University of Patras - Nemertes

University of Patras 

School of Engineering 

Department of Mechanical Engineering and 

Aeronautics 

A dissertation submitted to the 

Department of Mechanical Engineering and Aeronautics and the 

University of Patras in partial fulfillment of the requirements for the degree of 

DOCTOR OF PHILOSOPHY 

in 

Mechanical Engineering 

High-order discontinuous Galerkin 

discretization for flows with strong moving 

shocks 

Author: 

Dipl.-Ing 

Konstantinos Kontzialis 

Supervisor: 

Prof. 

John A. Ekaterinaris 

© Konstantinos Kontzialis. All rights reserved. 

Patras 2012

1 

Thesis committee: 

Prof. John A. Ekaterinaris, Department of Mechanical Engineering and 

Aeronautics, University of Patras 

Prof. Yannis Kallinderis, Department of Mechanical Engineering and 


Prof. Kyriakos Giannakoglou, Department of Mechanical Engineering, 

National Technical University of Athens 

Examining committee: 

Prof. John A. Ekaterinaris, Department of Mechanical Engineering and 


Prof. Yannis Kallinderis, Department of Mechanical Engineering and 


Prof. Kyriakos Giannakoglou, Department of Mechanical Engineering, 

National Technical University of Athens 

Prof. Vasilis Kostopoulos, Department of Mechanical Engineering and 


Prof. Dimitris Saravanos, Department of Mechanical Engineering and 


Prof. Efstratios Gallopoulos, Department of Computer Engineering and 

Informatics, University of Patras 

Prof. John Tsamopoulos, Department of Chemical Engineering, University of 

Patras

Acknowledgments 

The present thesis is the culmination of years of hard work. Many challenges and 

difficulties appeared during my research, which undoubtedly were the source of 

inspiration and creation in many aspects of my PhD work. 

I am deeply thankful to my supervisor Professor Yannis Ekaterinaris, a charismatic 

person, whose experience proved to be the source of guidelines and knowledge 

of unprecedented value. His patience, encouragement and spirit of giving, and most 

importantly his calm perseverance, were the key elements for the successful completion 

of my PhD research. 

I would like also to express my gratitude to Professor Yannis Kallinderis who 

was the first person to initiate me in the wonderful field of CFD. His love for excellence 

taught me that ideas mature only through continuous and hard work. 

I express also my gratitude to the support of all the persons whose love and 

presence were of importance for a deeper understating and continuous effort. 

Lastly, I am deeply thankful to my very supportive family for the love and 

encouragement to keep on and succeed in my goals. 

2

Summary 

Supersonic flows over both simple and complex geometries involve features over a 

wide spectrum of spatial and temporal scales, whose resolution in a numerical solution 

is of significant importance for accurate predictions in engineering applications. 

While CFD has been greatly developed in the last 30 years, the desire and necessity 

to perform more complex, high fidelity simulations still remains. 

The present thesis has introduced two major innovations regarding the fidelity 

of numerical solutions of the compressible Navier-Stokes equations. The first one is 

the development of new a priori mesh quality measures for the Finite Volume (FV) 

method on mixed-type (quadrilateral/triangular) element meshes. Elementary types 

of mesh distortion were identified expressing grid distortion in terms of stretching, 

skewness, shearing and non-alignment of the mesh. Through a rigorous truncation 

error analysis, novel grid quality measures were derived by emphasizing on the direct 

relation between mesh distortion and the quality indicators. They were applied 

over several meshes and their ability was observed to identify faithfully irregularlyshaped 

small or large distortions in any direction. It was concluded that accuracy 

degradation occurs even for small mesh distortions and especially at mixed-type 

element mesh interfaces the formal order of the FV method is degraded no matter 

of the mesh geometry and local mesh size. 

Therefore, in the present work, the high-order Discontinuous Galerkin (DG) 

discretization of the compressible flow equations was adopted as a means of achieving 

and attaining high resolution of flow features on irregular mixed-type meshes 

for flows with strong moving shocks. During the course of the thesis a code was 

developed and named HoAc (standing for High Order Accuracy), which can perform 

via the domain decomposition method parallel p-adaptive computations for 

flows with strong shocks on mixed-type element meshes over arbitrary geometries 

at a predefined arbitrary order of accuracy. In HoAc in contrast to other DG developments, 

all the numerical operations are performed in the computational space, 

for all element types. This choice constitutes the key element for the ability to perform 

p-adaptive computations along with modal hierarchical basis for the solution 

expansion. The time marching of the DG discretized Navier-Stokes system is per- 

3

formed with the aid of explicit Runge-Kutta methods or with a matrix-free implicit 

approach. 

The second innovation of the present thesis, which is also based on the choice 

of implementing the DG method on the regular computational space, is the development 

of a new p-adaptive limiting procedure for shock capturing of the implemented 

DG discretization. The new limiting approach along with positivity preserving limiters 

is suitable for computations of high speed flows with strong shocks around 

complex geometries. The unified approach for p-adaptive limiting on mixed-type 

meshes is achieved by applying the limiters on the transformed canonical elements, 

and it is fully automated without the need of ad hoc specification of parameters as 

it has been done with standard limiting approaches and in the artificial dissipation 

method for shock capturing. 

Verification and validation studies have been performed, which prove the correctness 

of the implemented discretization method in cases where the linear elements 

are adequate for the tessellation of the computational domain both for subsonic and 

supersonic flows. At present HoAc can handle only linear elements since most grid 

generators do not provide meshes with curved elements. 

Furthermore, p-adaptive computations with the implemented DG method were 

performed for a number of standard test cases for shock capturing schemes to illustrate 

the outstanding performance of the proposed p-adaptive limiting approach. 

The obtained results are in excellent agreement with analytical solutions and with 

experimental data, proving the excellent efficiency of the developed shock capturing 

method for the DG discretization of the equations of gas dynamics. 

4

To my beloved family, to whom I owe everything. 

