Ddif,− − FD

dif,+)

]dy

(4.71)


Similarly, contributions of the second diffusion term on the RHS of (4.70) to a vertical

edge are as follows

−∫e

FDdif,−

[(uD− + uD+

2

)− βx(uD− − uD+)− uD−

]dy

−∫e

−FDdif,+

[(uD− + uD+

2

)− βx(uD− − uD+)− uD+

]dy

= −∫e

[(uD+ − uD−

2

)(FD

dif,− + FDdif,+)− βx(uD− − uD+)(FD

dif,− − FDdif,+)

]dy

(4.72)

Combining the contributions from the first two diffusion terms in the RHS of (4.70)

by adding (4.71) and (4.72) gives

−∫e

τ(uD− − uD+)2dy ≤ 0 for τ ≥ 0 (4.73)

For the 3rd and 4th diffusion terms, although there is no integral and there are pth

derivatives, a treatment similar to that done for the 1st and second terms give rise to

identical cancellations and the contribution to a vertical edge from these terms is

−cτJ2p+1y

(∂puD−∂yp

−∂puD+∂yp

)2

≤ 0 for τ ≥ 0 and c ≥ 0 (4.74)

Therefore, since the advection and diffusion terms on the RHS of (4.70) give non-

positive contributions over each edge, their sum is also non-positive. Combining this

with the fact that b > 0 and therefore −b‖qD‖2 is also non-positive, we can conclude

thatd

dt

(‖uD‖2

)≤ 0 (4.75)

thus proving the stability of the FR scheme for the linear advection-diffusion equation

on all cartesian meshes for b > 0, c ≥ 0 and τ > 0.


4.3 Numerical Experiments

0 0.2 0.4 0.6 0.8 11.05

1.1

1.15

1.2

1.25

1.3

t/T

L2en

ergyofuD

c−/2

cDGcSDcHU

c = 0.01c = 10

0.279 0.279 0.2791 0.2791 0.2791 0.2791

1.1884

1.1885

1.1885

1.1885

1.1886

1.1886

1.1887

1.1887

t/T

L2en

ergyofuD

Figure 4.3: Plots of the evolution of the L2 norm of the solution for the FR approachusing upwind interface fluxes for different values of the VCJH parameter c. The ploton the right shows a zoomed version of the one on the left

In this section, we solve the 2D linear Advection-Diffusion equation numerically using

the FR approach to support our analysis in the previous sections. We show results

for both upwind and central fluxes. In the following numerical experiments, the

computational domain is chosen as Ω =

(x, y)| − 5 ≤ x, y ≤ 5

. A 20× 20 uniform

Cartesian quadrilateral mesh is used for the computations. The advection velocity is

chosen as a = [1 1]T and the diffusion coefficient b is chosen to be 0.01. The initial

condition is a centered Gaussian bump, i.e.,

u(x, 0) = e−(x2+y2)

A periodic boundary condition is used and the time period is 10. A fourth order

Runge Kutta method (RK44) with a constant time-step (dt = 0.01) is used for time-

stepping. The LDG formulation parameters used for this simulation are β = 0 and

τ = 1. Figures 4.3 and 4.4 show the plots of the evolution of the L2 energy of the

solution for upwind and central interface fluxes respectively. The physical diffusion

present in the problem dominates the energy dissipation, but trends similar to the

case of the linear advection equation (See 3.5) is observed with respect to the VCJH


parameter c.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.05

1.1

1.15

1.2

1.25

1.3

t/T

L2energy

ofuD

c−/2

cDGcSDcHU

c = 0.01c = 10

0.3356 0.3356 0.3356 0.3356 0.3356 0.3356 0.3356 0.3356

1.1768

1.1768

1.1768

1.1768

1.1768

1.1768

1.1769

t/T

L2energy

ofuD

c−/2

cDGcSDcHU

c = 0.01c = 10

Figure 4.4: Plots of the evolution of the L2 norm of the solution for the FR approachusing central interface fluxes for different values of the VCJH parameter c. The ploton the right shows a zoomed version of the one on the left

4.4 Conclusions

In this chapter, we investigated the stability of the FR approach for solving the

2D linear advection-diffusion equation on Cartesian meshes. The variation of the

transformation Jacobian matrix within each element and the fact that the (p + 1)th

derivative of a pth-degree tensor product formulation does not vanish are two complex-

ities that set apart quadrilateral elements from their 1D and triangular counterparts.

While we circumvent the first difficulty by confining ourselves to Cartesian meshes for

the stability proof, we have solved the second one successfully. In particular, we show

that the FR approach for linear advection-diffusion equation on a periodic domain is

stable on Cartesian meshes whenever the VCJH parameter c ≥ 0.

The trends of dissipation and stability of the schemes with c are similar to those

observed for the advection case although the physical diffusion could potentially

dominate these properties. We also believe that an extension of this analysis to


three dimensional Cartesian meshes with hexahedral elements should be straightfor-

ward.

Part II

Numerical Simulation of

Compressible Flows

91

Preamble

Discontinuous solutions pose a variety of problems to numerical solutions of PDEs,

particularly in the context of high order methods based on polynomial bases. Some of

the problems can be easily seen by considering the approximation of a jump discon-

tinuity by a polynomial basis. From Figure 4.5, there are three major effects brought

by a discontinuity in such polynomial based methods:

1. The order of accuracy (pointwise) is reduced to first order around the point of

discontinuity

2. Pointwise convergence is lost at the point of discontinuity

3. Oscillations are created about the point of discontinuity that are persistent as

the order increases.

This phenomenon is a classic phenomenon known as the Gibbs phenomenon and its

properties have been well understood ( [50, 51, 52]). Along with issue of reduced

accuracy, such oscillations also introduce instabilities in the numerical calculation.

While solving Euler and Navier Stokes equations, these oscillations can also lead to

negative density or pressure and lead to the loss of hyperbolicity of the problem. In

combustion problems, these oscillations could cause the value of a certain quantity

like the temperature to go beyond a critical value, causing false ignition etc.

These problems necessitate special handling of such situations and forms an impor-

tant part of any numerical method for compressible flow calculations. There have

been several shock capturing methods in the context of lower order finite difference

92

93

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

uh

x

P = 8

P = 16

P = 32

P = 64

Exact

(a) Approximation

−1 −0.5 0 0.5 1

10−4

10−3

10−2

10−1

100

101

log

(|u

h −

u|)

x

P = 8

P = 16

P = 32

P = 64

(b) Pointwise error

Figure 4.5: Figure shows the problem arising from representing a step discontinu-ity using a Legendre polynomial basis. (a) shows the polynomial representation forvarious orders. (b) provides a plot of the pointwise error for these approximate rep-resentations. It can be noticed that the convergence rate is reduced to first order

and finite volume methods that have been quite successful. But discontinuous solu-

tions pose significantly higher difficulties for high order finite element methods based

94

on polynomials. One of the major challenges facing such high order methods from

being adopted in the industry is the limited robustness of these methods, particu-

larly in the context of compressible flows. Therefore robust shock capturing methods

which are generalizable to all such high order finite-element based methods is of high

importance.

The work in this part has been presented from the following publications:

• Sheshadri A, Jameson A. A Robust Sub-Cell Shock Capturing Method for the

Numerical Simulation of Compressible Viscous Flows using Unstructured High

Order Methods (In the process of submitting to Journal of Computational

Physics)

• Sheshadri A,Jameson A. Shock Detection and Capturing Methods for High

Order Discontinuous-Galerkin Finite Element Methods [53]

• Sheshadri A, Crabill J, Jameson A. Mesh deformation and shock capturing

techniques for high-order simulation of unsteady compressible flows on dynamic

meshes [54]

• Manuel R. Lopez-Morales, Jonathan Bull, Jacob Crabill, Thomas D. Economon,

David Manosalvas, Joshua Romero, Abhishek Sheshadri, Jerry E. Watkins II,

David Williams, Francisco Palacios, and Antony Jameson. Verification and

Validation of HiFiLES: a High-Order LES unstructured solver on multi-GPU

platforms [31]

Chapter 5

Shock Capturing

5.1 Comparison of shock capturing methods

Several strategies have been used to handle shocks in numerical methods. The most

commonly used among these are Limiting, Artificial Viscosity methods, Filtering and

Reconstruction. In this section, we discuss a few important aspects of these methods

in the context of their application to the FR approach and discuss their advantages

and disadvantages.

5.1.1 Limiting

The concept of limiting has been widely used in finite volume methods with a lot

of success. Limiting deals with modifying or limiting the solution to satisfy certain

properties like Total Variation Diminishing (TVD) or Total Variation Bounded (TVB)

of the solution. The major advantage of these schemes is that they can potentially

totally eliminate oscillations around shocks. However the extension of such limiters

to high order finite element type methods for general unstructured grids becomes

very hard. While there have been several such extensions for structured grids [55,

95

CHAPTER 5. SHOCK CAPTURING 96

56, 57, 58, 59, 60], the extensions to unstructured grids are very complicated and

such methods become highly expensive for implementation on GPUs. But more

importantly, such limiters preclude the possibility of capturing sharp shocks within a

single element as they smear a shock across several elements. This would then require

additional mesh adaptation to get good resolution and the idea of simulating using

coarse meshes with high orders becomes hard. Also, the application of limiters can

severely reduce the accuracy of the solution in smooth regions near local extrema

unless special care is taken to avoid this. But this introduces the need for an apriori

estimation of the second derivative of the solution near smooth local extrema.

5.1.2 Artificial Viscosity Methods

The method of artificial viscosity is based on introducing an additional artificial dis-

sipation term in the PDE and controlling the amount of dissipation based on certain

detectors which try to distinguish between regions of smoothness and those with

shocks. The classic Jameson-Schmidt-Turkel (JST) schemes [61] use such a method

where artificial second and fourth order dissipation terms are added to the Euler

equations. Cook and Cabot [62, 63, 64] also proposed a non-linear artificial diffusion

scheme where artificial grid-dependent terms are added to the transport coefficients

such that they vanish in smooth regions. This approach was originally presented for

the case of uniformly spaced Cartesian meshes and was later extended to curvilinear

and anisotropic meshes by Kawai and Lele [65]. Premasuthan et al. [66, 67] employed

these ideas in the context of high order Spectral Difference methods.

Another such method adopted to high order DG methods is the sub-cell shock cap-

turing method using artificial viscosity by Persson and Peraire [68, 69]. For example,

the Euler equations are solved by adding the additional dissipation terms as follows:

∂u

∂t+ ∇ · F (u) = ∇ · (ε∇u) (5.1)

where u is the solution vector (ρ, ρu, ρv, ρw, ρE)T and F is the flux vector for the


Euler equation and ε represents the coefficient of the artificial viscosity term.

One important disadvantage of such a method is that they can potentially change

the order of the PDE. For example, while solving the Euler equation, the addition

of an artificial dissipation terms introduces the necessity for handling a second or

higher order derivative terms which is quite expensive in the context of high order

DG type methods. In cases where these terms are grid-dependent, for example in [64],

extensions to unstructured meshes are complicated. Apart from the ambiguities of

how to implement boundary conditions for this non-physical term, these extra terms

can introduce their own time-step limitations and in general increase the complexity

of the method due to requirement of handling a second order term.

Also, in DG type methods, one might have to render the coefficients of artificial

viscosity smooth across elements in order to avoid creating artificial oscillations in

the state gradients at element boundaries (See [70] for more details). Any method

used for that, whether it be solving an additional equation for the coefficients of

artificial viscosity as in [70], or using a vertex based method to enforce C0-continuity

of these coefficients like in [68, 69], significantly increase the computational cost of

the approach.

5.1.3 Filtering

The idea of filtering the solution is very commonly used for stabilizing the method

against weak non-linear instabilities like those caused by aliasing. Many such filters

are closely related to the method of artificial vsicosity. These filters also add artificial

dissipation to the solution, but as a separate step from solving the PDE. Let us make

this clear. Consider the 1D advection equation

∂u

∂t+∂f(u)

∂x= 0 (5.2)


Following [71] and [19], suppose we add an additional artificial dissipation term to

the discrete form of the above equation to get

∂uh

∂t+Dh(fh(uh)) = ε(−1)s+1

[∂

∂x(1− x2)

∂

∂x

]suh (5.3)

where the h superscript represents the quantity used in the numerical calculation and

Dh is the numerical differentiation operator. We have intentionally kept this general

here as we don’t want to get into too much detail about any one particular method

at this point. But we assume that we are using a DG-type method with Legendre

polynomial bases as this particular type of viscosity is ideal for such a situation.

Considering that this added term is unphysical and is only a numerical trick, we solve

this equation in a time-splitting fashion by advancing one time-step of

∂uh

∂t+Dh(fh(uh)) = 0 (5.4)

followed by∂uh

∂t= ε(−1)s+1

[∂

∂x(1− x2)

∂

∂x

]suh (5.5)

This is only an O(∆t) approximation to solving (5.3) , it does not really matter as

to how accurately we include this artificial term.

Since the solution inside each element is a polynomial, it can be expressed equivalently

in a basis of orthogonal polynomials of the same degree. We choose the Legendre

polynomial basis for this purpose [72, 73]. Legendre polynomials are special cases

of Jacobi polynomials and are eigenfunctions of the Sturm-Liouville operator (See

Sec 6.3.3 for more details about Jacobi polynomials). They are orthogonal with

respect to the L2 inner product with a unit weight in the interval [−1 , 1], i.e.,

1∫−1

Lm(x)Ln(x)dx =2

2n+ 1δmn (5.6)


where Lm(x) is the degree m Legendre polynomial. Let us consider

Pn(x) =

√2n+ 1

2Ln(x) (5.7)

which is the orthonormal version of the Legendre polynomials. The advantage of

using Legendre polynomials is that all Legendre polynomials of degree greater than

zero integrate to zero, i.e.,

1∫−1

Lm(x)dx = 0 for m 6= 0 (5.8)

which can be obtained by considering n = 0 in (5.6) and by noting that L0(x) = 1.

We discuss how this property benefits us in Section 5.3.3.

The solution uh expressed using the Legendre polynomial modes as

uh =P∑n=0

unPn(x) (5.9)

is referred to as the modal form of the solution. Substituting this into (5.5) and using

a simple Forward Euler time-stepping we get

uh,∗(x, t) ' uh(x, t) + ε∆t(−1)s+1

P∑n=0

un(n(n− 1))sPn(x) (5.10)

For ε = 1∆tP 2s , this is equivalent to

uh,∗(x, t) 'P∑n=0

σ(n

P)unPn(x) (5.11)

where σ(η) = 1− αη2s. i.e., In general, solving (5.10) is equivalent to applying a low


pass filter. σ(η) above is the filter function with the following properties.

σ(η) =

1, η = 0

≤ 1, 0 ≤ η ≤ 1

= 0, η > 1

The convergence properties of the solution under such a treatment can be found

in [71].

A popular alternative to the above filter function is the class of exponential fil-

ters

σ(η) = e−αη2s

We discuss this in more detail in Section 5.3. The major advantages of filtering,

especially in the context of application to high order methods are as follows:

1. Filtering is performed as a separate step from time-advancing the numerical

method for the PDE. This precludes the need for handling higher order terms

in the PDE and avoids imposing strong additional time-step restrictions due to

these terms.

2. It can be implemented as a simple matrix-vector product between a pre- com-

puted filter matrix and the numerical solution as

uh = Fuh

where F is the filter matrix. It is also an element-local operation which does

not need neighboring element information or the estimation of global quantities.

Therefore it is computationally very efficient and highly suitable for explicit-

time stepping methods on GPUs which have small time step limits in general.

3. Extensions of this approach to arbitrary elements in multiple dimensions is very

straightforward and does not add any additional complexity. This is because

the filter applies to polynomial modes and the solution in any element type can


be represented using some hierarchical polynomial basis.

4. The filtering framework can be used to handle both shock-capturing and alias-

ing. We could just create two filter matrices of different strengths: A strong

filter for shocks and a weak one for aliasing instabilities. The weak filter can

be applied everywhere if aliasing instabilities are expected and the strong fil-

ter only where shocks are present. This would need a method to predict the

location of shocks, which we introduce in 5.2 and discuss in full detail in 6.

The major disadvantage in using filters for shock capturing is that one needs strong

filters to stabilize against shocks and if such a filter is applied everywhere in the

domain including in smooth regions, it can lead to a severe loss of accuracy. However,

we propose a strategy which handles this issue effectively in 5.2.

5.1.4 Reconstruction Approaches

Recognizing the inherent issue with trying to represent a discontinuous solution using

polynomial bases, reconstruction approaches propose the use of other functions ratio-

nal functions (e.g., Pade forms). Here usually one seeks two element-local polynomials

RM and QL of order M and L respectively, such that

uh(x) =RM(x)

QL(x)x ∈ Ωk (5.12)

where M +L+ 1 ≤ P . There are several ways of determining the sense in which the

two expressions are the same, the most obvious one being

∀m ∈ [0,M + L] :

∫Ωk

(uhkQL −RM)Pm(x)dx = 0

Such approaches are better than straightforward filtering or limiting as they can not

only dramatically reduce Gibbs oscillations but can also provide more rapid conver-

gence even close to the point of discontinuity.


However, changing the basis of representation of the solution in certain elements only

is not only computationally tedious and possible more expensive, but one also needs

to be careful with how the properties of the numerical method change with such

modifications. Also, we believe that extension of such approaches to arbitrary high

order elements might get complicated and tedious, although we have not explored

this in detail.

5.2 Shock Capturing Strategy

In 5.1.3, we discussed the advantages of the method of filtering in the context of appli-

cation to high-order methods, particularly those with explicit time-stepping designed

to be implemented on GPUs. We also noted that the major disadvantage is that, in

order to capture strong shocks, one needs strong filters and if such a filter is applied

everywhere in the domain including in smooth regions, it can lead to a severe loss of

order of accuracy of the solution in the entire domain.