5

Nomenclature 

c k e(t) elemental degrees of freedom 

∆x e , ∆y e 

projections of the dual edges in the x, y directions 

∆x e,k , ∆y e,k 

grid metrics 

γ 

λ 

Λ 

Θ e 

J 

k 

adiabatic exponent 

thermal conductivity coefficient 

diagonal eigenvalue matrix 

auxiliary variable for gradient computation 

Jacobian matrix 

unit vector in an arbitrary direction 

L, R left and right eigenvectors 

n 

U 

u 

M kj 

R 

R e 

S k 

V k 

outward normal vector 

conservative variable vector 

contravariant velocity component 

mass matrix 

universal gas constant 

elemental residual 

line integral 

volume integral 

µ dynamic viscosity of the fluid 

6

7 

ω 

skewness angle 

e x xx, e x yy, . . . normalized error coefficients 

φ 

ρ 

τ 

θ 

shearing angle 

density 

two dimensional viscous stress tensor 

mesh rotation angle 

Ũ(x, t) solution over the computational domain 

Ũ e (x, t) FE interpolant 

Ω q st 

Ω e 

standard element configuration 

elemental space 

E(x, y) error in gradient computation 

b e k (x) elemental basis functions 

c 

c p 

c V 

d x , d y 

E 

e 

E x , E y 

local speed of sound 

specific heat at constant pressure 

specific heat at constant volume 

stretching factors 

total energy per unit volume 

internal energy per unit mass 

error in the computation of the x, y derivatives 

e x x, e x y, . . . error coefficients 

F i 

F v 

f x , f y 

H 

h 

inviscid flux tensor 

viscous flux tensor 

grid functions 

numerical flux 

(specific) enthalpy of the gas

8 

k 

L x , L y 

M 

Ma 

N 

p 

P r 

Boltzmann’s constant 

local element lengths 

Laplacian approximation 

Mach number 

order of approximation 

pressure 

Prandtl number 

Q, q mesh quality indices 

Q 1 , Q 2 , Q m , Q n 

number of quadrature points 

q x , q y 

Re 

s 

T 

heat fluxes 

Reynolds number 

entropy 

temperature 

u, v Cartesian velocity components 

u e 

v 

field value at the middle of each edge of the dual contour 

test or weighting function 

∇ h u, ∇u numerical and analytical value of the gradient 

S 

dual area

Acronyms 

TE Truncation Error 

FE Finite Element 

FD Finite Difference 

FV Finite Volume 

SD Spectral Difference 

SV Spectral Volume 

ENO Essentially non-oscillatory 

WENO Weighted essentially non-oscillatory 

DG Discontinuous Galerkin 

RK Runge-Kutta 

RKDG Runge-Kutta Discontinuous Galerkin 

TVB Total variation bounded 

SSP Strong stability preserving 

IC Initial conditions 

BC Boundary conditions 

EC Error coefficients 

CV Control Volume 

PETSc Portable Extensible Toolkit for Scientific Computing 

HoAc High Order Accuracy 

9

10 

GMRES Generalized Minimum Residual 

JFNK Jacobian Free Newton Krylov 

LDG Local Discontinuous Galerkin 

BO Baumann and Oden 

BR1 First Bassi-Rebay scheme 

BR2 Second Bassi-Rebay scheme 

NIPG Non-symmetric Interior Penalty Method 

DOF Degrees of freedom 

PDEs Partial differential equations 

ODEs Ordinary differential equations 

MPI Message Passage Interface 

GPUs Graphics Processor Units

Contents 

Contents 6 

1 Introduction 23 

1.1 Classical methods in CFD . . . . . . . . . . . . . . . . . . . . . . . . 27 

1.2 Why the DG method . . . . . . . . . . . . . . . . . . . . . . . . . . 31 

1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 

2 Governing equations of gas dynamics 37 

2.1 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 

2.1.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 

2.1.2 Polytropic ideal gas . . . . . . . . . . . . . . . . . . . . . . . . 39 

2.1.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 

2.1.4 Mathematical character of the Euler equations . . . . . . . . . 42 

2.2 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . 44 

2.2.1 Mathematical character of the Navier-Stokes equations . . . . 45 

2.3 Non-dimensional form of the Navier-Stokes equations . . . . . . . . . 46 

2.4 Integral form of the compressible flow equations . . . . . . . . . . . . 48 

3 Mesh quality measures for the FV method 49 

3.1 Gradient computation in the FV method . . . . . . . . . . . . . . . . 52 

3.2 The general form of the truncation error for the node based FV method 53 

3.3 Consistency condition for the node based FV method . . . . . . . . . 56 

3.4 Verification of the error coefficients analytic expressions . . . . . . . . 59 

11

CONTENTS 12 

3.5 Elementary Types of Mesh Distortion . . . . . . . . . . . . . . . . . . 60 

3.6 Stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 

3.6.1 Skewness and Shearing . . . . . . . . . . . . . . . . . . . . . . 63 

3.6.2 Mesh rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 

3.6.3 Mixed-type element mesh interfaces . . . . . . . . . . . . . . . 65 

3.7 Direct relation between truncation error and mesh distortion . . . . . 66 

3.8 Definition of an appropriate index of mesh distortion . . . . . . . . . 68 

3.8.1 Normalized error coefficients . . . . . . . . . . . . . . . . . . . 68 

3.8.2 Mesh quality index . . . . . . . . . . . . . . . . . . . . . . . . 69 

3.8.3 Analytic expressions of the quality index Q for each type of 

elementary mesh distortion . . . . . . . . . . . . . . . . . . . . 72 

3.8.4 Calibration of the mesh quality index Q . . . . . . . . . . . . 73 

3.9 Application of the mesh quality indices . . . . . . . . . . . . . . . . . 74 

3.9.1 Channel grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 

3.9.2 Airfoil grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 

3.9.3 Cylinder grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 

4 Discontinuous Galerkin discretization 87 

4.1 Spatial Discontinuous Galerkin discretization . . . . . . . . . . . . . . 88 

4.1.1 Discontinuous weak formulation for the Euler equations . . . . 89 

4.1.2 The Numerical flux . . . . . . . . . . . . . . . . . . . . . . . . 92 

4.1.3 Basis functions and standard elemental configuration . . . . . 95 

4.1.4 Elemental operations . . . . . . . . . . . . . . . . . . . . . . . 100 

4.1.5 Selection of Gauss type numerical integration . . . . . . . . . 102 

4.1.6 Mapping between physical and computational space . . . . . . 103 

4.2 DG discretization for the Navier-Stokes Equations . . . . . . . . . . . 105 

4.2.1 Computation of the viscous terms in the DG method . . . . . 105 

4.2.2 The LDG method . . . . . . . . . . . . . . . . . . . . . . . . . 107 

4.3 The treatment of initial and boundary conditions . . . . . . . . . . . 110 

4.4 Wall boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 111

CONTENTS 13 

4.5 Far field boundary conditions . . . . . . . . . . . . . . . . . . . . . . 113 

4.6 Application of Initial Conditions . . . . . . . . . . . . . . . . . . . . . 115 

5 Time discretization 117 

5.1 Runge-Kutta methods . . . . . . . . . . . . . . . . . . . . . . . . . . 118 

5.2 Implicit time marching . . . . . . . . . . . . . . . . . . . . . . . . . . 121 

5.3 Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 

5.4 Krylov subspace methods . . . . . . . . . . . . . . . . . . . . . . . . 123 

5.5 Jacobian-free Newton-Krylov Method . . . . . . . . . . . . . . . . . . 124 

5.6 Preconditioning of the JFNK method . . . . . . . . . . . . . . . . . . 126 

6 Unified limiting 128 

6.0.1 Limiting for quadrilateral elements . . . . . . . . . . . . . . . 131 

6.0.2 Limiting for triangular elements . . . . . . . . . . . . . . . . . 134 

6.1 Positivity preserving limiters for the DG method . . . . . . . . . . . . 136 

7 Numerical results 139 

7.1 Solution output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 

7.2 Code Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 

7.2.1 Convergence study for the Euler equations . . . . . . . . . . . 141 

7.2.2 Convergence study for the Navier-Stokes equations . . . . . . 143 

7.2.3 Inviscid flow over a cylinder . . . . . . . . . . . . . . . . . . . 144 

7.2.4 Standard Sod’s shock tube problem . . . . . . . . . . . . . . . 147 

7.2.5 Extreme Sod’s Riemann problem . . . . . . . . . . . . . . . . 149 

7.3 Supersonic flow over a cylinder . . . . . . . . . . . . . . . . . . . . . 151 

7.4 Double Mach Reflection of a strong shock . . . . . . . . . . . . . . . 157 

7.5 Inviscid flow at Mach number of 3 in a tunnel with a step . . . . . . . 163 

7.6 Diffraction of a strong shock over a backward facing step . . . . . . . 168 

7.7 Shardin’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 

7.7.1 Strong Mach 5 shock impingement on a triangle . . . . . . . . 185 

7.8 Flat plate boundary layer . . . . . . . . . . . . . . . . . . . . . . . . 189

CONTENTS 14 

7.9 Unsteady viscous flow over tandem airfoils . . . . . . . . . . . . . . . 191 

8 Conclusions and Future Work 197

List of Figures 

2.1 Control Volume (CV) Ω. . . . . . . . . . . . . . . . . . . . . . . . . . 48 

3.1 Median dual surface for evaluating first-order derivatives at grid point 

0 via the FV method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 

3.2 Mixed-type element channel mesh. . . . . . . . . . . . . . . . . . . . 60 

3.3 Distribution of the analytic (✷) and the numerical (✸) TE for the 

field u(x, y) = xy 2 on the mesh of Fig. 3.2 using the median dual. . . 61 

3.4 Difference between the analytic and the numerical TE for the field 

u(x, y) = xy 2 on the mesh of Fig. 3.2 using the centroid dual. . . . . 62 

3.5 Stretching for (a) structured and (b) unstructured meshes defined to 

facilitate comparison of accuracy degradation on them. . . . . . . . . 63 

3.6 Skewed mesh, depicting the deviation angle ω from the 180 ◦ angle 

between one of the two pairs of edges sharing point 0 (ω ∈ [0 ◦ , 90 ◦ ]). 