We tackle this issue by using a two-step approach to shock capturing:

• Step 1: Detect the locations of shocks using an effective shock detection mech-

anism. We have developed one such technique which we discuss in full detail

in 6. We show that this technique is capable of clearly distinguishing between

shocks and gradient-rich but smooth regions like those containing vortices and

boundary layers with enough separation of scales. The shock sensor we propose

is designed with the intention to completely eliminate the problem of careful

parameter selection and provide a “hands-free” method for detection of shocks.

• Step 2: Use modal filtering with an adequately strong filter only in elements

where strong shocks have been detected by the shock sensor/detector. If filtering

for aliasing instabilities is needed in other regions, a weak filter may be used in

those regions.

The above two-step approach selectively applies filtering only in regions where the


polynomial bases are inadequate in representing the solution well (near discontinu-

ities). In such regions, some artificial dissipation is necessary to limit unphysical

oscillations, maintain stability and also recover pointwise convergence at and around

the discontinuities, which is provided by the filters.

Although the shock sensor is designed to eliminate ambiguous parameter and thresh-

old selection, it also provides a measure of the size/strength of discontinuity which

allows for more complicated approaches with different strength filters applied at dif-

ferent ranges of the sensor if one wishes.

In 5.1.3, we already introduced the idea of filters. In 5.3, we describe the particular

exponential modal filters that we use and discuss its implementation. In Chapter 6,

we study the problem of shock detection in high order methods in detail and propose

a new shock detector based on the concentration property of Legendre polynomial

modes.

5.3 Modal Filters

We have already introduced the concept of filtering and reviewed its advantages and

disadvantages in 5.1.3. In 5.2, we discussed how the disadvantages can be overcome

with a careful selective application of the filters. In this section, we describe the

particular class of exponential filters in further detail and discuss parameter selection.

We have chosen the exponential modal filters for the filtering process. While there

are several other possible Fourier spectral-based filters as well as physical space filters

(mollifiers), this choice has worked very well for us in practice. Any other filtering

operation that can be applied to the solution in form a simple matrix-vector product

is equally efficient and it is a matter of choice.


5.3.1 Implementation

Consider the exponential filter

σ(η) = e−αηs

(5.13)

Note that from our discussion in 5.1.3, we need to require that this s must be even

(since we have dropped the factor of two to keep things less confusing).

The filter is applied to the modal form of the solution as follows:

u =P∑n=0

σ(n

P)unPn(x) (5.14)

This is equivalent to modifying the modal coefficients un to

˜un = σ(n

P)un (5.15)

Let Σ denote a (P + 1) × (P + 1) diagonal matrix with the nth diagonal element

containing σ(n−1P

). Then this operation within each element can be expressed as

˜u = Σu (5.16)

where u is the vector of modal coefficients of the solution within an element (an

arbitrary element Ωk).

In the FR approach and nodal-DG methods however, the solution is stored in a nodal

format. Therefore, we will need to first convert the vector of nodal solution values to

the modal form using

u = V−1u (5.17)

where V is the Vandermonde matrix. After applying the filtering on these computed

modal coefficients, we will have to compute back the filtered nodal solution using

u = V ˜u (5.18)


The above two operations can be combined into one pre-multiplication of the solution

vector by a filter matrix as follows

u = Fu where F = VΣV−1 (5.19)

This filter matrix can be computed and stored during the pre-processing step. There-

fore the application of the filter in an element where a shock has been detected reduces

to a single matrix pre-multiplication with the solution vector. This is just equivalent

to other elemental operations like extrapolation of the solution or flux correction in

terms of the time complexity. Also, matrix vector products are ideal for implemen-

tation on GPUs.

5.3.2 Extension to multiple dimensions

In multiple dimensions, a hierarchical modal basis is created along with its corre-

sponding Vandermonde matrix. However, while calculating η for the filter, we use

the degree of the mode and the overall degree of the polynomial basis. For example,

consider a P th order scheme on a quadrilateral element. The solution in such an

element is represented as

uh =P∑i=0

P∑j=0

uijPi(x)Pj(y) (5.20)

The filtering operation on this is performed as

uh =P∑i=0

P∑j=0

σ(i+ j

P

)uijPi(x)Pj(y) (5.21)

i.e., we do not bias one direction over another.


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α = 10; ηc = 0.25

η

σ(η

)

s = 2

s = 4

s = 8

s = 16

(a) Order of the filter

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s = 2; ηc = 0

η

σ(η

)

α = 0.1

α = 1

α = 10

α = 100

(b) Filter Strength

Figure 5.1: Figure shows the spectral action of the exponential filter as the parametersare varied. (a) shows this action for varying filter orders while (b) shows the samefor varying filter strengths


5.3.3 Parameter Selection

The basic exponential filter in (5.13) adds dissipation in the entire spectrum η = [0, 1].

A more general formulation of this filter is

σ(η) =

1, 0 ≤ η ≤ ηc = NcP

exp(−α(η−ηc1−ηc

)s), ηc ≤ η ≤ 1

(5.22)

where Nc is the modal number cutoff below which the lower modes are left untouched.

Again, the s here needs to be even for it to correspond to hyperviscosity term, i.e.,

s = 2 above is practically equivalent to adding a second order dissipative term to the

PDE. Therefore s is referred to as the order of the filter. α is the filter strength. η is

the modal fraction and η = n/P in 1D or (i+ j)/P in 2D.

Since the zeroth mode (n = 0) is untouched by the above filter and since all modes

other than the zeroth mode integrate to zero (See (5.8)), the averages of the quan-

tities are all conserved within each element through the filtering process. This is

important in order to not lose mass/energy through the filtering process which can

introduce bias into the shock speed and location estimates and affect the accuracy of

the solution.

While using the filter for shock capturing, we can set Nc such that the linear modes

are untouched by the filter. For example, for a quadrilateral element, Nc is set to 2 (for

the three modes 1, x and y). Also, in order to stabilize against strong shocks, we use a

second order filter in general. In viscous problems or problems with weak shocks, we

have had good experience with a fourth or sixth order filters as well. When the filter

is being used for aliasing instabilities, one can use even higher order filters. However,

we have not studied stabilization against aliasing in detail. The filter strength α can

be varied based on the problem at hand but we have generally used a filter strength

between 1 and 10. If we use Nc = 0, we generally use a low filter strength, and if

we set Nc to not affect linear modes, a stronger filter with higher α is used. A filter

strength of 1 with Nc = 0 and s = 2 is sufficient to stabilize almost all the testcases


we have run and is not gives very good solutions due to our ability to apply this filter

only in specific locations. The results are presented in Chapter 7. This makes for

a robust tool for getting very good results without requiring experimentation of the

parameters. If required, the parameters can be fine tuned in successive runs to get

better results.

5.4 Positivity Preservation

So far we have focused on stabilizing the method against strong shocks. However,

there is one issue we have not yet touched upon, i.e., the problem of unphysical

solutions. A common way in which this problem is encountered in flow problems

is through a negative density or pressure. Even if one argues that the problem of

negative solution may be temporary and might go away (which it rarely does), any

square root or exponentiation of such a negative quantity results in a NaN.

Such negative solutions are not necessarily associated with shocks. Indeed, the oscil-

lations created around a shock can get large enough when not treated properly and

can lead to negative density or pressure. But such problems are also encountered in

regions of low pressure or density where the high order polynomials can cause the

solution to overshoot beyond zero. Common examples are the stagnation point in the

wake of a bluff body, strong expansion fan around a corner etc.

Apart from physical situations where either the density or pressure is very low, this

problem is quite commonly encountered during start-up of high order codes. Starting

from a high Mach number uniform flow causes the flow to suddenly encounter the

body and can cause very large gradients. This is handled by starting with a low CFL

and increasing it after the initial large gradients have passed. But this is not very

reliable and might sometimes push the CFL to very small numbers.

The filters we have proposed to stabilize the method in the presence of strong shocks

reduces the oscillations but does not completely eliminate them. Also, in smooth


regions, the application of a strong filter might degrade the solution.

In order to tackle this issue, we use the high order positivity preservation ideas pro-

posed originally by Zhang and Shu [74, 75, 76, 77]. Their method was inspired by

earlier work of Tadmor who showed that entropy solutions of the Euler equations fol-

low a certain minimum entropy principle where the spatial minimum of the specific

entropy increases with time [78]. The work of Zhang and Shu were further extended

to unstructured grids and made simple to implement by Lv and Ihme [79]. These

methods are capable of preserving positivity of the elemental average of the density

and pressure at the next time-step if the corresponding averages at the current time-

step are positive. In smooth regions, this method preserves a high order accuracy,

while near discontinuities it can only guarantee first order accuracy.

In this section, we briefly describe the method and its adaptation to the FR approach.

Start by considering the 1D Euler equations:

∂w

∂t+

∂

∂x

(F (w)

)= 0 (5.23)

where

w =

ρ

ρu

ρE

and F (w) =

ρu

ρu2 + p

u(ρE + p)

and E = e+ 1

2u2 and p = (γ − 1)ρe = (γ − 1)(ρE − 1

2(ρu)2

ρ). The generalized entropy

function for the Euler equation is a convex function U s(w) with a corresponding

entropy flux F s(w) such that the following relation holds(∂U s

∂w

)T∂F

∂w=∂F s

∂w(5.24)

The so called entropy solutions of the Euler equation are solutions to (5.23) which

also satisfy the entropy inequality

∂U s

∂t+∂F s

∂x≤ 0 (5.25)


in the sense of distributions (weakly) for all entropy pairs (U s, F s). Tadmor [78]

showed that all entropy solutions satisfy a minimum principle with respect to the

specific entropy S = ln( pργ

):

S(x, t+ h) ≥ minS(y, t) : |y − x| ≤ ‖u‖∞h (5.26)

Now, using the fact that this specific entropy function is quasi-concave, one can show

that the set G below is convex.

G =

u =

ρ

ρu

ρE

ρ > 0 p > 0 and S ≥ S0 = minxS(w0(x))

(5.27)

Now consider the equation satisfied by the elemental averages in the FR method for

the Euler equation under forward Euler time-stepping:

wn+1j = wn

j − λ[h(w+j+1/2,w

−j+1/2)− h(w+

j−1/2,w−j−1/2)] (5.28)

where λ = ∆t∆x

and h is a positivity preserving flux under the CFL condition

λ‖(|u|+ c)‖∞ ≤ Cmax (5.29)

Now suppose qj(x) =(ρj(x), (ρu)j(x), (ρE)j(x)

)represent the polynomial vector of

the solution in element j, i.e., in Ωj. Also let xαj denote the Gauss-Lobatto quadrature

points such that the order of integration of the points is atleast P , the degree of the

solution polynomial in the element.

Then Zhang and Shu [77] showed that if qj(xαj ) ∈ G ∀j and α, then wn+1

j ∈ G∀junder the CFL condition

λ‖(|u|+ c)‖∞ ≤ ω1Cmax (5.30)


where ω1 is the weight of the first Gauss Lobatto quadrature point. They also propose

a limiter to enforce this condition while ensuring high order accuracy. However, we

use a slightly different limiting approach suitable to the FR approach and is described

in 5.4.1.

Although this was shown proven for a Forward Euler scheme, it can be directly

extended to higher order SSP time-stepping methods in a straightforward way by

noticing that they are just a convex combination of Forward Euler steps. Lastly, the

entropy condition is not necessary, i.e., just the positivity condition for ρ and p also

ensures a convex set and all the above results apply to that as well.

5.4.1 Adaptation to the FR approach

The limiting process proposed by Zhang and Shu is somewhat complicated and expen-

sive to implement in the context of FR approach, especially in multiple dimensions.

So we follow more along the lines of [79]. In the FR approach, we generally pick the

most optimal area quadrature points as the solution points (for example, the Gauss

quadrature points on quads and hexes). We also have the surface or edge quadrature

points on the element interfaces as flux points. Since we naturally have information at

these two sets of quadrature points, we enforce the positivity preservation condition

at all these points.

Let xdj and denote the vector containing all the solution and flux points in the element

j. While calculating elemental averages, we use the quadrature available in the form

of solution points. The limiting is done using the following steps:

1. Setup a small parameter or tolerance limit ε for ρ and p.

2. First limit the density, i.e.,

ρ(xdj ) = θ1(ρn(xdj )− ρnj ) + ρnj (5.31)


where

θ1 = min

ρnj − ε

ρnj − ρmin, 1

where ρmin = min ρnj (xdj ))

3. Limit the pressure to satisfy the entropy condition.

Let Wj =(ρ, (ρu)n, (ρE)n

)T, i.e., the solution vector after limiting just the

density. Now the limited solution vector at the solution and flux points is

obtained as

wj = Wj + θ2(wnj − Wj) ∀xdj (5.32)

where wnj is the solution average in the element computed using a quadrature.

Here the θ2 is obtained as follows:

θ2 =τ

τ − [p(wnj )− eS0

(ρ(wn

j ))γ

](5.33)

where τ is given by

τ = min

0,min

xdj

p(Wj)− eS

0j(ρ(Wj)

)γ(5.34)

where S0j is a (possibly) local entropy bound. However, computation of this

local entropy is quite expensive and not necessary. Preserving positivity in

pressure can be achieved by setting a global entropy bound of S0j → −∞.

The above steps have to be applied at each stage of a multi-stage scheme. This when

combined by an appropriate CFL condition [79], we are guaranteed that the average

solution wn+1 is going to be entropy bounded.

5.5 Time-Stepping scheme

In order for the stabilization properties of the positivity preserving limiter as well

as the filtering to extend successfully, we require a time-stepping scheme which is


a convex combination of the Forward Euler steps. There have been several articles

describing SSP schemes of different orders with various properties [80, 81, 82]. We

use the third order scheme (originally known as the TVD RK3 scheme) of Shu and

Osher [83]. In particular, we use the low storage ‘f-flavored’ version, i.e., we store the

residuals L(u(i)) of different stages instead of all u(i)s:

u(1) = u(0) + ∆tL(u(0)) (5.35)

u(2) = u(0) +1

4∆tL(u(0)) +

1

4∆tL(u(1)) (5.36)

u(3) = u(0) +1

6∆tL(u(0)) +

1

6∆tL(u(1)) +

2

3∆tL(u(2)) (5.37)

Chapter 6

Shock Detection

6.1 Introduction

In the previous chapter 5, we investigated several strategies traditionally adopted for

handling discontinuities and proposed a two-step approach of detecting locations of

shocks accurately and then applying a sufficiently strong filter locally in these regions

to stabilize the solution as well as eliminate the oscillations that develop around these

shocks. The success of this strategy in terms of preserving accuracy of the solution

largely depends on the quality of the shock detector.

As noted in [84] which compares several shock capturing methods in the context of

turbulence simulations, developing a universal shock sensor which works well on gen-

eral problems is challenging. We discuss the advantages and disadvantages of several

methods used in the past for shock detection in section 6.2. While a few shock sensors

have been quite successful in certain contexts, they are either not suitable for high or-

der DG type methods, require fine-tuning of several parameters or based on physical

quantities which are not generalizable to different flow problems or equations.

Drawing on certain techniques used for edge detection in images, MRI scans and sig-

nals [85], we propose a new robust detector that is purely based on the mathematical

114

CHAPTER 6. SHOCK DETECTION 115

properties of discontinuous functions and is generalizable to any equation or context.

This method does not require gathering neighboring data (i.e., it is completely ele-

ment local) or computing high order derivatives and can be implemented as a single

matrix vector multiplication on the solution field which is extremely efficient. The

method generalizes in a straightforward fashion to unstructured domains without the

need of any additional effort.

6.2 Comparison of Current Shock Sensors

Shock sensors in some form or the other were used starting from the classic JST

schemes [61]. Pressure gradients were used for detecting the locations of the shocks.

These schemes consisted of a second order dissipation term with a non-linear co-

efficient which was active mainly in regions around shocks and was responsible for

stabilizing the solution; and a linear fourth order dissipation term included to damp

high-frequency modes and allow the scheme to approach a steady state. This term

vanished near shocks and is generally active in smoother regions. Pressure based

sensors are not capable of detecting contact discontinuities in general. The sensor

based on the pressure gradient in the original paper has been modified by several

authors to suit various applications. [86, 87]. The sensor described by [87] has a an

additional multiplicative term based on vorticity and the divergence of the solution

and therefore is not applicable to problems without vorticity.

Another successfully used shock sensor based on physical quantities is the one by

Cook [64]. Cook proposed the addition of grid dependent artificial components to the

transport coefficients which vanish in smooth regions. The result was addition of non-

linear second order artificial dissipation terms. While this method was proposed for

uniform Cartesian meshes, Kawai and Lele [65] extended it to curvilinear anisotropic

meshes. However, the extension of such methods to unstructured grids especially in

the context of FEM-type methods is complicated. Moreover, these methods need

computation of higher order derivatives which is expensive in the context of FR and


other DG type methods which need to correct each derivative before calculating the

next.

There have been several sensors based on an estimation of the smoothness of the

solution in the context of ENO and WENO schemes [88, 89]. In DG methods, the

well established limiters proposed by Cockburn and Shu [55, 56, 57, 13] also utilize

a sensor based on an estimate of the global maximum of the second derivative of

the solution in the domain. This is integrated into the limiting mechanism. These

limiting methods work very well for low orders but designing them to maintain order

of accuracy at higher order requires a hierarchical application and is in general quite

complicated on unstructured meshes.

One of the sensors that is very suitable to high order DG type methods is the one

proposed by Persson and Peraire [68, 69]. They suggest a simple sensor based on

the ratio of the energy contained in the modal coefficients of the highest degree to

the overall energy. Near a discontinuity, the modal coefficients decay at a lower rate

and are therefore expected to contain a higher proportion of the energy compared to

smoother regions. The sensor is element local and can be computed inexpensively.

However, the main drawback of this sensor is that this sensor has a limited capability

of distinguishing between discontinuities and other flow features like vortices or in

high gradient regions. Moreover, the selection of a thresholds is hard and needs fine

tuning based on the problem in order to obtain quality results. More detailed analysis

of this sensor can be found in [90].

The method we propose is based on the concentration property which also utilizes

the fact that the modal coefficients of a discontinuous function decay at a slower rate

compared to smooth or continuous functions. However, this method uses concentra-

tion kernels combined with a non-linear enhancement technique to separate the scales

between regions of smoothness from those containing discontinuities. We adapt this

technique to the context of high order DG type methods and discuss how to elim-

inate parameter selection and yet obtain very reliable and robust shock detection.

The implementation is also very simple similar to the one by Persson [68]. The major


advantages of the shock sensor we propose are as follows:

• It is based only on the modal properties of discontinuous functions and not on

any physical quantity. Therefore it can be employed in any problem (including

multiphysics problems) to capture a discontinuity in any variable. For example,

we do not need to employ different strategies to capture shocks and contact

discontinuities.

• It is capable of clearly distinguishing between shocks and other gradient-rich

zones like vortex trails or boundary layers. This prevents the contamination of

the flow in important regions.

• It almost eliminates parameter selection, thereby giving a universal approach

that can be employed on a wide variety of problems.

• It is completely element-local and can be calculated through a single pre-

multiplication of the solution by a pre-computed matrix.

6.3 Concentration Method

The concentration method, proposed by Gelb, Cates and Tadmor [85] [91] [41] is a

general framework for recovering edges in piecewise smooth functions with finitely

many jump discontinuities. The approach is based on two main aspects: localiza-

tion using appropriate concentration kernels and separation of scales by nonlinear

enhancement. We first discuss the general idea behind concentration kernels and how

they are applied to recover the jump location and the jump size from the Fourier

modes of a periodic function. We then discuss their extension to polynomial modes,

particularly in the context of Legendre polynomials that are used by us in the Flux

Reconstruction context. We then introduce the idea of non-linear enhancement.


6.3.1 Concentration Kernels

Suppose we want to detect the edges in a piecewise smooth function with jump

discontinuities. For simplicity, consider a function f(.) with jump discontinuities

with well defined one-sided limits f(x±) = limx→x±

f(x). Let [f ](x) := f(x+)− f(x−).

Piecewise smoothness is determined using the criterion

Fx(t) :=f(x+ t)− f(x− t)− [f ](x)

t∈ BV [0, δ]∀x (6.1)

where BV [a, b] refers to the space of functions with bounded variation endowed with

the seminorm ‖φ‖ =b∫a

|φ′|dx. For example, f is piecewise smooth if it has finitely

many discontinuities and f ′(x±) is well-defined at these discontinuities.

Gelb, Cates, Tadmor and others involved in work distributed over several articles

have proposed the use of what they call concentration kernels to detect the locations

and sizes of the disconuities of f . These concentration kernels, denoted by Kε are

based on a small parameter ε and have the following concentration property

Kε ∗ f(x)→ [f ](x) as ε→ 0 (6.2)

where ∗ denotes the convolution operation. To guarantee this concentration property,

Kε is required to have the following three properties:

1. Kε needs to be an odd kernel, i.e.,

Kε(t) = −Kε(−t) (6.3)

2. It has to be normalized, i.e.,∫t≥0

Kε(t)dt = −1 +O(ε) (6.4)


3. It satisfies an admissible property∫t

tKε(t)φ(t)dt ≤ Const · ε‖φ‖BV (6.5)

If the above three conditions are satisfied, it was shown that the kernel then satisfies

the concentration property and the following error estimate holds:

|Kε ∗ f(x)− [f ](x)| ≤ Const · ε (6.6)

While many concentration kernels have been studied, the one which is of particular

interest to us is the conjugate Dirichlet kernel with general concentration factors, i.e.,

KσN(t) = −

N∑k=1

σ(k/N) sin kt (6.7)

where σ(.) are concentration factors which satisfy a regularity property

1∫0

σ(ξ)

ξdξ = 1 (6.8)

If a function f is available in the form of a partial Fourier sum or spectral projection

SN(f) = −N∑

k=−Nfke

ikx, then the action of this kernel on f is equivalent to its action

on its spectral projection:

KσN ∗ f(x) = Kσ

N ∗ SN(f) = iπ

N∑k=−N

sgn(k)σ(|k|/N)fkeikx (6.9)

Such kernels are referred to as ‘good’ kernels [92]. Note that if σ(ξ) = ξ, we also have

KσN ∗ SN(f) = SN(f)′. We then have the property

|KσN ∗ f(x)− [f ](x)| ≤ Const · ε (6.10)


6.3.2 Concentration kernel for spectral projection

To see how this property comes about in this case of a spectral projection, let us

consider a 2π periodic function f with a single jump discontinuity at c. Its Fourier

coefficients are given by [50, 93]

fk = [f ](c)e−ikc

2πik+O(

1

k2) (6.11)

Then using the concentration property of a Dirichlet Kernel localized at c [93, 85],

we can write

π

NSN(f)′ = π

N∑k=−N

ik

N

[f ](c)

eik(x−c)

2πik+O(

1

k2)

=[f ](c)

2N

N∑k=−N

1 +O(

1

k)

eik(x−c)

= [f ](x) +O(logN

N)

(6.12)

This allows for the detection of location of edges as well as the size directly using the

Fourier coefficients of the function. The above property can also be shown for other

general concentration factors of the form σ(ξ) = ξµ(ξ) for1∫0

µ(ξ)dξ = 1. Among

several choices for the form of the concentration factors, the exponential factors have

been found to be the most effective [85]. We shall discuss this in more detail in the

later sections.

6.3.3 Concentration Property for Jacobi Polynomials

In order to use the concentration property in the context of the FR approach, it

would be ideal to show that a similar property as (6.12) for the case of polynomial

modal representation. This is because the solution and fluxes in the FR approach

are represented using a Legendre polynomial basis (when using the Gauss-Legendre


points for the solution points). We start by considering a piecewise smooth function

f . The Jacobi expansion of this function is given by

SN(f) =N∑k=0

fkPk(x) (6.13)

where

fk =

1∫−1

f(x)ω(x)Pk(x)dx (6.14)

Pk(x) are the Jacobi polynomials [94, 73] which are the eigenfunctions of the Sturm-

Liouville equation

((1− x2)ω(x)P ′k(x))′ = −λkω(x)Pk(x) − 1 ≤ x ≤ 1 (6.15)

with weight functions ω(x) = (1 − x2)α and corresponding eigenvalues λk = λ(α)k =

k(k + 2α + 1).

Now, similar to (6.11), for a function with a single jump discontinuity, we have the

Jacobi co-efficients given by

fk = − 1

λk

1∫−1

f(x)((1− x2)ω(x)P ′k(x))′dx (6.16)

Without getting rigorous, we can see that, by splitting the integral into the two

separate portions [−1, c) and (c, 1] , and applying integration by parts on each, we

get boundary terms and integral terms involving f ′(x) in each separate region. These

integral terms, similar to the Fourier case can be shown to be O(1/λ2k). Therefore,

we can write

fk =1

λk[f ](c)(1− c2)ω(c)P ′k(c) +O(

1

λ2k

) (6.17)

Now, similar to the Fourier case, let us consider the conjugate partial sum upto N


modes (with a slightly different pre-factor):

π√

1− x2

N

N∑k=1

µ(k

N)fkP

′k(x) =

[f ](c)π√

1− x2

N(1− c2)ω(c)

N∑k=1

µ(k

N)

1

λk+O(

1

λ2k

)

P ′k(c)P

′k(x)

(6.18)

where the concentration factors σ(ξ) = ξµ(ξ) has been assumed. If we consider the

case of µ(ξ) = 1, we recover the Jacobi-modal projection of f ′ similar to the Fourier

case, i.e.,

π√

1− x2

NSN(f)′ =

π√

1− x2

N

N∑k=1

µ(k

N)fkP

′k(x)

= [f ](c)π√

1− x2

N(1− c2)ω(c)

N∑k=1

1

λk+O(

1

λ2k

)

P ′k(c)P

′k(x)

(6.19)

Similar to the Fourier case, this conjugate partial sum can be seen to possess a concen-

tration property, whereby the kernel concentrates near a jump discontinuity [85].

Concentration property for Jacobi expansion: Let SN(f) be the truncated Jacobi

expansion of f with −1 < α ≤ 0, where α is the exponent of the weight function

ωα(x) = (1−x2)α. Then the conjugate partial sum with concentration factors σ(ξ) = ξ

which is equal to SN(f)′(x) admits the property

∣∣π√1− x2

NSN(f)′(x)− [f ](x)

∣∣ ≤ Const

(1− x2)α/2+1/4· logNN

for − 1 +Const

N2< x < 1− Const

N2

(6.20)

Note that this property is only applicable for −1 < α ≤ 0 and is not applicable to a

small portion close to the edges of the domain.


The Legendre polynomial basis that we generally use in the FR approach is a special

case of the Jacobi polynomials with α = 0. Therefore, the property holds for the Leg-

endre basis as well. The benefit of this is that we do not have to build a set of Fourier

modes or build new polynomials since we can move from nodal to modal framework

via a Vandermonde matrix. We will make use of this property to build our shock

sensor. So far, we have a method which concentrates near the jump discontinuity.

As N ↑, this conjugate Jacobi partial sum gets increasingly better at distinguishing

between a point of discontinuity from one that is not. However, for relatively small

N, the difference may be small and it would be hard to clearly separate them. In

this regard, there is a method that allows for the separation of scales between these

points of discontinuities and other continuous points. This is very important because

the polynomial bases used in the FR approach are often as small as 4 or 5. Later, we

show results with various degrees of polynomials.


−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Step at x = 0

−1 −0.5 0 0.5 10

2

4

6

8

10

12

(b) Enhanced kernel portrait for (a)

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c) Step at x = 0.7

−1 −0.5 0 0.5 10

5

10

15

20

25

30

35

40

45

(d) Enhanced kernel portrait for (c)

Figure 6.1: Figure shows the portraits of the enhanced kernel (before clipping itbelow a threshold) for the step discontinuity. (a) and (b) show the step and its kernelportrait when the step is located in the center of the element. (c) and (d) show thesame when the step is located near the boundary


6.3.4 Non-Linear Enhancement

As discussed, for relatively small values of N , the concentrating effect of the conjugate

partial sum is not sufficient for clear detection and differentiation of edges. In such

a case, a non-linear enhancement can be used to further amplify the separation of

scales between the edges/discontinuities and the smooth regions. Consider the action

of a concentration kernel of smallness parameter ε on a function f with a jump-

discontinuity at c,

Kε ∗ f =

O(ε), if x 6= c

[f ](c) when x = c(6.21)

This is the asymptotic behavior of the kernel as ε ↓ 0. To further enhance the

separation of these two scales, consider

ε−q/2(Kε ? f(x))q ∼

εq/2, at a smooth point x 6= c

([f ](c))qε−q/2, at a discontinuity x = c(6.22)

By increasing the exponent q > 1, we can enhance the separation between the van-

ishing scale at the points of smoothness (O(ε)q/2) and the growing scale at the jump

(O(ε)−q/2). In our problems where we use the Legendre polynomial modes, the con-

centration property for Jacobi polynomials indicates that the smallness parameter ε

is given by ε ∼ logNN

.

By choosing a q > 1 (we typically need only go upto q = 5), we can enhance the

separation between the vanishing scale at the smooth points and growing scale at the

jumps. The final step is to choose a threshold Jcrit. If the value of the enhanced

kernel is above this, we determine that it is a point of discontinuity. Otherwise, we

conclude that it is a smooth point. This can be formalized as choosing the enhanced

kernel

KqN,J ? f(x) =

KN ? f(x), if ε−q/2|KN ? f(x)|q > Jcrit

0, otherwise.(6.23)


−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) C0-kink at x = 0

−1 −0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

(b) Enhanced kernel portrait for (a)

−1 −0.5 0 0.5 10.4

0.5

0.6

0.7

0.8

0.9

1

(c) 1.4 + cos(3 + 1.3sin(x))

−1 −0.5 0 0.5 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

(d) Enhanced kernel portrait for (c)

Figure 6.2: Figure shows the portraits of the enhanced kernel (before clipping itbelow a threshold). (a) and (b) show the function and its kernel portrait for a ramp(derivative discontinuity, or C0 function) while (c) and (d) show the same for a smoothfunction.


−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

x

Enhanced K

ern

el

step

ramp

(a) N = 2, q = 2

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

x

Enhanced K

ern

el

step

ramp

(b) N = 3, q = 2

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

x

Enhanced K

ern

el

step

ramp

(c) N = 5, q = 2

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

x

Enhanced K

ern

el

step

ramp

(d) N = 8, q = 2

Figure 6.3: Figure compares the portraits of the enhanced kernel for the step and rampfunctions for various polynomial orders. The sensor seems to be able to consistentlygenerate adequate separation of scales to distinguish between a jump and a rampacross all polynomial orders


−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

x

Enhanced K

ern

el

step

ramp

(a) q = 2, N = 3

−1 −0.5 0 0.5 10

1

2

3

4

5

6

7

x

Enhanced K

ern

el

step

ramp

(b) q = 3, N = 3

−1 −0.5 0 0.5 10

5

10

15

20

25

x

Enhanced K

ern

el

step

ramp

(c) q = 5, N = 3

−1 −0.5 0 0.5 10

50

100

150

x

Enhanced K

ern

el

step

ramp

(d) q = 8, N = 3

Figure 6.4: Figure compares the portraits of the enhanced kernel for the step andramp functions for different values of the non-linear enhancement exponent q. It isevident that higher the nonlinear exponent, the better the separation of scales


−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

x

Enhanced K

ern

el

polynomial

exponential

(a) q = 2

−1 −0.5 0 0.5 10

2

4

6

8

10

12

14

16

x

Enhanced K

ern

el

polynomial

exponential

(b) q = 5

Figure 6.5: Figure compares the portraits of the enhanced kernel using two differentconcentration factors - polynomial and exponential, for the step function case. (a)shows this comparison for a non-linear enhancement exponent q = 2 while (b) showsthe same for q = 2

where q is the enhancement exponent and Jcrit is an appropriately chosen threshold.

This additional step becomes very important for smaller values of N if we want a

good separation of scales between shocks and vortices, for example.

Figures 6.1 and 6.2 shows the values of the enhanced kernel, i.e., ε−q/2|KN ? f(x)|q

for different functions starting from a step or a jump discontinuity to a smooth func-

tion. We can see that the kernel clearly distinguishes between the points near the

discontinuity and other smooth points. Notice that the actual values of the kernel are

much higher for the discontinuous cases compared to the continuous ones. Figure 6.3

compares the step and the ramp (C0− kink) for different values of N . It can be seen

that the differentiation between a ramp and a step gets clearer as we increase the

value of N . This is expected as the concentration effect of the kernel improves as N

gets large.


6.4 Design and Implementation of the Shock

Detector

So far, we have investigated the concentration property and its extensions to polyno-

mials. We have also discussed how this can be effectively used to distinguish between

regions containing a discontinuity from smooth regions. In this section, we explore

the process of designing and implementing an effective shock detection technique in

the context of a high order polynomial based finite-element method.

6.4.1 Parameter Selection

We begin by examining the topic of selecting the threshold Jcrit appropriately. Since

N is already fixed by the polynomial order of the Legendre basis used to represent

the solution by the FR approach, the choices we have are in choosing the type of

concentration factors, the exponent of the non-linear enhancement and the critical

threshold Jcrit.

From Figure 6.3, we can see that for N > 4, there is sufficient differentiation between

the step and ramp functions for q = 2. But for small N , the difference is relatively

small and it might be hard to distinguish between them clearly. However, this can be

easily solved by using higher values of q. Figure 6.4 compares the difference between

the enhanced kernel portrait for the step and ramp for different values of the non-

linear enhancement exponent q. We can see that the difference between the step

and ramp keeps increasing as q increases, showing the effectiveness of the non-linear

enhancement. In practice, we have found that q = 5 is sufficient and is able to

distinguish between step and ramp easily for all cases. Therefore we use q = 5 in all

our future shock-capturing experiments.

In our description of the concentration method in the previous sections, we used

polynomial concentration factors. We used concentration factors of the form σ(ξ) =

ξµ(ξ). If µ(ξ) = rξr−1, they are called polynomial factors. We particularly used


r = 1. We can use higher order polynomial factors, however the exponential factors

are better than any polynomial factors. Exponential factors are of the form

µ(ξ) = Ce1

αξ(ξ−1) (6.24)

where

C =

1∫0

e−1

αζ(ζ−1)dζ (6.25)

and is meant to normalize the concentration factors. Figure 6.5 shows the difference in

performance between the polynomial (r = 1) and exponential factors. It is clear that

the exponential factor performs better than the polynomial one and this is consistent

across N and q.

Selecting an appropriate Jcrit is a matter of design. In our numerical CFD experi-

ments, we would like to identify only shocks and treat them differently than other

smoother parts of the domain. Solutions like the ramp often occur in compression

and rarefaction regions. It is important to differentiate between these. Therefore,

we first calculate the maximum value of the enhanced kernel for the step and the

ramp for the particular N and select the average of the two as our Jcrit. Before we

detect the shock in the element, we first normalize the appropriate quantity (usually

density) inside the element so that it lies between 0 and 1 and then use this Jcrit.

This enables us to maximally match a general situation to the numerical experiments

shown above.

6.4.2 Shock Sensor in 1D

Now let us summarize how we use the method of concentration as a shock detec-

tion mechanism in a high order finite element formulation in 1D. We assume the

method uses a polynomial basis derived from Jacobi or Chebyshev polynomials in

each element. We later discuss how to extend this idea to higher dimensions.


1. As a pre-processing step, at each solution or nodal point x in the element, we

compute πN

√1− x2µ( |k|

N)P ′k(x) for k = 0, 1, 2, .., N − 1 where N is the number

of solution points (or modes) in the element. Since all our elemental compu-

tations are performed in a reference or a parent domain, these solution points

are fixed (and are in [−1, 1]) which is perfectly suitable for using the concen-

tration method. This would just be a single NxN matrix. Let us call this the

concentration matrix.