The lengths of the edges are equal. . . . . . . . . . . . . . . . . . . . 64 

3.7 Sheared mesh, depicting displacement of the edges causing deviation 

from orthogonality expressed by the angle φ (φ ∈ [0 ◦ , 90 ◦ ]). The 

lengths of the edges are equal. . . . . . . . . . . . . . . . . . . . . . . 64 

3.8 Rotated structured mesh depicting non alignment of the edges with 

the axes of the global system (∆x ≠ ∆y and θ ∈ [0 ◦ , 45 ◦ ]). . . . . . . 65 

3.9 Common mixed-type element mesh interfaces. . . . . . . . . . . . . . 65 

3.10 High aspect ratio quadrilaterals typical of a structured mesh in a 

boundary layer region. . . . . . . . . . . . . . . . . . . . . . . . . . . 71 

3.11 Mesh quality index Q x for the stretched unstructured mesh (◦) and 

the stretched structured mesh (□) vs. displacement factor d x (d y = 0). 73 

3.12 Mesh quality index Q x and its corresponding normalized error coefficients 

vs. the skewness angle ω. . . . . . . . . . . . . . . . . . . . . . 74 

15

LIST OF FIGURES 16 

3.13 Mesh quality index Q y for the mixed-type element mesh interface of 

Fig. 3.9(a) vs. angle ψ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 75 

3.14 Locally stretched structured channel mesh: (a) mesh geometry, (b) 

index q and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . 78 

3.15 Locally stretched unstructured channel mesh: (a) mesh geometry, (b) 


3.16 Mixed-type element channel mesh: (a) mesh geometry, (b) index q 

and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 

3.17 O–type structured mesh around a NACA 0012 airfoil: (a) mesh geometry, 

(b) index q and (c) index Q. . . . . . . . . . . . . . . . . . . 81 

3.18 Unstructured mesh around a NACA 0012 airfoil: (a) mesh geometry, 

(b) index q and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . 82 

3.19 Mixed-type element mesh around a RAE 2822 airfoil: (a) mesh geometry, 

(b) index q and (c) index Q. . . . . . . . . . . . . . . . . . . 83 

3.20 Structured mesh around a cylinder: (a) mesh geometry, (b) index q 

and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 

3.21 Unstructured mesh around a cylinder: (a) mesh geometry, (b) index 

q and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 

3.22 Mixed-type element mesh around a cylinder: (a) mesh geometry, (b) 


4.1 Local reconstruction of the field at two adjacent elements. . . . . . . 90 

4.2 Transformation of the physical quadrilateral to the standard quadrilateral 

element configuration (η 1 , η 2 ∈ [−1, 1]). . . . . . . . . . . . . . 96 

4.3 Transformation of the physical domain triangle to the standard triangular 

element (ξ 1 , ξ 2 ∈ [−1, 1] with ξ 1 + ξ 2 ≤ 1) and standard 

quadrilateral element configuration (η 1 , η 2 ∈ [−1, 1]). . . . . . . . . . 97 

4.4 Basis functions over the standard square element: (a) (p, q) = (0, 0), 

(b) (p, q) = (1, 0), (c) (p, q) = (2, 0), (d) (p, q) = (0, 1), (e) (p, q) = 

(1, 1), (f) (p, q) = (2, 1), (g) (p, q) = (0, 2), (h) (p, q) = (1, 2), (i) 

(p, q) = (2, 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 

4.5 Basis functions over the standard triangular region: (a) (p, q) = (0, 0), 

(b) (p, q) = (1, 0), (c) (p, q) = (2, 0), (d) (p, q) = (0, 1), (e) (p, q) = 

(1, 1), (f) (p, q) = (0, 2). . . . . . . . . . . . . . . . . . . . . . . . . . . 99 

6.1 Mixed-element mesh patch for estimating parameter M. . . . . . . . 130


7.1 Computational mesh (blue lines) and visualization mesh (black lines) 

for a P 4 expansion of the solution over triangular and quadrilateral 

elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 

7.2 Convergence rate for the DG discretization of the Euler equations. . . 142 

7.3 Convergence rate for the DG discretization of the Navier-Stokes equations 

with the LDG method. . . . . . . . . . . . . . . . . . . . . . . . 144 

7.4 Mach contours for the inviscid flow over a cylinder using discontinuous 

output of the solution and the curvature based boundary conditions 

in [92]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 

7.5 Comparison between the numerically evaluated pressure coefficient 

and the analytical solution for the inviscid flow around a cylinder. . . 146 

7.6 Convergence history for the residual in the density field for the inviscid 

flow over a cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 

7.7 Meshes for the two dimensional Sod’s shock tube problem. . . . . . . 148 

7.8 Limited elements in time for the Sod’s shock tube problem using the 

quadrilateral mesh of Fig. 7.7a. . . . . . . . . . . . . . . . . . . . . . 148 

7.9 Comparison of the exact density and entropy variation at t = 0.25 

with the two-dimensional numerical solution for the Sod’s shock tube 

problem in the triangular channel mesh of Fig. 7.7b. . . . . . . . . . 149 

7.10 Density plot for the Sod’s problem with a pressure ratio of 100000. . 150 

7.11 Pressure plot for the Sod’s problem with a pressure ratio of 100000. . 150 

7.12 Entropy plot for the Sod’s problem with a pressure ratio of 100000. . 150 

7.13 Quadrilateral mesh for supersonic flow at Mach 2 around a cylinder. . 151 

7.14 Triangular mesh for supersonic flow at Mach 2 around a cylinder. . . 152 

7.15 Mixed type mesh for supersonic flow at Mach 2 around a cylinder. . . 152 

7.16 Pressure contour lines using 30 equally spaced intervals for flow at 

Mach 2 over a cylinder for the numerical solution obtained with a P 3 

approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 

7.17 Limited elements for the flow at Mach 2 over a cylinder for the numerical 

solution obtained with a quadrilateral mesh and a P 3 approximation; 

the blue elements are limited the adjacent green elements are 

P 1 expansions, the adjacent to green red elements are P 2 , and for the 

rest of the domain P 3 expansions are employed. . . . . . . . . . . . . 154


7.18 Limited elements for the flow at Mach 2 over a cylinder for the numerical 

solution obtained with a P 2 approximation; the blue elements 

are limited the adjacent red elements are P 1 expansions and for the 

rest of the domain P 2 expansions are employed. . . . . . . . . . . . . 155 

7.19 Comparison of the pressure distribution along the stagnation line obtained 

from P 2 and P 3 approximations of the numerical solutions 

with different meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 

7.20 Variation of the maximum value of parameter M for each characteristic 

field obtained during convergence to steady state of the numerical 

solution for the flow at Mach 2 over a cylinder with a quadrilateral 

mesh and P 2 solution expansion. . . . . . . . . . . . . . . . . . . . . 157 

7.21 Limited elements on the coarse quadrilateral mesh for the double 

mach reflection problem at t = 0.2. . . . . . . . . . . . . . . . . . . . 158 

7.22 Density contours on the coarse quadrilateral mesh for the double mach 

reflection problem at t = 0.2. . . . . . . . . . . . . . . . . . . . . . . . 159 

7.23 Density contours for the numerical solution computed on the rectangular 

mesh, h = 1/240, and a P 1 approximation for the double mach 

reflection problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 

7.24 Limited elements for the numerical solution computed on the rectangular 

mesh, h = 1/240, and a P 1 approximation for the double 

mach reflection. The blue elements are limited and for the rest of the 

domain a P 1 approximation of the solution is employed. . . . . . . . 160 

7.25 Limited elements for the numerical solution computed on the triangular 


reflection. The blue elements are limited and for the rest of the domain 

a P 1 approximation of the solution is employed. . . . . . . . . . 161 

7.26 Density contours for the numerical solution computed on the triangular 



7.27 Density contours for the numerical solution computed on the triangular 



7.28 Variation of the maximum value of parameter M for each characteristic 

field obtained during the time advancement of the numerical 

solution for the double mach reflection problem obtained with a uniform 

triangular element mesh with a P 2 approximation and h = 1/240.163


7.29 Limited elements for the numerical solution obtained on a rectangular 

mesh and P 1 approximation and h = 1/80 for the flow at Mach 

number of 3 in a wind tunnel with a forward facing step at t = 2.0. . 164 

7.30 Density field for the numerical solution obtained on a rectangular 



7.31 Limited elements for the numerical solution obtained on a fine triangular 

mesh and a P 2 approximation and h = 1/100 for the flow 

at M = 3.0 in a wind tunnel with a forward facing step at t = 4.0. 