2. Choose a representative quantity for detecting the discontinuity. For example,

this can be density. One could also take the conservative approach of detecting

using multiple quantities and taking the maximum over them.

3. In each element, normalize this chosen quantity to lie between 0 and 1, so that

the threshold chosen based on the step and the ramp applies to this solution.

This normalization can be done using

ρ(xi) =ρ(xi)− ρminρmax − ρmin

(6.26)

where ρmax and ρmin refer to the maximum and minimum density over the

solution points within the element. However, note that, if ρmax − ρmin is small

(measured w.r.t. to the corresponding freestream value), then normalizing will

magnify small jumps, so we do not normalize if ρmax−ρmin is below a threshold

fraction of the freestream value.

4. In each element, we convert the element-local nodal solution of a particular

representative quantity (say density) to modal form using the Vandermonde

matrix.

5. Use the modal coefficients to compute, at each solution point x,

KσN ∗ SN(f) =

π

N

√1− x2

N∑k=1

σ(|k|N

)fkP′k(x)

.


Note that this would be a simple matrix-vector multiplication of the concen-

tration matrix and the modal co-efficients. This is a O(N2) computation in

each element. Most of the elemental operations in this method are of the same

complexity (including computing modal co-efficients from nodal solution).

6. Evaluate the enhanced kernel and identify those points for which it is beyond a

threshold as points of discontinuity. The maximum value of the enhanced kernel

across all points in the element can be considered as the elemental shock-sensor

value.

7. If one or more points in an element have been identified as being points of discon-

tinuity, we can then filter the solution in this element using any suitable method

- addition of artificial viscosity, polynomial order reduction (corresponding to a

sharp modal filter), exponential filter etc.

6.4.3 Extension to 2D and 3D Tensor Product

Elements

In order to adopt the concentration sensor as a tool for shock capturing in CFD, we

need to extend the method to 2D and 3D without adding too much complexity. In

the context of edge detection in images and MRIs with Fourier data, the method is

extended to the square domains by considering orthogonal 1D slices and computing

slice-based Fourier co-efficients as follows:

fk(yλ) =N∑

l=−N

ˆfk,leilyλ and fl(xν) =

N∑k=−N

ˆfk,leikxν

where yλ = −π + πλN

and xν = −π + πνN

where λ, ν = 0, 1, 2..., 2N

Using these slice based Fourier co-efficients, 2 edge maps are created: SσN [f ](x, yλ) and

SσN [f ](xν , y). These edge maps can then be used to determine points of discontinuities.

In order to determine the orientation of the shock, similar edge maps are created for


(a) A step-discontinuous function in 2D

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

1

2

3

4

5

6

(b) x-slices

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

1

2

3

4

5

6

(c) y-slices

Figure 6.6: Figure shows the performance of the sensor on a generalized step in 2Don a quadrilateral element. (a) shows a general two-dimensional step. (b) showsthe enhanced kernels for the x-slices and (c) shows the enhanced kernel portraits fory-slices


(a) A ramp function in 2D

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

0.5

1

1.5

(b) x-slices

−1

−0.5

0

0.5

1

−1

0

1

−1

−0.5

0

0.5

1

(c) y-slices

Figure 6.7: Figure shows the performance of the sensor on a ramp in 2D on a quadri-lateral element (a) shows a two-dimensional ramp (b) shows the enhanced kernels forthe x-slices and (c) shows the enhanced kernel portraits for y-slices


(a) Circular cylinder

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

5

10

15

20

(b) x-slices

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

5

10

15

20

(c) y-slices

Figure 6.8: Figure shows the performance of the sensor on a cylindrical step in 2Don a quadrilateral element (a) shows a cylindrical step function in 2D (b) shows theenhanced kernels for the x-slices and (c) shows the enhanced kernel portraits fory-slices


slices with a stagger to the original slices and a difference is taken.

This method however is quite expensive and is not necessary for our case. There

is an important difference between the application and the motivation for use of

the concentration method. In the context of edge detection, one is interested in

finding the exact locations and sizes of edges or discontinuities. In our case, we are

only interested in finding those elements which contain a discontinuity within them.

Therefore, we just perform several one-dimensional shock detections along slices in

the element.

In a tensor product element like quads and hexes, each x, y and z slice can be

considered as a 1D element and a 1D enhanced kernel portrait can be computed

using the 1D-concentration method for each such slice. The maximum value of the

enhanced kernel among all points in all slices can then be assigned as a the value of

the elemental “shock sensor” this to the element.

This simple strategy trades the accuracy of location and orientation of the shock

for significant saving of computational time. Since our shock capturing methods are

based at an element level, we do not profit from those extra calculations. Unlike the

traditional applications of concentration method where it needs to be applied only

once to determine the edges in an image, we need to perform this operation every

time step.

Figure 6.6 shows the enhanced kernel portraits along the x-slices as well as y-slices

separately. Figure 6.7 shows the same for a ramp function. The step function in

6.6 is deliberately oriented in a diagonal fashion to show that the method is capable

of detecting two-dimensional discontinuities. Figure 6.8 shows a general case with

circular step discontinuity.


(a) Triangle

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(b) Triangle

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

10

20

30

40

(c) Triangle

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

5

10

15

20

25

30

(d) Triangle

Figure 6.9: Figure shows the performance of the sensor on a generalized step in 2D ona triangle element (a) shows a generic step function inside a triangular element. (b)shows the tensor product Gauss-Legendre points of the quadrilateral when collapsedonto the triangle by collapsing over the top left point (c) and (d) show the enhancedkernels for the x and y slices respectively


6.4.4 Extension to Triangles and Tetrahedra

On triangles and tetrahedra, the approach of considering rows and columns (or x and

y slices) is not directly applicable as the number of points in a slice decreases in a

triangular fashion. However, we make use of the idea of collapsing a quadrilateral to

a triangle in order to detect shocks on triangular elements.

Consider the standard triangle (−1,−1), (1,−1) and (−1, 1) and the standard quadri-

lateral element (−1,−1), (1,−1), (1, 1) and (−1, 1). As a preprocessing step, compute

the locations of the points in the triangle that correspond to the Gauss-Legendre

points in the quadrilateral element. This depends on which point we collapse the

standard quadrilateral about to get the standard triangle. A point (ξ, η) in the stan-

dard quadrilateral element gets mapped to the standard tirangle using the following

transformation:

xtri = N1(ξ, η)v1 +N2(ξ, η)v2 +N3(ξ, η)v3 +N4(ξ, η)v4 (6.27)

where

N1(ξ, η) =1

4(1− ξ)(1− η)

N2(ξ, η) =1

4(1 + ξ)(1− η)

N3(ξ, η) =1

4(1 + ξ)(1 + η)

N1(ξ, η) =1

4(1− ξ)(1 + η)

(6.28)

We compute the locations for two sets - one for collapsing about (−1, 1) and the other

about (1,−1). For the former, we set v3 and v4 to (−1, 1). For the latter, we set v2

and v3 to (1,−1). Let us call the (p+ 1)2 number of points in the triangular element

obtained by these two different collapsing methods as xltri and xrtri respectively.

Now, suppose we have the solution uδ inside the triangular element at some timestep.


We start by interpolating the solution on to xltri and xrtri using appropriate interpola-

tion matrices. Let us denote this by uδ(xltri) and uδ(xrtri) respectively. We treat these

as though they were solutions inside a quadrilateral element and apply the concen-

tration method along 1D-slices. From figure 6.9, it can be seen that for the collapsing

over the top-left point, only the x-slices of the quadrilateral element seem to have a

structured slice-like orientation in the triangular element. Similarly, from Fig 6.10, we

can see that the y-slices have a better line-like structure when we collapse about the

bottom right point. In general, if we consider both x and y slices for any one choice of

edge collapsing, shocks are detected adequately well. Although this shock detection

mechanism on triangles is a slightly more heuristic approach compared to the theo-

retical setting used for quadrilaterals and 1D, it works extremely well in practice, as

can be seen in 7.


(a) Triangle

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(b) Triangle

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

1

2

3

4

5

6

(c) Triangle

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

20

40

60

80

100

(d) Triangle

Figure 6.10: Figure shows the performance of the sensor on a generalized step in 2D ona triangle element. (a) shows a generic step function inside a triangular element. (b)shows the tensor product Gauss-Legendre points of the quadrilateral when collapsedonto the triangle by collapsing over the bottom right point (c) and (d) show theenhanced kernels for the x and y slices respectively

Chapter 7

Numerical Experiments

This chapter deals with the numerical investigation of the non-linear stability of the

FR approach. We study applications of the shock detection and capturing methods

discussed in the previous chapters to solving both Euler and Navier Stokes equations.

Robustness while solving non-linear problems is one of the biggest challenges facing

high-order methods. In this chapter, we simulate a variety of different problems in

1D as well as 2D and show that the sub-cell shock capturing methods proposed in

the previous chapters work very well. While some testcases are used to validate the

convergence to correct solutions as well has higher order of accuracy, we also simulate

a couple of hard problems for shock capturing. In all problems except for the one with

a step in a wind tunnel with Mach 3 flow 7.7, we use only shock detection and filtering

and do not need the positivity preservation limiter. We start by briefly describing

the governing equations for the different testcases, i.e., the Euler and Navier Stokes

equations. Exact intial and boundary conditions for each testcase is described in their

respective sections.

142

CHAPTER 7. NUMERICAL EXPERIMENTS 143

7.1 Sod Shock Tube Problem

This one-dimensional testcase was introduced by Sod [95] in order to compare the

numerical behavior of various flow solvers when applied to a problem with shocks and

contact discontinuities. First let us consider the governing equations, which are the

1D Euler equations:∂U

∂t+

∂

∂x

(F (U)

)= 0 (7.1)

where

U =

ρ

ρu

ρE

and F (U ) =

ρu

ρu2 + p

u(ρE + p)

where ρ represents the density, u represents the velocity, E is the specific total energy,

E = e+ 12u2, where e = CvT is the specific internal energy. The Pressure p is given by

p = (γ−1)(ρE− 12ρu2). Note that in Part I of this thesis, we have used p to represent

the polynomial order of the solution. In this chapter, we use p to denote the pressure

and P for the polynomial degree used for representing the solution. The domain for

the testcase is a one-dimensional shock tube of length 2 given by Ω = [−1, 1]. The

initial conditions for the problem are as follows:

ρ(x, 0) =

1 for x < 0,

0.125 for x ≥ 0

p(x, 0) =

1 for x < 0,

0.1 for x ≥ 0

u(x, 0) = 0

(7.2)

We run the problem from t = 0 to t = 0.4 where none of the characteristics of

the Riemann problem would have reached the boundary. This initial state can be

produced by having a diaphragm in the middle of the tube. The gas to the left and

right of the diaphragm is initially at rest. The pressure and density are discontinuous


across the diaphragm. At t = 0, the diaphragm is broken. Two types of singularities

then propagate through the gas:

• Contact discontinuities: The pressure and velocity are continuous, but the den-

sity and specific internal energy e are discontinuous.

• Shock waves: All quantities are in general discontinuous across the shock front.

We have setup two testcases for this problem. Both the testcases use the same

common parameters except for the filter strengths α. Testcase one uses a weak filter

alpha = 0.4 while Testcase 2 uses a stronger filter with α = 4. The parameters are

as follows:

1. Number of elements: N = 100

2. Polynomial order in each element: P = 5, i.e., a 6th order scheme.

3. Shock Sensor parameters: Exponential concentration factors, Non-linear en-

hancement exponent q = 2, Critical threshold set to the average of the max.

values for step and ramp for P = 5, i.e., Jcrit = 6.4.

4. Filter parameters: Filter order s = 2, Modal cut-off Nc = 0. Filter strength

α = 0.4 for testcase 1 and α = 4 for testcase 2. Filtering performed after each

complete time-step using an TVD RK3 scheme with a CFL of 0.5.

Table 7.1 shows a comparison of the norms of the difference between the analytical

and numerical solutions for the two testcases. Figures 7.1 and 7.2 show the plots

comparing the numerical and anaytical results for testcase 1 and 2 respectively. We

can see that weaker filter performs better than the stronger one in terms of the

errors. However the weaker filter allows for an overshoot/undershoot of the velocity

(and therefore the Mach number M). The stronger filter controls such overshoot but

smears the shock. Such a tradeoff is commonly encountered in numerical treatments

of discontinuous solutions and one needs to balance this in an unsteady problem if

one needs to get the best results with a given mesh. From the robustness point of

view alone, the default settings of s = 2, α = 1 for the filter works quite well for


a wide range of testcases. Combining a weak filter with the positivity preservation

scheme discussed in Section 5.4 can avoid issues of obtaining negative density or

pressure and prevent the code from ‘NaNing’. However this was not necessary for the

testcases shown here as both the density and pressure stay positive throughout the

simulation.

From the figures, it is evident that apart from stabilizing the solution, the filter is

applied at the correct locations of the shocks and and does not destroy the accuracy of

the solution far away from the shocks. The shocks as well as the contact discontinuity

are well captured by the numerical method.

Variable Weak Filter Strong FilterError in L2 Error in L∞ Error in L2 Error in L∞

ρ 0.007615 0.059184 0.017483 0.094786u 0.037212 0.512378 0.074807 0.613601p 0.006965 0.085243 0.015243 0.110720M 0.031808 0.431799 0.065749 0.510206

Table 7.1: Norms of the difference between the numerical and analytical solutions att = 0.4 for the two testcases.


7.1.1 Testcase 1: Weak Filtering

−1 −0.5 0 0.5 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

De

nsity

Analytical

Numerical

(a) Density

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

Ve

locity

Analytical

Numerical

(b) Velocity

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

Ma

ch

Analytical

Numerical

(c) Mach

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Pre

ssu

re

Analytical

Numerical

(d) Pressure

Figure 7.1: Figure shows the results from the simulation of the flow through a shocktube results at t = 0.4 for testcase 1 with a weak filter α = 0.4


7.1.2 Testcase 2: Strong Filtering

−1 −0.5 0 0.5 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

De

nsity

Analytical

Numerical

(a) Density

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

Ve

locity

Analytical

Numerical

(b) Velocity

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

x

Ma

ch

Analytical

Numerical

(c) Mach

−1 −0.5 0 0.5 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Pre

ssu

re

Analytical

Numerical

(d) Pressure

Figure 7.2: Figure shows the results from the simulation of the flow through a shocktube results at t = 0.4 for testcase 2 with a strong filter α = 4

7.2 Shock-Entropy Interaction

In this section, we investigate the shock-entropy interaction problem introduced by

Shu and Osher [96]. The density solution to this problem consists of a fine structure

which is better resolved with higher order methods. We use this testcase to illustrate


that high order methods can be advantageous even in cases with shocks. We solve

the Euler equations (7.1) in a one-dimensional computational domain given by Ω =

[−5 5]. The testcase starts with a Mach 3 shock wave moving to the right into a

stationary fluid with a sinusoidal density variation.

−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Density

Reference Solution

P = 0; N = 4800 (NDOF = 4800)

(a) P = 0;N = 4800; NDoF = 4800

−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Density

Reference Solution

P = 7; N = 100 (NDOF = 800)

(b) P = 7;N = 100; NDoF = 800

−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Density

Reference Solution

P = 7; N = 200 (NDOF = 1600)

(c) P = 7;N = 200; NDoF = 1600

−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Density

Reference Solution

P = 7; N = 600 (NDOF = 4800)

(d) P = 7;N = 600; NDoF = 4800

Figure 7.3: Plots of the density at t = 1.8 for different combinations of polynomialdegree P and number of elements N

The initial conditions are as follows.

ρ = 3.857143; u = 2.629369; p = 10.33333 for x ≤ 4 (7.3)

ρ = 1 + ε sin 5x; u = 0; p = 1 for x > 4 (7.4)


If ε = 0, this is a pure Mach 3 right-moving shock wave. For this testcase we consider

ε = 0.2. As the shock wave passes through the density perturbation, it produces os-

cillations which develop into shocks of smaller amplitudes. A shock capturing scheme

that adds too much dissipation can damp out these oscillations, thereby losing the

fine structure of the density solution. We solve this problem with different values of

the polynomial order and number of elements to compare the performance of schemes

of various orders.

The other parameters used for this simulation are as follows:



values for step and ramp for each P .


α = 1 for all the different simulations. Filtering performed after each complete

time-step using an TVD RK3 scheme.

Figure 7.3 shows the density obtained from simulations at t = 1.8 for four different

cases along with the reference solution obtained from Shu and Osher [96]. The refer-

ence solution is obtained using a large Number of Degrees of Freedom (NDoF) and

can be considered as the exact solution. The eighth order method (P = 7) with just

100 elements ( NDoF 800) performs comparably to the first order method (P = 0)

with 4800 elements ( NDoF 4800) in terms of accuracy. As the number of elements is

increased further for the eighth order scheme, the accuracy improves and the method

recovers the exact solution with NDoF of 4800. This shows that, for problems with

fine structures like in most turbulent flows, high order methods can provide a signif-

icant benefit, even when there multiple shocks of different amplitudes.

Figure 7.4 compares two high order simulations with the same NDoF. Figure 7.4(b)

shows a magnified version of the plot zoomed in on the region in front of the shock

consisting of the oscillations and the small amplitude shocks. With a higher NDoF,

both fourth and eight order schemes perform very well and recover the exact solution.


Therefore we have chosen a relatively low NDoF in order to illustrate the differences.

The plots show that the eight order method performs slightly better than the fourth

order method, showing us that it can be beneficial to go to higher orders and that the

lower robustness of the schemes as the order is increased can be tackled effectively

with a robust shock capturing scheme.


−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Density

Reference Solution

P = 3; N = 200

P = 7; N = 100 (NDOF = 800)

(a) Density at t = 1.8

0.5 1 1.5 2 2.53

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

x

Density

(b) Magnified version of (a)

Figure 7.4: (a) shows a comparison of the plots of the density at t = 1.8 from a fourthorder and an eighth order scheme with the same NDoF. (b) shows a magnified versionof (a)


−5 0 5−0.5

0

0.5

1

1.5

2

2.5

3

x

Velo

city

P = 7; N = 600

(a) Velocity at t = 1.8

−5 0 50

2

4

6

8

10

12

x

Pre

ssure

P = 7; N = 600

(b) Pressure at t = 1.8

Figure 7.5: (a) shows a comparison of the plots of the density at t = 1.8 from a fourthorder and an eighth order scheme with the same NDoF. (b) shows a magnified versionof (a)


7.3 Blast Wave Problem

This testcase was introduced originally by Woodward and further explored by Wood-

ward and Colella [97] to illustrate the strong relationship between the accuracy of the

overall flow solution and the thinness of discontinuities on the grid. It involves mul-

tiple interactions of strong shocks and rarefactions with each other and with contact

discontinuities.

The equations solved are the 1D Euler equations (7.1) similar to the Sod shock tube

problem. The domain for the testcase is again a one-dimensional shock tube of unit

length given by Ω = [0, 1]. The intial conditions for the problem are as follows:

ρ(x, 0) = 1

u(x, 0) = 0

p(x, 0) =

1000 for x < 0.1,

0.01 for 0.1 ≤ x < 0.9,

100 for x ≥ 0.9,

(7.5)

Reflecting boundary conditions are used at the left and right walls. Two strong blast

waves develop and collide, producing a new contact discontinuity. In [97], a special

version of the scheme which treats the three regions as different fluids to track the

contact discontinuity carefully is used. We do not use any such special treatment.

The numerical solution in [97] is computed with a very fine grid (with 3096 zones)

which is adapted to the location of interaction of the two blast waves in order to

capture the dynamics very well. During the evolution of the flow, their scheme also

further refines the mesh by 8 times near the discontinuity in both space and time.

For a full description of the evolution of the flow, we refer the reader to [97].

The parameters are as follows:

1. Number of elements: N = 774. The elements are distributed in a fashion similar


0 0.2 0.4 0.6 0.8 1−10

−5

0

5

10

15

x

Ve

locity

(a) Velocity

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

De

nsity

x

(b) Density

(c) Woodward Colella results

Figure 7.6: Figure shows a comparison of our blastwave simulation results with thosefrom Woodward and Colella at t = 0.026. (a) shows the velocity and (b) shows thedensity. (c) shows the same quantities from the reference

to that in [97], i.e., we use a denser distribution in [0.68, 0.81].

2. Polynomial order in each element: P = 2, i.e., a 3rd order scheme.




values for step and ramp for P = 2, i.e., 238.3.

4. Filter parameters: Modal cut-off Nc = 0. Filter strength α = 1. Filter order

was set to be s = 2, Filtering performed after each complete time-step using an

TVD RK3.

0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10

12

14

16

x

Velo

city

(a) Velocity

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x

De

nsity

(b) Density

(c) Woodward Colella results

Figure 7.7: Figure shows a comparison of our blastwave simulation results with thosefrom Woodward and Colella at t = 0.038. (a) shows the velocity and (b) shows thedensity. (c) shows the same quantities from the reference.


Figure 7.6 shows our numerical solution as well as the reference solution from [97]

at time t = 0.026. Figure 7.7 shows a similar comparison at t = 0.038. Detailed

descriptions of the flow features shown in the figures can be found in [97], but in

general, the solutions match very well, except we have some smeared regions com-

pared to the reference solution. Considering the additional adaptive refinement that

they perform, this is expected. These results show the stabilizing capability of the

shock capturing method when applied to hard problems with multiple shocks, contact

discontinuities and interactions between them and its ability to not lose accuracy in

smoother regions.

7.4 Inviscid Transonic Flow: Structured

Quadrilateral Mesh

In the previous sections, we examined 1D testcases in order to validate our shock

capturing methodology. Now we move on to 2D testcases used in aircraft design.

This testcase also presents the first steady state case as the two 1D testcases were

unsteady. The numerical simulation of transonic flow over an airfoil is very important

for airfoil shape optimization as many aircraft design conditions are transonic. While

this problem has been used as a standard testcase for many years, Vassberg and

Jameson [98] have performed a detailed study of the grid convergence of flow solvers

like FLO82 [99, 100], OVERFLOW [101] and CFL3D [102] on a few different Mach

number and angle of attack configurations of the NACA 0012 airfoil.

The governing equations for this flow are the 2D Euler equations given by

∂U

∂t+∂F

∂x+∂G

∂y= 0 (7.6)


where

U =

ρ

ρu

ρv

ρE

, F (U ) =

ρu

ρu2 + p

ρuv

u(ρE + p)

and G(U) =

ρv

ρuv

ρv2 + p

v(ρE + p)

where u represents the velocity in the x direction and v represents the velocity in

the y direction. The other symbols have the same meaning as in the 1D Euler equa-

tions (7.1).

The computational domain is 200 chord-lengths diametrically. We have used a 64×64 O-mesh obtained by mapping a quasi circle to the airfoil geometry using the

Karman-Trefftz conformal transformation [103, 104]. All elements are quadrilateral.

Figure 7.8 shows both an outer view of the mesh as well as a zoomed in version.

The freestream conditions (which are used as the initial conditions as well) are as

follows:

ρ = 1 u = 0.799809 v = 0.017451 p = 0.714285 (7.7)

i.e., Ma = 0.8 AoA = 1.25 (7.8)

At the airfoil, a slip wall boundary condition that enforces a zero normal velocity is

used. At the farfield boundaries, at characteristic boundary condition is used.

The parameters used in the calculation are as follows:

1. Number of elements: N = 64× 64 = 4096

2. Polynomial order in each element: P = 4, i.e., a 5th order scheme. The total

number of degrees of freedom NDOF = 102, 400.

3. Shock Sensor parameters: Linear concentration factors, Non-linear enhance-

ment exponent q = 5, Critical threshold set to the average of the max. values

for step and ramp for P = 4, i.e., 16.52.


(a) Farfield mesh

(b) Mesh around airfoil

Figure 7.8: Figure shows the 64x64 structured quadrilateral mesh employed for thistestcase. (a) shows the farfield mesh while (b) provides a closer view of the mesh nearthe airfoil


(a) Mach

(b) Sensor

Figure 7.9: Figure shows the results for the simulation of an inviscid flow over aNACA 0012 airfoil at Ma = 0.8 and AoA = 1.25. (a) shows a flood plot of thedensity with the mesh overlaid on top and (b) shows the sensor distribution. Thecritical sensor threshold beyond which we filter is 16.52



α = 1.

The numerical simulation is performed using ZEFR, a code developed by Josh Romero

at the Aerospace Computing Laboratory at Stanford, which utilizes the Direct Flux

Reconstruction method (DFR) [37, 36]. We have simulated this problem with and

without multigrid and we converge to the same solution, as measured by the CD and

CL convergence. We use p-multigrid only with a V-cycle that cycles through all the

polynomial orders from P to 0. At the coarsest level (i.e. P = 0), we perform three

steps.

Figure 7.9 shows the Mach contours and the sensor values of the converged solution.

We have plotted the maximum value of the sensor for each element. If this maximum

value is beyond the predetermined threshold, we apply filtering in that element. It is

evident that the sensor is localized around the shock. It must also be noted that the

sensor threshold beyond which we apply filtering is 5.024. The mesh overlaid on top

of Fig. 7.9 shows how the shock is captured within an element, without necessitating

smearing over multiple elements. Figure 7.10(a) shows the distribution of the Pressure

coefficient Cp. We get a very good match with the plot obtained using FLO82 using a

4096 x 4096 grid in [98] which is reproduced in 7.10(b). Note that FLO82 includes

the influence of a point vortex on the far-field boundary condition which might make

the results differ slightly.

We also incur some amount of oscillations around the shock. This is quite common

while capturing shocks with high order methods and has been discussed by several

authors [68, 69, 70, 105]. While using artificial viscosity based methods to capture

shocks, these authors believe that having discontinuous artificial viscosity coefficients

over element boundaries is one of the major reasons for these oscillations and en-

forcing continuity of these artificial viscosity coefficients is important for alleviating

them. This enforcement of continuity of artificial viscosity coefficients is performed

using rather tedious and expensive procedures, like solving an additional diffusion

equation for the viscosity coefficient as in [70, 105] or by building C0-continuous


0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x

−C

P

Top

Bottom

(a) Pressure coefficient

(b) Pressure coefficient from FLO82

Figure 7.10: (a) shows the distribution of the Coefficient of Pressure over the airfoilwall for the converged solution resulting from a simulation of an inviscid flow overa NACA 0012 airfoil at Ma = 0.8 and AoA = 1.25. (b) shows the result obtainedusing FLO82 run with 4096× 4096 points


surfaces through other algorithms [69, 90]. In our experience, we find that these os-

cillations across element boundaries are usually small and do not affect the stability

and therefore can be handled using reconstruction at the end of the simulation as a

post-processing operation since the high order information is preserved.

7.4.1 Convergence to Steady State

Figure 7.11 shows the convergence history of the numerical solution run with p-

multigrid. Fig. (a) shows the L1-norm of the residual of the density equation over

the entire domain, i.e., ‖∇ · (ρu)‖L1 . We can see that the residual does not converge

to machine zero. Indeed, it saturates in the order of 10−3 − 10−2. However, note

the clear convergence of the coefficient of lift CL and CD in parts (c) and (d). This

discrepancy is due to the method used to add the dissipation needed to capture the

shock. As the filtering is performed as a post-processing step and is not included in

the solved PDE as an extra term, the residual of the original PDE (for any of the

four equations) does not converge to machine zero like in methods where the artificial

dissipation and therefore the residual is built into the equation.

Figure 7.12(b) gives the mean and standard deviations of CL and CD between 250,000

and 400,000 iterations.. We converge to a CL and CD value very close to the continuum

value of the OVERFLOW [101] code which reaches that value beyond 4096x4096

points. This is a great example which serves to show that robust shock capturing

along with maintaining high order accuracy is possible. By selectively treating only

a few elements where a shock is present prevents a strong filter or dissipation added

to smooth regions.

Since the residual does not provide a reasonable criterion for convergence in a general

problem, in these cases it is better to consider the rate of change of solution normalized

by the free stream value, i.e., ‖ρn+1 − ρn‖L∞/(ρ∞∆t). We can see that this quantity

decreases to relatively small values. Most importantly, the CL and CD converge

clearly. In fact we reach within a 3 digit accuracy within 50,000 iterations, which is


0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

10−3

10−2

10−1

100

Iterations

De

nsity R

esid

ua

l

(a) Residual History, i.e., ‖∇ · (ρu)‖L1

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Iterations

‖∆ρ‖∞/ρ∞∆t

(b) ‖ρn+1 − ρn‖L∞/(ρ∞∆t)

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

Iterations

CL

(c) CL

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

0.015

0.02

0.025

0.03

Iterations

CD

(d) CD

Figure 7.11: Figure shows the convergence history for the simulation of an inviscidflow over a NACA 0012 airfoil at Ma = 0.8 and AoA = 1.25. (a) shows the residualchange with time. (b) shows the L∞ norm of the change in the density from one time-step to the next, normalized by the time-step value. (c) shows the convergence historyof the coefficient of lift CL and (d) shows the convergence history of the coefficient ofdrag CD

a positive sign.


(a) OVERFLOW Results

CL CDMean 0.352052 0.022468Std 0.000178 0.000039

(b) Results from our numerical simu-lation

Figure 7.12: (a) shows the CL and CD for this case obtained using OVERFLOW. Itgives a table of CL and CD for OVERFLOW at various number of degrees of freedom.(b) shows the values obtained by us for the case described in this section. We getvery close to their continuum values with NDOF = 102,400


7.4.2 Convergence Acceleration

0 1 2 3 4 5 6 7 8

x 105

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

Iterations

CL

Without Multigrid

With Multigrid

(a) CL

0 1 2 3 4 5 6 7 8

x 105

0.015

0.02

0.025

0.03

Iterations

CD

Without multigrid

With multigrid

(b) CD

Figure 7.13: Figure compares the convergence histories of the CL and CD with andwithout employing p-multigrid. The p-multigrid simulation converges about eighttimes faster than the simulation without it in terms of the number of iterations

In Fig. 7.13 we show a comparison of the convergence of the CL and CD with and

without multigrid. It is evident that the multigrid provides almost an order of mag-

nitude reduction of the number of iterations. Even with the higher work performed

per time-step, we observe a reasonable speed-up. The acceleration in terms of iter-

ations is generally around 6-8x and about 2x in the wall clock time. However, the

convergence rate is still slow relative to low order codes like FLO82 which employ

the JST scheme [61, 99, 106]. In order to improve this convergence, we can utilize a

trick employed in the JST scheme, i.e., to add a high order dissipation term in the

smooth regions. In the JST scheme, a fourth order dissipation term is added in the

smooth regions while a second order dissipation term is used to capture shocks. To

this end, we apply a fourth order filter, i.e., s = 4, α = 1 in regions where the sensor

is not active or below the shock threshold.

Figure 7.14 shows a comparison of the convergence history of the relative change in

density over the time-step value. It is evident that such a filtering can provide a great

benefit in terms of accelerating convergence. The application of such a fourth order


0 1 2 3 4 5 6

x 105

10−12

10−10

10−8

10−6

10−4

10−2

100

Iterations

‖∆ρ‖∞/ρ∞∆t

p−Multigrid only

p−Multigrid + 4th order filter

Max. density change/Time−step

Figure 7.14: Figure compares the convergence histories of the shows the L∞ norm ofthe change in the density from one time-step to the next, normalized by the time-stepvalue, for two cases: One with p-multigrid only and the other with p-multigrid and aJST-style fourth order filter

filter in smooth regions will affect the order of accuracy of the scheme. However, one

could potentially use this solely for initial convergence acceleration and later switch

off the filtering or experiment with higher order filters. We do not go into the details

of optimizing this convergence acceleration here.

7.5 Inviscid Transonic Flow: Unstructured

Triangle Mesh

In the previous section, we examined the performance of the shock capturing tech-

niques for a two-dimensional steady state flow problem on a quadrilateral mesh. Here

we consider a similar testcase but with a completely unstructured triangle mesh in

order to examine the shock detection techniques we proposed for triangles in 6.4.4.


(a) Farfield Mesh

(b) Mesh near the airfoil

Figure 7.15: Figure shows the unstructured triangle mesh composed of 11,464 ele-ments employed for this testcase. (a) shows the farfield mesh while (b) provides acloser view of the mesh near the airfoil

We also consider a symmetric testcase here in order to test whether the shock cap-

turing methods affect solution symmetry on a symmetric testcase with a symmetric

mesh.

We again consider the NACA 0012 airfoil and a circular computation domain of


radius of 100 chord-lengths from the center of the airfoil. The mesh contains 11,784

triangular elements of second order. Figure 7.15 shows the farfield mesh as well as a

zoomed in view of the mesh near the airfoil.

The freestream conditions are as follows:

ρ = 1 u = 0.8 v = 0.0 p = 0.714285 (7.9)

i.e., Ma = 0.8 AoA = 0 (7.10)

At the airfoil, a slip wall boundary condition that enforces a zero normal velocity is

used. At the farfield boundaries, at characteristic boundary condition is used.

The numerical simulation is performed using ZEFR which is based on the Direct Flux

Reconstruction (DFR) method. [37, 36] Here, triangle elements are considered as

collapsed quadrilateral elements and the scheme uses the same methodology used for

the quadrilateral elements for triangles as well, except for handling the collapsed edge

carefully. In 6.4.4, for shock detection on triangles, we started with the solution on

a triangular domain and obtained the solution on a standard quadrilateral element

using the same idea of considering at as a collapsed triangle. In this collapsed edge

method, since we already have the solution on the (collapsed) quadrilateral element on

the Gauss-Legendre points, it saves us this extra step of converting from the triangle

solution points to the Gauss-Legendre points on the collapsed quadrilateral element.

The rest of the shock detection process proceeds as described in 6.4.4.

The parameters used in the simulation are as follows:

1. Number of elements: N = 11, 464

2. Polynomial order in each element: P = 3, i.e., a 4th order scheme, i.e., NDOF =

(P+1)2 ·N = 188544. Note that this is because triangles are handled in the form

of collapsed-edge quadrilaterals in ZEFR. The general FR method can handle

triangles separately [43, 40].


(a) Density

(b) Sensor

Figure 7.16: Figure shows the results for the simulation of an inviscid flow over aNACA 0012 airfoil at Ma = 0.8 and AoA = 0. (a) shows a flood plot of the densityand (b) shows the sensor distribution. The sensor cutoff for this case is 2.53






α = 1.

Figure 7.16 shows the density contours as well as the elemental sensor values from the

resulting converged numerical solution. It is evident that the sensor works very well

in this case, possibly even better than the quadrilateral case due to the unstructured

nature of the mesh.

Figure 7.17 shows a comparison of the CD obtained by our numerical simulation and

those obtained by [98] using OVERFLOW. We see that we get a slightly better than their

case 3 which uses more than 1 million degrees of freedom with just a fifth of their

degrees of freedom.

Figure 7.18 shows the distribution of the coefficient of pressure on the airfoil wall.

We get a very good match with the plot obtained using FLO82 using a 4096 x 4096

grid in [98]. Note that FLO82 includes the influence of a point vortex on the far-field

boundary condition while we do not.


(a) OVERFLOW

Mean StdCD 0.008400 0.000373

(b) Our numerical simulation

Figure 7.17: (a) shows CL and CD for OVERFLOW at various number of degrees offreedom. (b) shows the values obtained by us for the case described in this section.We get very good results with a much smaller number of degrees of freedom.