Blue elements: limited solution. Green elements P 1 expansion. Red 

elements P 2 expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . 165 

7.32 Density field for the numerical solution obtained on a fine triangular 



7.33 Mixed-type element mesh for the inviscid flow at Mach number of 3 

in a tunnel with a step. . . . . . . . . . . . . . . . . . . . . . . . . . . 166 

7.34 Density field for the numerical solution obtained on a mixed-type 

element mesh and a P 1 approximation for the flow at Mach number 

of 3 in a wind tunnel with a forward facing step at t = 4.0. . . . . . . 167 

7.35 Limited elements at t = 0.1 using a mixed-type element mesh for the 

inviscid flow at Mach number of 3 in a wind tunnel with a forward 

facing step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 

7.36 Variation of the parameter M for the TVB limiter applied on the 

mixed-type element mesh for the inviscid flow at Mach number of 3 

in a wind tunnel with a forward facing step. . . . . . . . . . . . . . . 168 

7.37 Sample meshes with density contours at t = 0.8 for the diffraction 

of a strong shock over a backward facing step using a P 2 solution 

expansion. The black square shows where the computational domain 

was clipped for showing the underlying computational mesh. . . . . . 169 

7.38 Limited elements for the diffraction of strong shock over a backward 

facing step at t = 2.0. Blue elements: limited solution. Green elements 

P 1 expansion. Red elements P 2 expansion. . . . . . . . . . . . 170 

7.39 Density contours for the diffraction of strong shock over a backward 

facing step at t = 2.0 with 30 equally spaced intervals from 0.06 to 

7.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171


7.40 Pressure contours for the diffraction of strong shock over a backward 

facing step at t = 2.0 with 30 equally spaced intervals from 0.09 to 

31.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 

7.41 Plot of density and pressure along the line defined by points [0, 7] and 

[13, 7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 

7.42 Numerically computed pressure (left) and density (right) gradient 

field for the diffraction of strong shock over a backward facing step 

at t = 2.0, using a rectangular elements mesh. . . . . . . . . . . . . . 173 

7.43 Numerically computed pressure (left) and density (right) gradient 

field for the diffraction of strong shock over a backward facing step 

at t = 2.0, using a triangular elements mesh. . . . . . . . . . . . . . . 174 

7.44 Variation of the second derivative estimate for the diffraction of a 

strong shock over a backward facing step using a P 2 expansion on a 

rectangular elements mesh. . . . . . . . . . . . . . . . . . . . . . . . . 174 

7.45 Plot of density and pressure along a line defined by the points [2, 5.5] 

and [5, 4] at time t = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . 175 

7.46 Vorticity field for the diffraction of a Mach 5 shock over a backward 

facing step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 

7.47 Density contours for Schardin’s problem with 30 equal spaced intervals 

from 0.6 to 2.4 using a P 2 solution expansion . . . . . . . . . . . 179 

7.48 Pressure contours for Schardin’s problem with 30 equal spaced intervals 

from 0.4 to 2.14 using a P 2 expansion. . . . . . . . . . . . . . . . 180 

7.49 Numerical Schlieren for Schardin’s problem using a P 2 solution expansion. 

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

7.50 Shadowgraph for the Schardin’s problem at t = 128µs. . . . . . . . . 182 

7.51 Numerical Schlieren for the Shardin’s problem on the finest mesh and 

using P 2 solution expansion. . . . . . . . . . . . . . . . . . . . . . . . 183 

7.52 Numerical Schlieren for Schardin’s problem on the refined triangular 

mesh with mesh element size in the vortex region equal to h 1 = 0.001. 184 

7.53 Numerical Schlieren for Schardin’s problem on the refined triangular 

mesh with mesh element size in the vortex region equal to h 2 = 0.0005.185 

7.54 Density contours using a P 2 expansion for the flow of a strong shock 

moving at Mach number of 5 around a triangle with 30 equal spaced 

intervals from 0.28 to 13.28. Results are depicted at non-dimensional 

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187


7.55 Limited elements for the flow of a strong shock moving at Mach number 

of 5 around a triangle along with the gradient of pressure over 30 

equally spaced intervals from 0 to 1000. Results are depicted at nondimensional 

time. Blue elements: limited solution. Green elements 

P 1 expansion. Red elements P 2 expansion. . . . . . . . . . . . . . . . 188 

7.56 Comparison of the U-velocity profile with the Blasius solution for the 

flow over a flat plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 

7.57 Comparison of the V-velocity profile with the Blasius solution for the 

flow over a flat plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 

7.58 Velocity field vectors and vorticity field for the flat plate boundary 

layer solution using a P 3 solution expansion. . . . . . . . . . . . . . . 191 

7.59 Visualization mesh for a P 4 computation for the two tandem NACA 

0012 airfoils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 

7.60 Lift and drag coefficients for the forward airfoil. . . . . . . . . . . . . 194 

7.61 Lift and drag coefficients for the aft airfoil. . . . . . . . . . . . . . . . 195 

7.62 Velocity contours for the flow over NACA0012 tandem airfoils. . . . . 196 

7.63 Vorticity contours for the flow over NACA0012 tandem airfoils. . . . 196

List of Tables 

2.1 Reference quantities used for the non-dimensionalization of the Navier- 

Stokes equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 

3.1 Expressions for the error coefficients for each type of mesh distortion. 66 

3.2 Expressions for the error coefficients for the mixed-type element mesh 

interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 

3.3 Central interface point displacements for improving the accuracy of 

the u y derivative evaluation for the mixed-type element mesh interfaces. 67 

3.4 Characteristic local lengths for the mixed-type element mesh interfaces. 70 

3.5 Mesh quality index Q expressions for the elementary types of mesh 

distortion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 

4.1 Basis functions for a third-order approximation over the standard 

quadrilateral region. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 

4.2 Basis functions for a third-order approximation over the standard 

triangular region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 

4.3 Types of Gauss numerical integration. . . . . . . . . . . . . . . . . . . 102 

4.4 DG methods and their interiors fluxes taken for Arnold et. al [6]. . . 108 

5.1 Butcher’s table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 

7.1 Results for the sensitivity study regarding the far field boundary distance. 

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

22

Chapter 1 

Introduction 

The present work focuses on the numerical solution of the compressible flow equations 

for flows with strong moving shocks and in the a priori evaluation of the 

geometrical quality of the mesh used for the discretization of the computational domain. 

The high-order Discontinuous Galerkin (DG) method is used for discretizing 

the system of the compressible Navier-Stokes equations in space along with highorder 

explicit and implicit discretization schemes for unsteady problems, in order to 

achieve high spatial and temporal resolution of the complex flow features appearing 

in flows with strong moving shocks. 

The numerical solution of the equations of compressible flow is an extremely 

powerful technique for the analysis of flows over a wide range of speeds around complex 

geometries. Until the mid 1980s, two methodologies prevailed the study of fluid 

motion, namely the experimental and the theoretical methodology. The former uses 

and develops techniques for the understating of the physical phenomena associated 

with a flow field. However, experiments are costly and time consuming and in some 

cases, even impossible to perform due to technical difficulties and physical limitations. 

An example of such cases is the study of the flow field over a re-entry vehicle. 

The theoretical approach attempts to solve the governing equations of fluid flow, or 

simplified versions of them, using mathematical methods and advanced numerical 

algorithms since analytical solutions exist only for very few simple cases of limited 

23

CHAPTER 1. INTRODUCTION 24 

practical interest. 