0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x

−C

P

Top

Bottom

(a) Pressure coefficient at the airfoil

Figure 7.18: Figure shows the distribution of the Coefficient of Pressure over theairfoil wall for the converged solution resulting from a simulation of an inviscid flowover a NACA 0012 airfoil at Ma = 0.8 and AoA = 0


7.6 Viscous Supersonic Flow: Hybrid Mesh

This testcase has been chosen mainly to highlight the capability of the shock capturing

technique we use to perform well in situations with other gradient-rich features like

vortices and boundary layers. We choose a viscous supersonic flow case over the

NACA 0012 airfoil. This is not a standard testcase with results for validation. We

therefore use this case just to exhibit the fact that the shock sensor has minimal

interference with the boundary layer or the vortices.

The computational domain is similar to that of the previous two testcases, i.e., it

consists of a circular domain of radius 100 chord-lengths. For this testcase, to show

the full capability of the shock capturing method to run on general hybrid meshes,

we have created a mixed mesh with a structured quadrilateral grid around the airfoil

and an unstructured triangle mesh in the farfield. The mesh consists of a total of

72,790 with 7,252 6-node second-order triangles and 65,538 9-node quadrilaterals. The

quadrilateral near-body mesh extends to three chord-lengths in the normal directions

as well as along the wake in the streamwise direction. The particularly high resolution

has been chosen to eliminate any excessive dissipation from the numerical scheme

apart from the shock capturing, so that the effects of the shock capturing scheme on

the flow features can be separated out and investigated.

In this case, we have chosen a supersonic flow of Mach 1.2 at a 2 angle of attack. We

use a no slip boundary condition at the airfoil boundary and characteristic boundary

conditions at the farfield boundary. We use Sutherland’s model for viscosity. The

freestream flow conditions which are also used as the initial conditions of the problem

are as follows:

Ma = 1.2 AoA = 2 Re = 60, 000 (7.11)




(a) Farfield triangle mesh

(b) Near-body quad mesh

Figure 7.19: Figure shows the hybrid structured and unstructured mesh composed of65,538 9-node second order quadrilateral elements and 7,252 6-node second triangleelements employed for this testcase. (a) shows the farfield unstructured triangle meshwhile (b) provides a closer view of the structured quadrilateral mesh near the airfoil

2. Polynomial order in each element: P = 3, i.e., a 4th order scheme.



for step and ramp for P = 3, i.e., at 12.765.



α = 1. Filtering performed after each complete time-step using a TVD RK3

scheme.

Figure 7.20: Figure shows the flood plot of the density for the simulation of a super-sonic viscous flow over a NACA 0012 airfoil at Ma = 1.2, AoA = 2 and Re = 60,000at t = 3.85s

Two important things to note for this problem are that we are using the non-linear

enhancement exponent of 5 and a filter order of 6. Firstly, we use a stronger non-

linear enhancement exponent in order to obtain a clear separation of scales between

the vortical regions or boundary layers and the shocks. Since the problem already

contains viscosity, we use a weaker filter of only order 6 except during startup, where

we used a filter of order 2 for the first 0.1s of the problem and then increase the filter

order to 6. This is due to the issue of starting with a supersonic initial condition in the


Figure 7.21: Figure shows a plot of the sensor distribution obtained from the simu-lation of a supersonic viscous flow over a NACA 0012 airfoil at Ma = 1.2, AoA = 2

and Re = 60,000 at t = 3.85s

entire domain. In fact, since the filtering is highly localized to around shocks, using a

filter order of 2 throughout the calculation does not affect the solution significantly,

but we increase the order of the filter to show that viscous cases need only weaker

filtering due to the inherent second order dissipation present in the system.

Figure 7.20 shows the density flood plot at a flow time of 3.85s. At this point, the

airfoil has begun to shed vortices, but the lambda shock is just about to start forming

behind the tail shocks. The sensor plot in 7.21 shows that the detector is not adding

dissipation in the vortex trail. The high values of the sensor we see compared to

previous testcases is due to the higher non-linear enhancement exponent of 5.


(a) Flow time t = 3.85s

(b) Flow time t = 5.50s

Figure 7.22: Figure shows snapshots of evolution of the vortex trail shed from theairfoil for the simulation of a supersonic viscous flow over a NACA 0012 airfoil atMa = 1.2, AoA = 2 and Re = 60,000


Fig. 7.22 shows zoomed in versions of plots focusing on the evolution of the vortex

trail. It can be seen that the vortex trail is captured in great detail. In Fig. (b) we

can see the beginnings of the formation of a lambda shock. Even the region near the

intersection of the lambda shock and the vortex trail has been captured well. Finally,

fig. 7.23 shows the further evolution of the lambda shock as it moves outward.



(b) Flow time t = 20.75s

Figure 7.23: Figure shows snapshots of evolution of the vortex trail shed from theairfoil as well as the λ-shock for the simulation of a supersonic viscous flow over aNACA 0012 airfoil at Ma = 1.2, AoA = 2 and Re = 60,000


7.7 Mach 3 Wind Tunnel With a Step

In this testcase, we simulate a hard two-dimensional testcase with a sharp corner,

multiple interactions of shocks, expansion fans and contact discontinuities. In this

case, we also encounter the need for the positivity preserving limiter during the startup

due to the sharp corner and the expansion fan driving the density and pressure very

low. The sharp corner is a singular point of the flow and the flow is seriously affected

by large numerical errors generated just in the neighborhood of this singular point if

nothing special is undertaken [97]. However, we take no special action at this corner

and yet get very good results comparable to the ones shown in [97].

Figure 7.24 shows the computational domain. The inflow boundary conditions at the

left which are also used as the initial conditions are as follows:

ρ = 1.4 u = 3.0 v = 0.0 p = 1 (7.12)

i.e., Ma = 3.0 (7.13)

0.6

3

0.2

1Ma = 3

Figure 7.24: Figure shows the computational domain which is a rectangular windtunnel with a step. The flow is from the left at Ma = 3



(b) Flow time t = 1s

Figure 7.25: Figure shows a flood plot of the density along with 30 contour linesplotted according to (7.14). (a) shows the density plot at t = 0.5s while (b) showsthe same for t = 1s

The sharp corner is known to generate gross errors if it is not handled specially.

Woodward and Colella [97] propose an artificial boundary condition at the corner.

This involves resetting the density and entropy values above the step immediately to

the right of the corner to the values to the left of the step; for the first few zones. This

boundary condition is equivalent to stating that the state around the corner is steady,


which is clearly inaccurate during the startup of the flow. This artificial condition is

just enforced in order to compare different methods in [97] without incurring these

errors. However, while using high-order unstructured grids, such a resetting of the

solution is hard and can be mesh-dependent. Although we use a structured mesh in

this computation, we are interested in finding the effects of not handling the corner

and studying the rest of the shock capturing and stabilizing properties.

Since we are not artificially handling the sharp corner, we need to increase the reso-

lution around the corner if we have to hope for a solution where the boundary layer

above the step does not contaminate the solution significantly. Therefore, we perform

this calculation on a much denser grid. We find that without any special treatment of

the corner, we are able to get very good matches with the results shown in [97]. We

are able to maintain fine shock structures and prevent Kelvin-Helmholtz instabilities

instigated by numerical errors that are often seen in this problem behind the shock

stem near the top.

We use a computational domain with 360x240 quadrilateral elements with P = 2,

i.e., a third order scheme. The mesh is compressed towards the corner in order to

get a higher resolution near the corner. We are able to get good results with coarser

meshes as well but it needs certain care and manipulation of the sensing and filtering

parameters in order to prevent the formation of a large enough boundary layer that

interacts with the shocks and changes the solution significantly. This dense mesh

allows us to capture all flow features precisely without special handling of the step,

evaluate post-shock oscillations, development of Kelvin-Helmholtz instability etc. For

this problem, we also had to use the positivity preserving limiter to handle formation

of negative density and pressure during the initial stages. Another difference from

the previous cases is that we set the sensor threshold to the maximum value of the

ramp instead of the average of the maximum values of the ramp and step. This is

necessary to handle the sharp turning of the flow at the corner. While the positivity

enforcing limiter is capable to handling temporary negative densities and velocities, it

does not handle the oscillations causing these negative values and they tend to build

up over time. Therefore it is necessary to handle these oscillations through the filter


and if we still need to enforce positivity, we then use this limiter.





1. Number of elements: N = 63004




ment exponent q = 5, Critical threshold set to the value of the ramp for P = 2,

i.e., 103.924.



scheme.

The details about the evolution of the flow can be found in [97]. In each figure we

plot the density field along with 30 contours between the values given by

ρmax + ρmin2

± 29

60(ρmax + ρmin) (7.14)

The same method is followed for the Pressure as well. Figures 7.25 7.26 7.27 show the

density evolution at various times. Finally 7.28 shows the density as well as the sensor

at t = 0.4. As can be seen in Fig. 7.28, The shock stem at the top left is captured in

a relatively fine fashion and yet we do not see the formation of any Kelvin Helmholtz

instability along the contact discontinuity behind it which is generally instigated by

numerical errors behind this stem. The shock sensor has been cutoff at the threshold

to show only those regions which are affected by the sensor at this time instant.

Finally, Fig. 7.29 shows the Pressure as well as the Adiabatic constant or the Entropy

A = p/ργ. We do seem to form a thin entropy boundary layer over the step but it

does not appear to contaminate the solution significantly. We obtain the various

shock locations in accordance to the results obtained in [97].






(a) Density at flow time t = 4s

(b) Sensor at flow time t = 4s

Figure 7.28: Figure shows the results obtained from the simulation of the flow in aMach 3 wind tunnel with a step at t = 4s. (a) shows a flood plot of the density alongwith 30 contour lines plotted according to (7.14). (b) shows the sensor distribution.The cutoff value is 103.924


(a) Pressure at flow time t = 4s

(b) Entropy at flow time t = 4s

Figure 7.29: Figure shows the results obtained from the simulation of the flow in aMach 3 wind tunnel with a step at t = 4s. (a) shows a flood plot of Pressure alongwith 30 contour lines plotted in a similar fashion to (7.14). (b) shows the Entropy orthe Adiabatic constant p/ργ


7.8 Shock Wave-Laminar Boundary Layer

Interaction

(a) Schlieren Visualization of the flow field

(b) Synopsis of the flow field

Figure 7.30: Fig.(a) shows a Schlieren visualization of the flow field obtained in Degrezet al. [107]. Fig.(b) shows a synopsis of the flow field

In this section, we consider a two dimensional shock wave boundary layer interaction

testcase for which both experimental and computational results are available for com-

parison. Interactions between shock waves and boundary layers occur frequently in


Figure 7.31: Figure shows the computational domain along with the boundary con-ditions imposed

transonic flows over wings and in high-speed inlets like those of ramjets. It is impor-

tant for a shock capturing scheme to capture this interaction well without affecting

the boundary layer significantly. We choose the testcase studied experimentally and

computationally in Degrez et al. [107]. This testcase was also proposed in the 4th In-

ternational High Order Workshop (HOW). The only difference between the testcase

studied in Degrez et al. and that proposed at the HOW is in the length of the com-

putational domain, which is slightly wider in the latter case to alleviate the effects of

the outflow boundary condition implementation on the flow field and the separation

point. We use this wider computational domain. Figure 7.30 shows the Schlieren

visualization from the experiments in [107] along with a synopsis of the flow field,

while 7.31 shows the computational domain in the HOW testcase.

The freestream Mach number is 2.15 and the freestream Temperature is 288.15K.

The Reynolds number based on the freestream quantities and the distance between

the leading edge of the plate and the abscissa of impingement of the inviscid shock

(xsh) with the plate is 105. The angle between the incident shock wave and the x-axis

is σ = 30.8. In this configuration, the flow remains stationary and two-dimensional.


The position of the shock wave is set through supersonic inlet conditions with a

uniform state corresponding to a Mach number of 2.15 for y ≤ y0 (inlet0) with

y0 = 1.2 tanσ. For y > y0, i.e., inlet1, the quantities are set such that they satisfy

the Rankine Hugoniot conditions across the oblique shock.A non-reflecting boundary

condition is imposed at the top of the domain (top), while an outflow condition is set

at the outlet ( outlet). A no-slip adiabatic condition is imposed at the plate and a

symmetric boundary condition is imposed for x ≤ 0, y = 0, i.e., for sym.

The mesh used for this simulation is composed of 19,000 quadrilateral elements with

higher density of elements near the leading edge of the plate and near xsh.









scheme.

Figure 7.32 shows the results obtained from the simulation. It is evident that the

sensor is not active in the boundary layer. Since this case has weak oblique shocks,

the filtering action subsides the sensor below its threshold value everywhere as can be

seen in the figure. In Figure 7.33, we compare the pressure distribution obtained from

our simulation to the results submitted to the High Order Workshop by University

of Bergamo which uses a modal DG scheme with the Godunov method for inviscid

fluxes. We compare with their P = 6 simulation with 11,041 elements.


(a) Flood plot of the density

(b) Sensor plot

Figure 7.32: Figure shows the steady state results. (a) shows the non-dimensionaldensity and (b) shows the sensor. The density is normalized by its freestream value


0 0.5 1 1.5 21

1.1

1.2

1.3

1.4

1.5

1.6

x

P/P

∞y = 0 (wall)

Our Simulation

U. Bergamo

(a) y = 0 (plate)

0 0.5 1 1.5 21

1.1

1.2

1.3

1.4

1.5

1.6

y = 0.1

x

P/P

∞

Our Solution

U. Bergamo

(b) y = 0.1

Figure 7.33: Figure shows a comparison of the pressure distribution at the plateand at y = 0.1 between our simulation and the results submitted to the High OrderWorkshop by University of Bergamo


7.9 Inviscid Transonic Flow over a Sphere

In this final testcase, we simulate the transonic flow over a sphere. This is not a

standard testcase, but we use this mainly to demonstrate that the shock capturing

methodology can be easily extended to three dimensions. The mesh is composed of

approximately 87,000 hexahedral elements. The radius of the sphere is 0.75 and the

computational domain is spherical and extends to a radius of 50. The mesh gets

coarsened with increasing radial distance in a geometric fashion. The mesh over the

sphere is shown in 7.34. The mesh is generated by splitting the sphere into 6 faces

and extruding the mesh.

Figure 7.34: Figure shows the mesh at the walls of the sphere

The inflow Mach number is 0.7 and is along the x-direction, i.e., the angle of attack

is zero. At the surface of the sphere, a slip-wall boundary condition is imposed and

at the farfield boundary, a farfield boundary condition is imposed. At this Mach

number, a normal shock is formed over the sphere. The parameters used for this


simulation are as follows:








scheme.

Figure 7.35 shows a flood plot of the Mach number over the sphere and Figure 7.36

shows the Mach number distribution along the z = 0 slice. The Mach number reaches

a value of 1.52 in front of the shock. The shock is located approximately at an angle

of 114.1 degrees from the horizontal or x axis when measured from the front of the

sphere where the flow impinges.


Figure 7.35: Figure shows the Mach number distribution over the surface of the spherewith the mesh overlaid on top


Figure 7.36: Figure shows the Mach number flood plot along the z = 0 slice

Chapter 8

Conclusions

In Part I of this dissertation, a detailed investigation of the stability of the FR for-

mulation on tensor product elements was performed and it was shown that the FR

formulation utilizing the 1D VCJH correction functions on tensor product elements

provides stable schemes for both linear advection and advection-diffusion equations

on Cartesian meshes whenever the VCJH parameter c is non-negative. While stable

schemes had been formulated in 1D and on simplex elements earlier, the stability

properties of the tensor product formulation were not studied in detail thus far due

to certain major difficulties.

In order to overcome these difficulties, a norm different from the one used in 1D and

for simplex elements was formulated and it was shown that this partial Sobolev norm

is non-increasing as long as c ≥ 0 and certain (common) conditions are satisfied.

Since the solution is represented using a polynomial basis, norm equivalence can then

be invoked to show that the L2 energy of the solution cannot grow in an unbounded

fashion, thereby proving stability of the numerical scheme. The L2 norm of the

solution can however temporarily increase without contradicting this analysis and

such a temporary increase can indeed be observed for numerical experiments involving

central fluxes at interfaces. In such cases, the Sobolev norm of the solution remains

a constant while the solution and its derivatives exhange energy with each other and

197

CHAPTER 8. CONCLUSIONS 198

an oscillatory behavior is observed for the L2 norm of the solution.

In addition to proving stability, the newly formulated norm for tensor product ele-

ments also displays an explicit dependence on the VCJH parameter c, which is unlike

the 1D or simplex element cases where stability analysis only predicted a bound for

c, above which stability of these schemes is guaranteed. This explicit dependence

provides an intuition for the results obtained through numerical experiments. In par-

ticular, the higher dissipation and correspondingly higher stability of the schemes

as c increases can be anticipated directly from the results of the stability analysis.

Also, in 1D, the stability analysis breaks as c is decreased below a certain value c−,

but it is not clear whether the schemes would blow up as c is decreased below this

value. The results obtained in Chapter 3 show that when c becomes negative, there

is a competing effect between stable and unstable contributions and that the schemes

become less stable as c is decreased.

Although the analysis is focused on quadrilateral elements, the extension to hexa-

hedral elements, i.e., 3D Cartesian meshes is believed to be straightforward when

utilizing the new norm formulated for tensor product elements. This answers the last

major open question regarding the linear stability of the FR formulation.

In Part II of this dissertation, focus is shifted towards nonlinear instabilities aris-

ing due to discontinuous solutions or shocks. Discontinuous solutions carry multiple

threats to a numerical scheme. Apart from destabilizing the numerical scheme, they

can also leave behind persistent oscillations which can cause non-physical solutions or

lead to loss of accuracy around the discontinuous regions, or even farther away from

them if not handled appropriately. While a wide variety of methods have been devel-

oped in CFD for the treatment of shocks, very few are truly suitable for high order

unstructured methods. With this as a motivation, a shock detection and capturing

method that can be used by any finite element type method and for any nonlinear

PDE has been proposed.