The advent of the digital computer and its enormous development over the 

last 50 years both in terms of speed and memory capabilities, lead to the creation 

of a third methodology as a branch of the theoretical approach for studding fluid 

flow, that of the numerical simulation, which is known as Computational Fluid Dynamics 

(CFD). The numerical solution of the governing equations does not have the 

limitations of the experimental and theoretical methodology. However, it innately 

possesses problems originating from the philosophy and practice of the numerical 

solution. 

The philosophy behind a CFD solution to a specific problem is to ”break” 

its mathematical continuum representation into a discrete form and numerically 

compute a solution. However, this approach possesses an error and the factors that 

contribute to this are the following: 

ˆ Discretization of the system of equations of fluid flow. 

ˆ Discrete representation of the geometry of the body around which the flow 

field is simulated. 

ˆ Round-off errors in computer arithmetic and floating point representation of 

numbers. 

ˆ Initial conditions and boundary conditions. 

ˆ Iterative solution procedures, which are stopped after some specified tolerances 

have been met. 

ˆ Modeling errors. 

Iterative solution procedures are applied when an implicit time discretization 

of the nonlinear system describing the motion of fluids is performed, but they are 

terminated after predefined tolerances are met leading to solutions which are close to 

the true solution of the system. The initial and especially the boundary conditions 

have a profound impact on the numerical results and their choice and implementation


may be the key for the success of a numerical solution for a given flow problem. The 

limitations imposed by the computer technology in arithmetic operations, requires 

special attention on how the mathematical operations are performed in a computer 

code. The discrete representation of the geometry in a computer model means that 

practically one is solving a problem over a polyhedral and not on a curved continuum 

geometry as it is in reality. Simplifications made for the mathematical form of the 

governing equations in order to be practical to solve them for a setting, simply means 

that an accurate physical model is not used. Taking all these factors along with the 

discretization methodology employed for ”breaking” the continuum approach, it is 

evident that their impact on the discretization process has to be accounted for in 

order to a priori minimize, as much as possible, their effect on the final solution. 

Emphasis on achieving predictive simulations as demonstrated in [94, 109] has 

caused research in numerical simulation to focus on the solution algorithms and the 

discretization error. The present research work aims at providing tools that increase 

the fidelity of the discretization of the equations of gas dynamics. The Finite Volume 

(FV) method is currently wide spread in CFD, however its successful application 

strongly depends on the quality of the mesh used for discretizing the physical space. 

For those reasons, a mesh induced error analysis based on a novel approach of 

the present thesis has been conducted in order to equip the CFD community with 

better and more accurate metrics for the appropriateness of a mesh for a CFD 

solution using the FV method. The present work assesses grid quality by computing 

appropriate metrics of the elements. However, there are two distinguishing aspects 

of it: (i) the metrics indicating quality are derived directly from related analytic 

forms of the truncation error and (ii) can give analytic expressions for reducing 

the discretization error via re-shaping of the elements. The primary issue with TE 

analysis is the complexity of the related expressions, especially for multi-dimensions 

and for general mixed-type element mesh topologies. The present work addresses 

this complexity barrier via employment of symbolic mathematics software [101]. 

The present work analyzes a generally-distorted mesh (structured, unstructured 

and mixed-type) into elementary distortions (stretching, skewness, shearing, 

rotation), as well as three common types of interfaces, and those were directly related


to the TE. The derived analytic expressions are relatively simple and amenable to 

future work on improving the mesh. The derived quality indices can be also applied 

for assessing the mesh quality for a higher order discretization method as demonstrated 

in [99]. 

Moreover, due to the limits of the FV method in terms of resolution, especially 

for flows with complex features with strong moving shocks and the difficulty in generating 

a good structured grid around complex geometries for Finite Difference (FD) 

discretizations, a DG code has been developed and named HoAc, which stands 

for High Order Accuracy. The developed code uses PETSc [58, 59] for performing 

parallel computations using the domain decomposition method. It can handle 

mixed-type element meshes employing hierarchical modal basis functions [83] up to 

arbitrary order of accuracy, for the discretization of the compressible Navier-Stokes 

equations. Explicit and implicit time schemes are used provided by PETSc and a 

novel shock capturing procedure has been developed, implemented and thoroughly 

tested. All the computations are performed in the standard element configuration, 

thus eliminating the complexity and difficulties arising by direct application of the 

solution algorithms in the physical space. 

The novel shock capturing approach proposed in this work uses the basic TVB 

limiter proposed by Cockburn and Shu [82]. However, it overcomes the ambiguity 

associated with the a priori specification of a parameter that estimates the second 

derivative of the solution. Application of the TVB limiter for the canonical elements 

of the computational domain, in a dimension per dimension fashion, renders our 

approach essentially the same with the original TVB limiter [82] that until now 

was used for rectangular elements. For the canonical element of the computational 

domain, tensor product orthogonal polynomial bases are constructed as described in 

detail in [83]. However, in contrast to the original TVB limiter, the second derivative 

of the numerical solution is estimated for each field, and it is not set to a constant 

value, (as suggested in the original paper [82]). Note that in the original approach 

[82] often and for cases with strong shocks and large pressure ratios, the suggested 

value must be altered, while in our approach no user intervention is required. Recent 

applications for flows with strong discontinuities resulting from high pressure ratios


[156] have demonstrated that the constant value of the second derivative, needs to 

be arbitrarily readjusted for each field in order to avoid divergence. Application of 

the limiter in the transformed domain has the additional advantage that limiting for 

rectangular, quadrilateral, and triangular elements is essentially the same, since the 

inverse transformation and the collapsed coordinates system introduced in [83] is 

used to recover the element in the physical space. The proposed limiting approach 

is accurate, effective, and can be applied on distorted mixed-type meshes without 

readjusting parameters for a wide range of flow problems discussed in the results 

section. The extension of the proposed limiting approach for three dimensions is 

straightforward [90]. 

In the following sections of the introduction chapter, the classical methods used 

in CFD are described along with comments on what extent they are affected by the 

underlying mesh employed for their application, and their ability to extend them in 

orders of accuracy higher than two. Furthermore, the selection of the discretization 

method employed in the current work is advocated and the overview of the thesis is 

presented. 

1.1 Classical methods in CFD 

Discretization techniques have been developed since the early 1900s due to the numerous 

efforts for developing algorithms for the numerical solution of partial differential 

equations (PDEs). Currently, the most popular and well established methods 

in CFD are the Finite Difference (FD) [97], the Finite Volume (FV) [96] and the 

Finite Element (FE) methods [83], and most CFD methods used in practice are at 

most second-order accurate in space [145, 148]. A brief description of each method 

follows. 

Finite Difference methods 

The most basic discretization technique for a system of PDEs is that of the FD 

method. A structured canonical grid, through the use of a generalized coordinate


transformation is employed, where the method approximates the analytical form of 

the derivatives appearing in the differential equations with finite differences leading 

to a discrete representation of the differential form, which can be solved numerically. 

There exists a plethora of approaches for the discrete estimation of the derivatives 

and some excellent text books describing these methodologies are those from Hirsch 

[63, 64], Anderson [4] and Tannehill et al. [132] to mention a few. 

The main advantages of the FD methods are that they are easy to program 

and efficient in terms of computational cost. Due to their efficiency and simplicity 

they have been used for computationally intensive problems. It is possible to 

construct high-order extensions of the FD methods, by adding more grid points in 

the computational stencil, since the accuracy of the method is determined by the 

estimation of the derivatives [54, 61, 95, 147]. However, in order to retain high-order 

accurate numerical solutions with FD discretizations it is required to have smooth 

canonical meshes in the physical domain. 

A major drawback, however, of the FD method is that it strictly requires structured 

grids for its application, thus limiting its use on simple geometrical configurations. 

It is possible to apply the method for problems accompanied with complex 

geometries. However, this is a difficult task, which practically turns the FD method 

inappropriate for computing flows around complex configurations. In principle it is 

possible to construct a FD scheme on unstructured grids [98], but this requires a 

reconstruction of a polynomial function from grid points, which is a complex problem 

even in the two dimensional case. Moreover, the high-order versions of the FD 

method require a regular, smooth grid for stability and accuracy reasons. Consequently, 

FD methods are mostly used for fundamental studies on simple geometries. 

Usually, in order to ensure stability of the scheme the order is reduced near and at 

the boundaries, which results in a special grid refinement close to the boundaries so 

that to compensate for the loss of accuracy, or use of one sided differencing schemes 

that must be employed in these regions.


Finite Volume methods 

The FV method is quite popular in the CFD community. The starting point for a 

FV discretization of a system of partial differential equations is the integral form 

of the system. A tessellation of the physical space in elements is performed and in 

each element the governing equations are discretized. In contrast to the FD method, 

where derivatives are being estimated, fluxes through the element boundaries are 

evaluated. There is a plethora of choices of how these fluxes can be chosen. A very 

popular approach for hyperbolic type systems is the upwind method [64, 96] where 

the flux estimate is based on the structure of the local wave propagation, leading 

to robust solution algorithms. Another approach, which however requires an ad hoc 

specification of parameters for stability reasons and shock capturing, is the artificial 

dissipation method [69]. 

One significant advantage of the FV method is its ability to perform well on 

structured, unstructured, and mixed-type element meshes in two and three dimensions. 

This permits the application of the method on problems involving complex 

geometries. Formally, the method is first order accurate, but with a reconstruction 

procedure [14, 96] of the solution at the element boundaries, its accuracy may be 

extended to higher order. However, in practice this reconstruction is quite computationally 

expensive and difficult to perform, even for moderate orders of accuracy 

(third-order for instance) [47], and most FV codes on unstructured meshes have 

resolution properties belonging to the second-order accurate schemes. 

The FV as the FD method requires the construction of regular and smooth 

meshes. Moreover, due to the difficulty in constructing a higher order FV scheme, 

use of the second-order accurate FV method for practical flow problems necessitates 

the construction of fine meshes especially for the solution of the Navier-Stokes 

equations.


Finite Element methods 

FE methods use a tessellation of the physical space in elements where the weak 

form of the governing equations is solved. Specifically, the weak form is derived 

by multiplying the system of the PDEs with a weighting function and integrating 

over the elemental space. Then, a polynomial expansion of the solution is chosen 

and the discrete from of the system is derived. The choice of the test (also known 

as weighting) and the expansion functions dictates which type of FE method is 

being employed. Versions of FE methods are the Galerkin [21, 24, 83], also known 

as the Bubnov-Galerkin method, the Petrov-Galerkin [62, 65], also known as the 

generalized Galerkin method and the least-squares FE methods [114, 115]. Highorder 

versions for FE methods can be constructed by increasing the order of the 

polynomial used for the solution expansion. Increase of the order of accuracy in 

FE methods comes however at the expense of computing cost, algorithmic and 

programming complexity. 

The Bubnov-Galerkin method uses the same function space both for the test 

function and the solution expansion, leading to a symmetric representation of the 

weak formulation, and it is applicable to any type of partial differential equations. 

The Petrov-Galerkin version of the FE method is used for solving partial differential 

equations with odd terms. In this version the formulation method is the 

same as the normal Galerkin method. However, the test function and the solution 

function approximations are chosen not to be the same, hence they must be approximated 

separately and thus, the representation of the weak form looses symmetry. 

The least squares FE method originates from the idea of least squares approximation 

developed by Gauss. In this version of the FE method the residual of the 

system obtained from the solution expansion is set as the test function, leading to 

a symmetric form of the weak representation. 

FE methods may be classified into two categories: continuous and discontinuous. 

Continuous FE methods employ an expansion set, which is continuous over 

the entire domain of the physical space being tessellated into non overlapping elements. 

On the other hand, discontinuous FE methods perform an approximation of


the solution of the system of PDEs at each element separately permitting it to be 

discontinuous across inter element boundaries. 

Higher-order versions of the FE method do not exhibit strong dependence on 

the mesh constructed for a field simulation. This is due to the local support of the 

solution from the local expansion at every element. In the present work, the DG 

method which belongs to the version of Bubnov-Galerkin FE methods is used and 

it will be presented in detail in a following chapter. 

1.2 Why the DG method 

Accurate prediction of aerodynamic and thermal loads on aerospace vehicles is a crucial 

stage of the design cycle. Wind tunnel investigations and theoretical predictions 

are used to estimate these loads. Experimental measurements of high speed flows 

are costly, difficult to perform, and usually unable to reach realistic flight Reynolds 

numbers. On the other hand, highly accurate theoretical predictions with DNS or 

LES are limited to simple flow configurations [2] and low Reynolds numbers. Most 

of these calculations are performed with structured grids that are more efficient 

computationally for high resolution simulations. For flows without discontinuities, 

high-order compact FD schemes [54, 147] can be used to provide the necessary highorder 

spatial accuracy at a reasonable computing cost. On the other hand, flows with 

strong discontinuities can be efficiently computed on structured grids using compact 

schemes with filters [131, 153] or the FD ENO [73, 130] and weighted ENO (WENO) 

[128] discretizations. The extension of ENO [1] or WENO [110] methodology to unstructured 

meshes is, however, not straightforward and becomes computationally 

intensive. Similarly, the high-order FV methods for three dimensional calculations 

are quite intensive in memory and computational time, and difficult to implement 

[47]. 

Use of standard second-order accurate in space FV and FD methods for the 

prediction of aerodynamic and thermal loads over more complex configurations, as 

mentioned before, is currently widespread. These methods, however, require very


high grid densities for the accurate computation of the complex flow features such 

as those generated from the interaction of shock waves with boundary layers even 

on relatively simple geometrical configurations. Use of unstructured meshes could 

reduce computing requirements for accurate computation of complex shock interactions, 

because they allow selective refinement in critical flow regions. Recently, it 

was demonstrated [149], that for these flows only the combination of selective grid 

refinement and high-order accuracy can achieve the desired level of accuracy at a 

reduced computing cost for complex shock wave interactions. In addition, Shi et 

al. [128] clearly demonstrated that resolution of complex flow features, generated 

by the impingement of a strong shock on a compression ramp, could be equally well 

achieved by grid refinement (doubling the resolution along each coordinate direction) 

or with the increase of the order of accuracy. The results of Shi et al. [128], 

obtained with the WENO scheme, indicate that increase of the order of accuracy 

of the spatial discretization scheme by an order of two is equivalent to doubling the 

resolution of the mesh in every direction, as far as the resolution of smooth but 

complex flow features is concerned. Similar conclusions concerning the enhancement 

of resolution were drawn for high-order discretizations with the DG method 

[53, 137–139] and the closely related Spectral Volume (SV) method [5, 149]. 

The main advantages of the DG [28, 41, 44] method compared to other available 

high-order methods is that, as FE type method, it has a compact stencil, it is 

applicable to arbitrary unstructured meshes, and it can handle hanging nodes and 

curved geometries in a natural manner. Therefore, this method is suitable for grid 

adaptation, and appropriate for the computation of flows with strong discontinuities. 

These significant advantages of the DG method over other high-order methods 

come unfortunately at the expense of implementation complexity, higher computing 

cost, and reduction of the allowable, by stability limitations, time step for marching 

in time. Computation in regions of strong shocks and other discontinuities with 

the DG method could be stabilized with the application of classical total variation 

bounded (TVB) limiters [28, 44]. In regions where the TVB limiter is activated the 

formal order of accuracy of the method drops. More recently, alternative approaches 

for the computation of flows containing discontinuities with high-order accurate DG


space discretizations were proposed by applying filters [100, 138], or by adding numerical 

dissipation [105, 142]. However, use of these approaches increases the overall 

computational cost of the DG method and does not guarantee that high accuracy 

is retained at the shock, where the computation essentially drops to lower order 

of accuracy, so that preservation of monotonicity is ensured. Filters and artificial 

dissipation in addition require ad hoc specification of tuning parameters. 