The higher resolution available within a cell or element is high order methods allow

for a sub-cell resolution of shocks. To achieve this in a computationally efficient


fashion, filtering has been proposed as the shock capturing tool. Filtering is a non-

intrusive approach where the original PDE remains unaffected and the solution is

filtered as a post-processing step after every (or every few) time-steps. This provides a

major advantage in terms of computational efficiency over artificial viscosity methods,

especially in the context of explicit time-stepping methods and is also very suitable for

implementation on GPUs since all operations can be cast in the form of matrix-matrix

multiplications.

The disadvantage of filtering over artificial viscosity methods is the lack of an efficient

approach for varying the amount of dissipation across the domain in a smooth fashion.

In order to tackle this effectively, a robust shock detection mechanism which can

clearly distinguish between shocks and other gradient rich regions like vortices and

boundary layers becomes necessary. In this regard, a novel shock detection technique

inspired by the method of concentration used in image detection was developed. While

the concentration property of Fourier expansions is used in image edge detection,

our method utilizes a similar concentration property of Jacobi polynomials to detect

regions with shocks. In comparison to image edge detection, the number of polynomial

modes available in the context of CFD is often much lower. A clear guideline for

selecting parameters so as to effectively handle this has been laid out and the method

has been shown to work very well even at polynomial degrees as low as 2.

A positivity preserving limiter along with a Strong Stability Preserving (SSP) time-

stepping schemes have been used to provide robustness against formation of negative

or unphysical solutions along with the shock capturing tools.

The following major conclusions can be drawn from the numerical experiments:

• The results from the Shu Osher shock-entropy interaction problem (Section 7.2)

show that simulations using the same number of degrees of freedom perform

better when higher order polynomials are used in spite of there being multiple

shocks. This highlights that although it might be harder to stabilize a high

order scheme compared to its low order counterparts, and the order of accuracy

is reduced near the shock, if the shock capturing is handled effectively and in a


sub-cell fashion, the benefit of higher accuracy away from the shocks provided

by high order methods can be retained.

• The transonic inviscid flow cases over NACA 0012 airfoil (Sections 7.4 and 7.5)

show shock capturing capability in two dimensional structured as well as un-

structured grids on both quadrilateral and triangular elements. They also ex-

hibit that the filtering setup can easily be used for convergence acceleration

apart from shock capturing.

• The blast wave (Section 7.3) and forward facing step (Section 7.7) simulations

exhibit the robustness of the method at highly adverse flow conditions and the

capacity to recover accurate solutions without the need for mesh adaptation.

• The supersonic viscous flow over the NACA 0012 airfoil (Section 7.6) and the

shock-wave boundary layer interaction (Section 7.8) cases show that the shock

detector is capable of clearly distinguishing between shocks and regions with

vortices or boundary layers and that the shock capturing method works well

when multiple shocks are present along with such viscous flow structures.

• The transonic flow over a sphere (Section 7.9) shows that the shock detection

and capturing methods can be extended to 3D in a straightforward fashion.

Although not shown in any numerical simulations, this shock detection and capturing

technique can be used for other nonlinear PDEs as well since it is based only on

detecting discontinuities using the smoothness (or the lack thereof) of the solution.

The framework can also be utilized for stabilizing against aliasing instabilities, which

is a common application for filters that we use for shock capturing.

Appendices

201

Appendix A

Partial Sobolev Norm

Let us reconsider the partial Sobolev norm used in Theorem 3.3.5 in Chapter 3.

‖uD‖2W 2p,2δ

=N∑k=1

(∫Ωk

[(uDk )2+

c

2

((∂puDk∂ξp

)2+(∂puDk∂ηp

)2)

+c2

4

( ∂2puDk∂ξp∂ηp

)2]dΩk

)(A.1)

Notice that the norm has uD in the physical domain while the derivatives are with

respect to the reference coordinates. We can use the additional notation we introduced

for the Cartesian mesh geometry to rewrite this completely in the physical domain

as follows

‖uD‖2W 2p,2δ

=N∑k=1

∫Ωk

[(uDk )2 +

c

2

(J2px

(∂puDk∂xp

)2+ J2p

y

(∂puDk∂yp

)2)

+c2

4J2px J

2py

( ∂2puDk∂xp∂yp

)2]dΩk

(A.2)

From our analysis of stability, we can see that we are mainly interested in c ≥ 0, since

for c < 0, the Θextra term contributes towards instability in the simplest case of a

uniform Cartesian mesh. However, as an exercise, it is interesting to investigate the

range of c for which the above is a norm.

202

APPENDIX A. PARTIAL SOBOLEV NORM 203

In Equation A.2 we write the norm completely in the physical domain. However for

algebraic manipulations, it is better to write the norm completely in the reference

domain. Since the norm in the domain Ω is a sum of the norms inside each element,

it is sufficient to consider the norm in a single element (kth element) as follows

‖uDk ‖2W 2p,2δ

=

1∫−1

1∫−1

[(uDk )2 +

c

2

((∂puDk∂ξp

)2+(∂puDk∂ηp

)2)

+c2

4

( ∂2puDk∂ξp∂ηp

)2]dξdη (A.3)

where we have intentionally left out the constant factor Jk that multiplies all the

terms since we are interested in investigating when the norm is non-negative and a

positive multiplicative factor does not affect this. Till now the transformed solution

uD has been represented using the pth degree tensor-product Lagrange polynomial

basis. However, we can equivalently expand our solution in a pth degree tensor product

Legendre polynomial basis:

uD =

p∑i=0

p∑j=0

Li(ξ)Lj(η)uij (A.4)

where uij represent the modal coefficients. This is usually referred to as the modal

form while the Lagrange expansion is referred to as the nodal form of the solution.

We can change from one form to the other using the corresponding Vandermonde

matrix. An important difference between the Lagrange and Legendre polynomials

is that the nth Legendre polynomial is of degree n unlike the Lagrange polynomials

which are all of degree p. Now we substitute the above expression for uD into the


norm definition A.3. The first term can be written as follows

1∫−1

1∫−1

(uD)2dξdη =

p∑i=0

p∑m=0

p∑j=0

p∑n=0

uijumn

1∫−1

1∫−1

Li(ξ)Lm(ξ)Lj(η)Ln(η)dξdη

=

p∑i=0

p∑j=0

u2ij

1∫−1

1∫−1

L2i (ξ)L

2j(η)dξdη

+

p∑i=0

p∑m=0m 6=i

p∑j=0

p∑n=0

uijumn

1∫−1

1∫−1

Li(ξ)Lm(ξ)Lj(η)Ln(η)dξdη

+

p∑i=0

p∑j=0

p∑n=0n6=j

uijuin

1∫−1

1∫−1

L2i (ξ)Lj(η)Ln(η)dξdη

(A.5)

By using the orthogonality property of the Legendre polynomials, we get

1∫−1

1∫−1

(uD)2dξdη =

p∑i=0

p∑j=0

(2

2i+ 1

)(2

2j + 1

)u2ij (A.6)

Now the pth ξ-derivative can be written in terms of Legendre polynomials as follows

∂puD

∂ξp=dpLp(ξ)

dξp

p∑j=0

Lj(η)upj = app!

p∑j=0

Lj(η)upj (A.7)

where one may recall that ap is the leading coefficient of Lp. Note that we have

used the fact that the pth derivative of Ln(ξ) for n < p is 0 in the above expression.

Therefore we have

1∫−1

1∫−1

(∂puD∂ξp

)2dξdη = 2(app!)

2

p∑j=0

1∫−1

L2j(η)u2

pjdη = 2(app!)2

p∑j=0

(2

2j + 1

)u2pj (A.8)


Similarly we have the pth η-derivative

1∫−1

1∫−1

(∂puD∂ηp

)2dξdη = 2(app!)

2

p∑i=0

(2

2i+ 1

)u2ip (A.9)

Now we consider the last term of the norm

∂2puD

∂ξp∂ηp=dpLp(ξ)

dξpdpLp(η)

dηpupp = (app!)

2upp (A.10)

Therefore1∫

−1

1∫−1

(∂2puD

∂ξp∂ηp

)2

dξdη = 4(app!)4u2

pp (A.11)

From equations A.6, A.8, A.9 and A.11, we can see that the norm inside the kth

element can be written as

‖uDk ‖2W 2p,2δ

=

p−1∑i=0

p−1∑j=0

(2

2i+ 1

)(2

2j + 1

)u2ij

+

p−1∑j=0

[(2

2p+ 1

)(2

2j + 1

)+c

22(app!)

2

(2

2j + 1

)]u2pj

+

p−1∑i=0

[(2

2p+ 1

)(2

2i+ 1

)+c

22(app!)

2

(2

2i+ 1

)]u2ip

+

[4

(2p+ 1)2+

2(app!)2

2p+ 1c+ (app!)

4c2

]u2pp

(A.12)

In order for this to be a norm, we need the co-efficients of each uij have to be non

negative. Therefore we have the following 2 conditions,

1

2p+ 1+c

2(app!)

2 ≥ 0 =⇒ c ≥ −2

(2p+ 1)(app!)2(A.13)

4

(2p+ 1)2+

2(app!)2

2p+ 1c+ (app!)

4c2 ≥ 0 (A.14)


The first condition is the same as the one obtained for the 1D case in [29]. The LHS

of the second condition is a convex quadratic with a negative discriminant, therefore

condition 2, i.e., Equation A.14 is always satisfied. Therefore, the condition on c for

A.3 to be a norm is the same as obtained in 1D.

Appendix B

Proof of Θadv ≤ 0

In Chapter 3, we obtained

d

dt‖uD‖2 = Θadv + cΘextra

where

‖uD‖2 =N∑k=1

(∫Ωk

[(uDk )2 +

c

2

((∂puDk∂ξp

)2

+

(∂puDk∂ηp

)2)+c2

4


)2]dΩk

)

is a broken Sobolev norm of the solution in the entire domain,

Θadv =N∑k=1

(−∫Ωk

uD(∇ · fD)dΩk −∫Γk

uD(fC · n)dΓk

)(B.1)

207

APPENDIX B. PROOF OF ΘADV ≤ 0 208

and

Θextra =N∑k=1

(J2p+1yk

[1

2

∂puDR∂yp

∂pFDR

∂yp− ∂puDR


∂yp

]k

+ J2p+1yk

[− 1

2

∂puDL∂yp

∂pFDL

∂yp− ∂puDL


∂yp

]+ J2p+1

xk

[1

2

∂puDT∂xp

∂pGDT

∂xp− ∂puDT


∂xp

]+ J2p+1

xk

[− 1

2

∂puDB∂xp

∂pGDB

∂xp− ∂puDB


∂xp

])(B.2)

There, we mentioned that the term Θadv is similar to the one obtained by Castonguay

et al. [38] for triangles and that a similar approach can be used to show that this term

is non-positive. We then proceeded to show that the other term Θextra is non-positive.

Here, we present the proof for the non-positivity of the Θadv term for completeness.

Let us consider the integrand of the first term inside the summation of Θadv:

uD(∇ · fD) =1

2∇ · (uDfD), (B.3)

which is due to the fact that we have a linear advective flux fDk = auD. Using this,

and the divergence theorem, we can rewrite (B.1) as

Θadv =N∑k=1

(− 1

2

∫Γk

uD(fD · n)dΓk −∫Γk

uD(fC · n)dΓk

)(B.4)

Although the above summation is over elements, since it consists only of boundary

integrals, it can be rewritten as a sum over all interfaces by aggregating the contri-

butions to each edge from its neighboring elements. Since we have a periodic domain

and since the FR approach works along 1D lines on quadrilaterals and does not dis-

tinguish between x and y directions, it is sufficient to show that the contribution of

these terms to a generic edge is non-positive.

Now, (fC)|Γe ·n =((f ∗− fD) ·n

)|Γe where Γe represent a generic element interface.


Therefore, this becomes

Θadv =N∑k=1

(1

2

∫Γk

uD(fD · n)dΓk −∫Γk

uD(f ∗ · n)dΓk

)(B.5)

Now let us reconsider the expression for the common numerical flux given by the Lax

Friedrichs flux

f ∗ = fD+λ

2

(max

u∈[uD− ,uD+ ]

∣∣∣∣∂f∂u · n∣∣∣∣)[[uD]] (B.6)

Rewriting this for the case of the linear advection equation, we get

f ∗ = fD+λ

2|a · n|[[uD]] (B.7)

Now we consider a generic vertical edge and accumulate the contributions to Θadv

from the two neighboring elements. To do this, let − and + subscripts denote the

element on the left and right of this vertical boundary respectively. For the element

on the left, this edge is its right boundary and for the right element, it is the left

boundary. Then we have

(f ∗ · n−) =1

2a(uD− + uD+) +

λ

2|a|(uD− − uD+) (B.8)

and (f ∗ ·n+) = −(f ∗ ·n−) Combining the above, the contributions to a vertical edge

become∫Γe

[1

2a(uD−)2 − a

2uD−(uD−+uD+)− λ

2|a|uD−(uD− − uD+)− 1

2a(uD+)2

+a

2uD+(uD− + uD+) +

λ

2|a|uD+(uD− − uD+)

]dΓe

(B.9)

All the terms not containing λ conveniently cancel out leaving us with∫Γe

[− λ

2|a|uD−(uD− − uD+) +

λ

2|a|uD+(uD− − uD+)

]dΓe (B.10)


which further simplifies to∫Γe

−λ2|a|(uD− − uD+)2dΓe ≤ 0 for λ ≥ 0 (B.11)

Since we considered a generic edge and showed the contribution to Θadv is non-

positive, and since we consider a periodic domain where every edge can be considered

as an internal interface, we can conclude that

Θadv ≤ 0 (B.12)

Bibliography

[1] P. E. Vincent and A. Jameson. Facilitating the Adoption of Unstructured High-

Order Methods Amongst a Wider Community of Fluid Dynamicists. Math.

Model. Nat. Phenom., 6(3):97–140, may 2011.

[2] Sanjiva K Lele. Compact Finite Difference Schemes with Spectral-like Resolu-

tion. J. Comput. Phys., 103, 1992.

[3] Timothy J. Barth and Paul O. Frederickson. Higher Order Solution of the

Euler Equations on Unstructured Grids using Quadratic Reconstruction. In

28th Aerosp. Sci. Meet., Reno Nevada, 1990.

[4] Timothy J Barth and Herman Deconinck. High-order methods for computational

physics, volume 9. Springer Science & Business Media, 2013.

[5] Michel Delanaye and Yen Liu. Quadratic Reconstruction Finite Volume

Schemes bn 3D Arbitrary Unstrtictured Polyhedral Grids. Technical report,

1999.

[6] Ami Harten, Bjorn Engquist, Stanley Osher, and Sukumar R. Chakravarthy.

Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III. J.

Comput. Phys., 131(1):3–47, feb 1997.

[7] R. Abgrall. On Essentially Non-oscillatory Schemes on Unstructured Meshes:

Analysis and Implementation. J. Comput. Phys., 114(1):45–58, sep 1994.

211

BIBLIOGRAPHY 212

[8] Carl F. Ollivier-Gooch. Quasi-ENO Schemes for Unstructured Meshes Based on

Unlimited Data-Dependent Least-Squares Reconstruction. J. Comput. Phys.,

133(1):6–17, may 1997.

[9] Xu-Dong Liu, Stanley Osher, and Tony Chan. Weighted Essentially Non-

oscillatory Schemes. J. Comput. Phys., 115(1):200–212, nov 1994.

[10] Changqing Hu and Chi-Wang Shu. Weighted Essentially Non-oscillatory

Schemes on Triangular Meshes. J. Comput. Phys., 150(1):97–127, mar 1999.

[11] Oliver Friedrich. Weighted Essentially Non-Oscillatory Schemes for the Inter-

polation of Mean Values on Unstructured Grids. J. Comput. Phys., 144(1):194–

212, jul 1998.

[12] W. H. Reed and T. R. Hill. Triangular Mesh Methods for the Neutron Transport

Equation. In Proc. Am. Nucl. Soc., 1973.

[13] Bernardo Cockburn and Chi-Wang Shu. The Local Discontinuous Galerkin

Method for Time-Dependent Convection-Diffusion Systems. SIAM J. Numer.

Anal., 10(2):443–461, 2000.

[14] J. Peraire and P.-O. Persson. The Compact Discontinuous Galerkin (CDG)

Method for Elliptic Problems. SIAM J. Sci. Comput., 30(4):1806–1824, jan

2008.

[15] Jim Douglas and Todd Dupont. Interior Penalty Procedures for Elliptic and

Parabolic Galerkin Methods. In Comput. Methods Appl. Sci., pages 207–216.

Springer Berlin Heidelberg, Berlin, Heidelberg, 1976.

[16] F. Bassi and S. Rebay. A High-Order Accurate Discontinuous Finite Element

Method for the Numerical Solution of the Compressible Navier–Stokes Equa-

tions. J. Comput. Phys., 131(2):267–279, mar 1997.

[17] B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of

discontinuous Galerkin, mixed and continuous Galerkin methods for second

BIBLIOGRAPHY 213

order elliptic problems, 2008.

[18] B Cockburn, G E Karniadakis, C W Shu, and M Griebel. Discontinuous

Galerkin Methods Theory, Computation and Applications. Lectures Notes in

Computational Science and Engineering, vol. 11. Inc. Marzo del, 2000.

[19] Jan S. Hesthaven and Tim Warburton. Nodal Discontinuous Galerkin Meth-

ods: Algorithms, Analysis, and Applications. Springer Publishing Company,

Incorporated, 2010.

[20] David A. Kopriva and John H. Kolias. A Conservative Staggered-Grid

Chebyshev Multidomain Method for Compressible Flows. J. Comput. Phys.,

125(1):244–261, apr 1996.

[21] Yen Liu, Marcel Vinokur, and Z.J. Wang. Spectral difference method for un-

structured grids I: Basic formulation. J. Comput. Phys., 216(2):780–801, aug

2006.