High resolution of strong shocks can also be achieved with the selective increase 

of grid resolution (h-refinement) and use of lower order spatial discretizations 

with TVB limiters at the regions of discontinuities. This approach appears quite 

appropriate for the resolution of shocks with DG discretizations and was applied 

successfully for hexahedral meshes in [142]. In addition hp-refinement schemes [103] 

have appeared in the literature. 

The DG method has become popular in recent years due to its high-order of 

accuracy, and the ability to simulate flows around complex geometries with the ease 

of increasing the order of approximation, while keeping a compact stencil. The DG 

method has higher computational cost compared with standard second-order FV 

methods. However, the compact stencil of the DG method provides an essential advantage 

for achieving high-order accuracy and parallelization of the algorithms, both 

with the use of domain decomposition through message passage interface (MPI), and 

graphics processor units (GPUs) [50, 81, 87]. In this work, parallelization of the DG 

method through domain decomposition and use of MPI was performed. 

Accurate predictions of skin friction and thermal loads of high speed complex 

flows in both simple and nontrivial geometries, require high resolution computations. 

High-order DG discretizations possess features that make them very attractive for 

computation of complex flows with strong shocks. A key ingredient that would 

make the DG method more suitable for these computations, is the straightforward 

application of p-adaptive procedures that ensure robust and accurate capturing of 

discontinuities along with high-order of accuracy in the smooth parts of the flow. In 

this spirit, low order expansions and high mesh density could be used in regions of 

discontinuities. On the other hand, resolution of smooth but complex flow features, 

such as vortices and shear layers that usually evolve in time, could be obtained


with relatively uniform mesh density using higher order expansions. Flow features 

where p-adaptivity is required could either be detected in the solution as regions 

of high gradients, such as density and pressure variations in vortical regions, or by 

a-posteriori error estimates of the FE solution [10, 108, 144]. However, these aspects 

of the DG method have not been addressed satisfactorily yet, and utilization of p- 

adaptivity is not currently widespread. Therefore, the ability of the DG method 

to compute flows with discontinuities in a unified manner for arbitrary mixed-type 

meshes requires further development. Moreover, limiting in three dimensions for 

arbitrary unstructured meshes has not been demonstrated. The DG method reduces 

the solution accuracy near discontinuities. This affects the resolution properties of 

the method and much of its potential could be lost when limiting is extended over 

large regions. Therefore, narrowing of the regions requiring limiting and use of p- 

adaptive procedures is expected to enhance the potential and broaden utilization of 

the DG method. 

The slope limiters for the computation of flows with discontinuities used so 

far with the DG method have been demonstrated mainly for rectangular elements. 

A total variation bounded (TVB) limiting procedure for the DG method was constructed 

for solving scalar, one-dimensional, hyperbolic conservation laws by Shu 

[129] and Cockburn and Shu [82]. The extension of the method to one-dimensional 

systems has been applied with satisfactory results [40]. The basic idea behind the 

TVB limiter was to use a slope limiter to preserve the monotonicity of the solution 

averages. The same nonlinear TVB limiting was also applied for two dimensional 

problems [27, 32, 41], for second (P 1 polynomial expansion) and third-order (P 2 ) 

accurate computations with rectangular meshes. One ambiguity of this limiting 

procedure is that it relies on a priori estimates of the numerical solution’s second 

derivative. Moreover, the application for triangular meshes [40, 82] is more elaborate 

and straightforward application of the method for mixed-type meshes has not 

been addressed yet. 

In an effort to extend the TVB limiter for higher order accurate solutions, Qiu 

and Shu [120] applied the same TVB limiter for the one-dimensional Euler equations, 

by reconstructing the solution at every element with a Hermite WENO scheme using


the average solution at every element. Applying this idea they achieved a third-order 

monotone solution. Similar approaches were followed by Zhu and Qiu [157, 158] and 

by Luo et al. [100] for multidimensional calculations. These limiting approaches were 

demonstrated for meshes with rectangular elements and for triangular meshes using 

the approach suggested by Cockburn [40], which is not straight forward to extend 

in three dimensions. In addition, the approach of Zhu [157, 158] implies increase of 

the stencil width, and one of the main advantages of the DG method is lost. Biswas 

et al. [29] proposed an alternative implementation of limiting, where the solution 

moments are altered in order to preserve monotonicity. Krivodonova, [91] applied 

this concept with promising results, in computations for flows with strong shocks 

and with order of accuracy higher than two for rectangular elements. On the other 

hand Jaffre [68] and Van der Ven [143] added artificial dissipation to stabilize the 

DG method. 

From the above it is concluded in the present thesis that the DG method as implemented 

and enhanced with the newly developed limiting approach is suitable for 

the computation of high speed complex flow fields around complex geometries. The 

present work has made a significant improvement to the application of TVB limiter 

leading to a p-adaptive shock capturing scheme with outstanding performance. 

1.3 Thesis outline 

Chapter 2 includes a brief presentation of the equations of gas dynamics, along with 

some basic thermodynamic variables. The mathematical character of the system 

of Euler and Navier-Stokes equations is briefly presented, which dictates the type 

of boundary conditions to be used in the far field boundaries of the computational 

domain and the non-dimensional form of the equations, which is used in all the 

computations in this work is shown. The mesh induced error study is performed 

in Chapter 3 and the derived novel mesh quality measures for mixed-type element 

meshes are presented. These measures are subsequently used in the construction 

of the new limiter. Several applications of the quality measures are also presented, 

which confirm their ability to inspect the quality of a mesh for a FV discretiza-


tion. The DG discretization of the Euler and Navier-Stokes equations is presented 

in Chapter 4, where the expansion bases used in the present work are presented in 

detail. Also, the numerical operations are described for performing all the computations 

in the computational space along with the transformation of the elemental 

physical configuration to its standard representation in the computational space. 

Furthermore, the Local Discontinuous Galerkin (LDG) methodology is presented 

for discretizing the system of Navier-Stokes equations in the DG framework. Chapter 

5 deals with the time marching of the DG discretization of the compressible 

equations of gas dynamics and the limiting procedure developed for shock capturing 

is presented in Chapter 6. In Chapter 7 numerical results for problems with strong 

moving shocks are presented along with verification cases that confirm the correctness 

of the implemented DG discretization and the limiting procedure in HoAc. 

Finally, in Chapter 8 the conclusions of the current thesis are presented.

Chapter 2 

Governing equations of gas 

dynamics 

The governing equations of inviscid and viscous compressible flow, along with some 

basic thermodynamic relations, will be presented. They are expressed in a Eulerian 

reference frame and they are based on the following conservation laws: 

ˆ Conservation of Mass 

ˆ Conservation of Momentum 

ˆ Conservation of Energy 

and mathematically they are derived by using Reynolds transport theorem. Their 

mathematical character will be stated, which defines the specification and application 

of the boundary conditions to be used for a numerical solution. The presented 

equations will be given first in differential form followed by their non-dimensionalization 

and concluding with their integral form. 

37

CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 38 

2.1 Euler equations 

The motion of a compressible fluid in two dimensions, without any viscous and 

thermal conduction effects, is described by the system of Euler equations. The 

differential form of this system in Cartesian coordinates is: 

L(U) = ∂U 

∂t + ∇ · F i(U) = 0, (2.1) 

where U is the conservative variable vector or state vector: 

⎡ ⎤ 

ρ 

U = 

ρu 

⎢ 

⎣ρv 

⎥ 

⎦ , (2.2) 

E 

and F i (U) = [f i 

g i ], is the inviscid flux tensor with vector components: 

⎡ ⎤ ⎡ ⎤ 

ρu 

ρv 

f i = 

ρu 2 + p 

⎢ 

⎣ ρuv 

⎥ 

⎦ , g i = 

ρuv 

⎢ 

⎣ ρv 2 + p 

⎥ 

⎦ . (2.3) 

(E + p)u 

(E + p)v 

This set of four equations is closed with the equation of state for a perfect gas. 