[22] Ht T Huynh. A Flux Reconstruction Approach to High-Order Schemes In-

cluding Discontinuous Galerkin Methods. AIAA Pap., 2007-4079(June):1–42,

2007.

[23] H T Huynh. A Reconstruction Approach to High-Order Schemes Including

Discontinuous Galerkin for Diffusion. In 47th AIAA Aerosp. Sci. Meet., 2009.

[24] Y. Allaneau and A. Jameson. Connections between the filtered discontinuous

Galerkin method and the flux reconstruction approach to high order discretiza-

tions. Comput. Methods Appl. Mech. Eng., 200(49-52):3628–3636, dec 2011.

[25] D. De Grazia, G. Mengaldo, D. Moxey, P. E. Vincent, and S. J. Sherwin. Con-

nections between the discontinuous Galerkin method and high-order flux recon-

struction schemes. Int. J. Numer. Methods Fluids, 75(12):860–877, 2014.

[26] P. Zwanenburg and S. Nadarajah. Equivalence between the Energy Stable

Flux Reconstruction and Filtered Discontinuous Galerkin Schemes. J. Comput.

BIBLIOGRAPHY 214

Phys., 306:343–369, 2016.

[27] G. Mengaldo, D. De Grazia, P. E. Vincent, and S. J. Sherwin. On the Con-

nections Between Discontinuous Galerkin and Flux Reconstruction Schemes:

Extension to Curvilinear Meshes. J. Sci. Comput., 67(3):1272–1292, jun 2016.

[28] Gianmarco Mengaldo. Discontinuous spectral/hp element methods: develop-

ment, analysis and applications to compressible flows. PhD thesis, 2015.

[29] P. E. Vincent, P. Castonguay, and A. Jameson. A New Class of High-Order

Energy Stable Flux Reconstruction Schemes. J. Sci. Comput., 47(1):50–72,

2011.

[30] Z.J. Wang and Haiyang Gao. A unifying lifting collocation penalty formulation

including the discontinuous Galerkin, spectral volume/difference methods for

conservation laws on mixed grids. J. Comput. Phys., 228(21):8161–8186, nov

2009.

[31] R L Manuel, Jonathan Bull, Jacob Crabill, Joshua Romero, Abhishek She-

shadri, Jerry E Watkins Ii, David Williams, Francisco Palacios, and Antony

Jameson. Verification and Validation of HiFiLES : a High-Order LES unstruc-

tured solver on multi-GPU platforms. AIAA Aviat., (June):1–27, 2014.

[32] Peter Vincent, Freddie D. Witherden, Antony M. Farrington, George Ntemos,

Brian C. Vermeire, Jin S. Park, and Arvind S. Iyer. PyFR: Next-Generation

High-Order Computational Fluid Dynamics on Many-Core Hardware. In 22nd

AIAA Comput. Fluid Dyn. Conf., Reston, Virginia, jun 2015. American Insti-

tute of Aeronautics and Astronautics.

[33] F.D. Witherden, B.C. Vermeire, and P.E. Vincent. Heterogeneous computing

on mixed unstructured grids with PyFR. Comput. Fluids, 120:173–186, 2015.

[34] Brian C Vermeire, Freddie D Witherden, and Peter E Vincent. On the Utility of

GPU Accelerated High-Order Methods for Unstructured Grids: A Comparison

Between PyFR and Industry Standard Tools. In AIAA Aviat., 2015.

BIBLIOGRAPHY 215

[35] Kartikey Asthana and Antony Jameson. High-Order Flux Reconstruction

Schemes with Minimal Dispersion and Dissipation. J. Sci. Comput., 62(3):913–

944, 2014.

[36] J. Romero, K. Asthana, and A. Jameson. A Simplified Formulation of the Flux

Reconstruction Method. J. Sci. Comput., 67(1):351–374, apr 2016.

[37] Joshua Romero and Antony Jameson. Extension of the Flux Reconstruction

Method to Triangular Elements using Collapsed-Edge Quadrilaterals.

[38] P. Castonguay, P. E. Vincent, and A. Jameson. A new class of high-order energy

stable flux reconstruction schemes for triangular elements. J. Sci. Comput.,

51(1):224–256, 2012.

[39] D.M. Williams, P. Castonguay, P.E. Vincent, and A. Jameson. Energy stable

flux reconstruction schemes for advection–diffusion problems on triangles. J.

Comput. Phys., 250:53–76, 2013.

[40] D. M. Williams and A. Jameson. Energy Stable Flux Reconstruction Schemes

for Advection–Diffusion Problems on Tetrahedra. J. Sci. Comput., 59(3):721–

759, jun 2014.

[41] Anne Gelb and Dennis Cates. Detection of edges in spectral data III-refinement

of the concentration method. J. Sci. Comput., 36(1):1–43, 2008.

[42] P. Castonguay, D.M. Williams, P.E. Vincent, and A. Jameson. Energy stable

flux reconstruction schemes for advection–diffusion problems. Comput. Methods

Appl. Mech. Eng., 267:400–417, 2013.

[43] David Williams. Energy Stable High-Order Methods For Simulating Unsteady,

Viscous, Compressible Flows On Unstructured Grids. PhD thesis, Stanford

University, 2013.

[44] P. Castonguay. High-Order Energy Stable Flux Reconstruction Schemes for

Fluid Flow Simulations on Unstructured Grids. PhD thesis, 2012.

BIBLIOGRAPHY 216

[45] Antony Jameson. A Proof of the Stability of the Spectral Difference Method

for All Orders of Accuracy. J. Sci. Comput., 45(1-3):348–358, oct 2010.

[46] A. Jameson, P. E. Vincent, and P. Castonguay. On the non-linear stability of

flux reconstruction schemes. J. Sci. Comput., 50(2):434–445, 2012.

[47] Abhishek Sheshadri and Antony Jameson. On the Stability of the Flux Recon-

struction Schemes on Quadrilateral Elements for the Linear Advection Equa-

tion. J. Sci. Comput., 67(2):769–790, 2016.

[48] Abhishek Sheshadri and Antony Jameson. Erratum to: On the Stability of

the Flux Reconstruction Schemes on Quadrilateral Elements for the Linear

Advection Equation. J. Sci. Comput., 67(2):791–794, may 2016.

[49] P. E. Vincent, P. Castonguay, and A. Jameson. A New Class of High-Order

Energy Stable Flux Reconstruction Schemes. J. Sci. Comput., 47(1):50–72, apr

2011.

[50] H. S. Carslaw. A historical note on Gibbs’ phenomenon in Fourier’s series and

integrals. Bull. Am. Math. Soc., 31(8):420–425, oct 1925.

[51] David Gottlieb, Chi-Wang Shu, Alex Solomonoff, and Herve Vandeven. On the

Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial

sum of a nonperiodic analytic function. J. Comput. Appl. Math., 43(1-2):81–98,

nov 1992.

[52] D. Gottlieb and J.S. Hesthaven. Spectral methods for hyperbolic problems. J.

Comput. Appl. Math., 128(1):83–131, 2001.

[53] Abhishek Sheshadri and Antony Jameson. Shock detection and capturing meth-

ods for high order Discontinuous-Galerkin Finite Element Methods. 32nd AIAA

Appl. Aerodyn. Conf. AIAA 2014-2688 Downloaded, (June):1–11, 2014.

[54] Abhishek Sheshadri, Jacob Crabill, and Antony Jameson. Mesh deformation

BIBLIOGRAPHY 217

and shock capturing techniques for high-order simulation of unsteady compress-

ible flows on dynamic meshes. In SciTech 2015, number January, pages 1–15,

2015.

[55] Bernardo Cockburn and Chi-Wang Shu. TVB Runge-Kutta Local Projection

Discontinuous Galerkin Finite Element Method for Conservation Laws II. Gen-

eral Framework. Math. Comput., 52(186):411–435, 1989.

[56] Bernardo Cockburn, San Yih Lin, and Chi Wang Shu. TVB runge-kutta local

projection discontinuous galerkin finite element method for conservation laws

III: One-dimensional systems. J. Comput. Phys., 84(1):90–113, 1989.

[57] B Cockburn, S Hou, and S W Shu. The Runge–Kutta Local Projection

Discontinuous Galerkin Finite Element Method for Conservation Laws IV:

The Multidimensional Case. Math. Comp., 54(190):545–581, 1990.

[58] Bernardo Cockburn and Chi-Wang Shu. The Runge–Kutta Discontinuous

Galerkin Method for Conservation Laws V. J. Comput. Phys., 141(2):199–224,

1998.

[59] L. Krivodonova, J. Xin, J. F. Remacle, N. Chevaugeon, and J. E. Flaherty.

Shock detection and limiting with discontinuous Galerkin methods for hyper-

bolic conservation laws. Appl. Numer. Math., 48(3-4):323–338, 2004.

[60] Lilia Krivodonova. Limiters for high-order discontinuous Galerkin methods. J.

Comput. Phys., 226(1):879–896, 2007.

[61] Antony Jameson, Wolfgang Schmidt, and Eli Turkel. Numerical Solution of the

Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping

Schemes. In AIAA 14th Fluid Plasma Dyn. Conf., 1981.

[62] Andrew W. Cook and William H. Cabot. A high-wavenumber viscosity for

high-resolution numerical methods. J. Comput. Phys., 195(2):594–601, 2004.

BIBLIOGRAPHY 218

[63] Andrew W. Cook and William H. Cabot. Hyperviscosity for shock-turbulence

interactions. 2005.

[64] Andrew W Cook. Artificial Fluid Properties for Large-Eddy Simulation of

Compressible Turbulent Mixing. 2007.

[65] S. Kawai and S.K. Lele. Localized artificial diffusivity scheme for discontinuity

capturing on curvilinear meshes. J. Comput. Phys., 227(22):9498–9526, nov

2008.

[66] Sachin Premasuthan. Towards an Efficient And Robust High Order Accurate

Flow Solver For Viscous Compressible Flows. PhD thesis, 2010.

[67] Sachin Premasuthan, Chunlei Liang, and Antony Jameson. Computation Of

Flows with Shocks Using Spectral Difference Scheme with Artificial Viscosity.

In 48th AIAA Aerosp. Sci. Meet., 2010.

[68] Po Persson and Jaime Peraire. Sub-cell shock capturing for discontinuous

Galerkin methods. AIAA Pap., pages 1–13, 2006.

[69] Per-olof Persson. Shock Capturing for High-Order Discontinuous Galerkin Sim-

ulation of Transient Flow Problems. In 21st AIAA Comput. Fluid Dyn. Conf.,

pages 1–9, 2013.

[70] Garrett E. Barter and David L. Darmofal. Shock capturing with PDE-

based artificial viscosity for DGFEM: Part I. Formulation. J. Comput. Phys.,

229(5):1810–1827, 2010.

[71] Mark H. Carpenter, David Gottlieb, and Chi Wang Shu. On the Conserva-

tion and Convergence to Weak Solutions of Global Schemes. J. Sci. Comput.,

18(1):111–132, 2003.

[72] Richard Courant and David Hilbert. Methods of mathematical physics. Vol. 1.

Wiley, 1989.

BIBLIOGRAPHY 219

[73] Milton Abramowitz and Irene A. Stegun. Handbook of mathematical functions

: with formulas, graphs, and mathematical tables. Dover Publications, 1970.

[74] Xiangxiong Zhang and Chi Wang Shu. On maximum-principle-satisfying high

order schemes for scalar conservation laws. J. Comput. Phys., 229(9):3091–3120,

2010.

[75] Xiangxiong Zhang and Chi Wang Shu. Positivity-preserving high order discon-

tinuous Galerkin schemes for compressible Euler equations with source terms.

J. Comput. Phys., 230(4):1238–1248, 2011.

[76] Xiangxiong Zhang and Chi-Wang Shu. Numerische Mathematik A minimum

entropy principle of high order schemes for gas dynamics equations. Numer.

Math, 121:545–563, 2012.

[77] Xiangxiong Zhang and Chi Wang Shu. Positivity-preserving high order finite

difference WENO schemes for compressible Euler equations. J. Comput. Phys.,

231(5):2245–2258, 2012.

[78] Eitan Tadmor. A minimum entropy principle in the gas dynamics equations.

Appl. Numer. Math., 2(3-5):211–219, 1986.

[79] Yu Lv and Matthias Ihme. Entropy-bounded discontinuous Galerkin scheme

for Euler equations. J. Comput. Phys., 295:715–739, 2015.

[80] Sigal Gottlieb and Chi-Wang Shu. Total variation diminishing Runge-Kutta

schemes. Math. Comp., 67(221):73–85, 1998.

[81] Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor. Strong Stability-Preserving

High-Order Time Discretization Methods *. Soc. Ind. Appl. Math., 43(1):89–

112, 2001.

[82] Sigal Gottlieb, David I. Ketcheson, and Chi-Wang Shu. High Order Strong

Stability Preserving Time Discretizations. J. Sci. Comput., 38(3):251–289, mar

2009.

BIBLIOGRAPHY 220

[83] Chi-Wang Shu and Stanley Osher. Efficient Implementation of Essentially Non-

oscillatory Shock-Capturing Schemes. J. Comput. Phys., 439471:439–471, 1988.

[84] Eric Johnsen, Johan Larsson, Ankit V Bhagatwala, William H Cabot, Parviz

Moin, Britton J Olson, Pradeep S Rawat, Santhosh K Shankar, Bjorn Sjogreen,

H C Yee, Xiaolin Zhong, and Sanjiva K Lele. Assessment of high-resolution

methods for numerical simulations of compressible turbulence with shock waves.

J. Comput. Phys., 229:1213–1237, 2009.

[85] Anne Gelb and Eitan Tadmor. Enhanced spectral viscosity approximations for

conservation laws. Appl. Numer. Math., 33(1):3–21, 2000.

[86] R.C Swanson and Eli Turkel. On central-difference and upwind schemes. J.

Comput. Phys., 101(2):292–306, aug 1992.

[87] F Ducros, V Ferrand, F Nicoud, C Weber, D Darracq, C Gacherieu, and

T Poinsot. Large-Eddy Simulation of the Shock/ Turbulence Interaction. J.

Comput. Phys., 152:517–549, 1999.

[88] D.J Hill and D.I Pullin. Hybrid tuned center-difference-WENO method for

large eddy simulations in the presence of strong shocks. J. Comput. Phys.,

194(2):435–450, 2004.

[89] Sergio Pirozzoli. Conservative Hybrid Compact-WENO Schemes for Shock-

Turbulence Interaction. J. Comput. Phys., 178(1):81–117, may 2002.

[90] A. Klockner, T. Warburton, and J. S. Hesthaven. Viscous Shock Capturing in

a Time-Explicit Discontinuous Galerkin Method. Math. Model. Nat. Phenom.,

X(X):1–27, 2011.

[91] Anne Gelb and Eitan Tadmor. Detection of Edges in Spectral Data. Appl.

Comput. Harmon. Anal., 7(1):101–135, 1999.

[92] Kourosh Raeen. A Study of The Gibbs Phenomenon in Fourier Series and

Wavelets. 2008.

BIBLIOGRAPHY 221

[93] H. S. (Horatio Scott) Carslaw and 1870-1954. Introduction to the theory of

Fourier’s series and integrals, 1930.

[94] G Szego. Orthogonal Polynomials. Number v. 23 in American Mathematical

Society colloquium publications. American Mathematical Society, 1959.

[95] Gary A Sod. A survey of several finite difference methods for systems of non-

linear hyperbolic conservation laws. J. Comput. Phys., 27(1):1–31, apr 1978.

[96] Chi-Wang Shu and Stanley Osher. Efficient Implementation of Essentially

Non-oscillatory Shock-Capturing Schemes, II. In Upwind and High-Resolution

Schemes, pages 328–374. Springer Berlin Heidelberg, Berlin, Heidelberg, 1989.

[97] Paul Woodward and Phillip Colella. The numerical simulation of two-

dimensional fluid flow with strong shocks. J. Comput. Phys., 54(1):115–173,

1984.

[98] J C Vassberg and A Jameson. In pursuit of grid convergence for two-dimensional

Euler solutions. J. Aircr., 47(4):1152–1166, 2010.

[99] Antony Jameson and Timothy J Baker. Solution of the euler equations for

complex Problems. AIAA, 1929:293–302, 1983.

[100] A. Jameson and T. Baker. Multigrid solution of the Euler equations for aircraft

configurations. In 22nd Aerosp. Sci. Meet., Reston, Virigina, jan 1984. American

Institute of Aeronautics and Astronautics.

[101] Robert H. Nichols and Pieter G. Buning. User’s Manual for OVERFLOW 2.2.

[102] Sherrie L. Krist, Robert T. Biedron, and Christopher L. Rumsey. CFL3D User’s

Manual (Version 5.0), 1998.

[103] L. M. (Louis Melville) Milne-Thomson. Theoretical aerodynamics. Dover Pub-

lications, New York, 4th ed. edition, 1973.

[104] L. M. Milne-Thomson. Applied Aerodynamics, 1940.

BIBLIOGRAPHY 222

[105] Garrett E Barter. Shock capturing with PDE-based artificial viscosity for an

adaptive, higher-order discontinuous Galerkin finite element method. PhD the-

sis, 2008.

[106] A Jameson. Analysis and Design of Numerical Schemes for Gas Dynamics, 2:

Artificial Diffusion and Discrete Shock Structure. Int. J. Comput. Fluid Dyn.

Compo Fluid Dyn, 5(5):1–2, 1995.

[107] G. Degrez, C. H. Boccadoro, and J. F. Wendt. The interaction of an oblique

shock wave with a laminar boundary layer revisited. An experimental and nu-

merical study. J. Fluid Mech., 177(-1):247, apr 1987.

Date post:	06-Apr-2020
Category:	Documents
Upload:	others
View:	3 times
Download:	0 times

AN ANALYSIS OF STABILITY OF THE FLUX RECONSTRUCTION FORMULATION WITH APPLICATIONS TO...

Documents