2.1.1 Energy 

The total energy E per unit volume is the sum of the internal and kinetic energy of 

the flow: 

E = ρe + 1 2 ρ(u2 + v 2 ). (2.4)


The term 1 2 ρ(u2 + v 2 ) is the kinetic energy, while e is the internal energy per unit 

mass . Internal energy includes translational, rotational and vibrational energy and 

other forms of energy in more complicated situations. In the present work the gas 

is assumed to be in local chemical and thermodynamic equilibrium and the internal 

energy is a function of pressure and density: 

e = e(p, ρ). (2.5) 

2.1.2 Polytropic ideal gas 

For an ideal gas the internal energy is a function of temperature only, e = e(T ). 

The temperature T is related to pressure p and density ρ by the ideal gas law: 

p = RρT, (2.6) 

where R is a constant obtained by dividing the universal gas constant R by the 

molecular weight of the gas. To a good approximation, the internal energy is simply 

proportional to the temperature T : 

e = c V T, (2.7) 

where c V is the specific heat at constant volume . Gases that obey Eqs. (2.6) and 

(2.7) are called calorically or thermally perfect. If energy is added to a fixed volume 

of a thermally perfect gas, then the change in internal energy and temperature is 

related via: 

de = c V dT. (2.8) 

However, if the gas is allowed to expand at a constant pressure, not all of the 

energy goes into increasing the internal energy. The work done by expanding the 

volume 1 by d( 1) is pd( 1 ) and the following relation is obtained: 

ρ ρ ρ


de + pd( 1 ρ ) = c pdT ⇔ d(e + p ρ ) = c pdT, (2.9) 

where c p is the specific heat at constant pressure . The quantity: 

h = e + p ρ , (2.10) 

is called the (specific) enthalpy of the gas . For a polytropic gas, c p is constant, so 

Eq. (2.9) yields: 

h = c p T, (2.11) 

and the enthalpy is simply proportional to the temperature. By the ideal gas law 

results: 

c p − c V = R. (2.12) 

The equation of state of a polytropic gas turns out to depend only on the ratio of 

the specific heats: 

and also called the adiabatic exponent . 

γ = c p 

c V 

, (2.13) 

The internal energy in a molecule is typically split up between various degrees 

of freedom (translational, rotational, vibrational, etc.). How many degrees of freedom 

exist, depends on the nature of the gas. The general principle of equipartition 

of energy requires that the average energy in each of these is the same leading to 

each degree of freedom contributing an average energy of: 

1 

2 kT,


per molecule, where k is Boltzmann’s constant and is equal to 1.3806503·10 −23 m 2 kg/s 2 

in SI units . This gives a total contribution of: 

a 

2 kT, 

per molecule if there are a degrees of freedom. Multiplying this by n, the number 

of molecules per unit mass (which depends on the gas), gives: 

e = a nkT. (2.14) 

2 

The product nk is precisely the gas constant R, which in comparison with Eq. 

(2.7), gives: 

c V = a R. (2.15) 

2 

From Eq. (2.12), results that: 

c p = 

( 

1 + a ) 

R, (2.16) 

2 

which leads to the following expression for the adiabatic constant: 

γ = c p 

c V 

= a + 2 

a . (2.17) 

For air a = 5 and γ = 7 2 

= 1.4. Noting that: 

T = 

p 

Rρ , (2.18) 

leads to: 

e = 

p 

(γ − 1)ρ . (2.19)


Using Eq. (2.19) in Eq. (2.4), gives the common form of the total energy for a 

thermally perfect gas: 

E = 

p 

γ − 1 + 1 2 ρ(u2 + v 2 ). (2.20) 

and the pressure is related to total energy and the conservative variables through 

the equation of state: 

) 

p = (γ − 1) 

(E − u2 + v 2 

. (2.21) 

2ρ 

2.1.3 Entropy 

Entropy is a fundamental thermodynamic quantity and indicates the degree to which 

its internal energy is available for producing usable work. The specific entropy is 

given by: 

( ) p 

s = c V log + constant. (2.22) 

ρ γ 

from which follows : 

p = κe ( s 

c V ) ρ γ , (2.23) 

where κ is a constant. 

2.1.4 Mathematical character of the Euler equations 

The system of Euler equations is hyperbolic, because the eigenvalues of the directional 

Jacobian matrix [64]: 

K = ∂F i · k, (2.24) 

∂U


are real and distinct for any direction represented by the unit vector k . The diagonal 

eigenvalue matrix is: 

⎡ 

⎤ ⎡ ⎤ 

uk x + vk y u 

Λ = 

uk x + vk y 

⎢ 

⎣uk x + vk y + c 

⎥ 

⎦ = u 

⎢ 

⎣u + c 

⎥ 

⎦ , (2.25) 

uk x + vk y − c u − c 

where u, v are the Cartesian velocity components , u is the contravariant velocity 

component along the k direction, and c is the local speed of sound given by the 

following equation: 

c = 

√ γp 

ρ . (2.26) 

The right and left eigenvectors of the directional Jacobian matrix in Eq. (2.24) 

are the following: 

⎡ 

R = 

⎢ 

⎣ 

⎤ 

ρ 

ρ 

k x 0 

2c 

2c 

ρu 

− ρkx 

2c 2 

ρv 

− ρky ⎥ 

2c 2 ⎦ 

q 1 k x 

uρk 

2 y − vρk x R 1 R 2 

ρu 

uk x ρk y + ρkx 

2c 2 

ρv 

vk x −ρk x + ρky 

2c 2 

, (2.27) 

⎡ 

L = 

⎢ 

⎣ 

− ukx 

ρ 

− ukx 

ρ 

k x − kx 

c 2 γ−1 

2 q 2 

− ukx 

ρ 

− vky 

ρ 

+ vky 

ρ 

− vky 

ρ 

+ (γ−1) 

2ρc q 2 

k xρ 

uk x(γ−1) 

c 2 

vk x(γ−1) 

c 2 

− kx(γ−1) 

c 2 

k y 

ρ 

− kx ρ 

0 

− u(γ−1) 

ρc 

k y 

ρ 

− v(γ−1) 

ρc 

+ (γ−1) q 2ρc 2 − kx − u(γ−1) − ky − v(γ−1) 

ρ ρc ρ ρc 

γ−1 

ρc 

γ−1 

ρc 

⎤ 

, (2.28) 

⎥ 

⎦ 

with :


q 1 = 

√ 

u2 + v 2 

, 

2 

q 2 = 2(u 2 + v 2 ) − q 1 , 

R 1 = qρ 

4c + uρk x 

+ vρk y 

2 2 

R 2 = qρ 

4c − uρk x 

− vρk y 

2 2 

2.2 Navier-Stokes equations 

+ ρc 

2γ − 2 , 

+ ρc 

2γ − 2 . 

The system of Navier-Stokes equations describes real compressible flows, which 

posses both heat conduction and viscosity effects. The differential form of this 

system is: 

N (U) = ∂U 

∂t + ∇ · F i(U) = ∇ · F v (U, ∇U), (2.29) 

where U and F i (U) is the conservative state vector and the inviscid flux tensor 

respectively, and F v (U) = [f v 

vector components : 

g v ], is the viscous flux tensor with the following 

⎡ 

⎤ ⎡ 

⎤ 

0 

0 

f v = 

τ xx 

⎢ 

⎣ τ 

⎥ 

xy ⎦ , g v = 

τ yx 

⎢ 

⎣ τ 

⎥ 

yy ⎦ , (2.30) 

uτ xx + vτ xy + q x uτ yx + vτ yy + q y 

and τ is the two dimensional viscous stress tensor : 

τ = 

[ 

τxx τ xy 

τ yx τ yy 

] 

. (2.31)


Assuming a Newtonian fluid, for which the Stokes hypothesis is valid, then τ 

is symmetric and a linear function of the velocity gradients: 

τ xx = − 2 µ∇ · u + 2µ∂u 

3 ∂x , 

(2.32a) 

τ yy = − 2 µ∇ · u + 2µ∂v 

3 ∂y , 

where µ is the dynamic viscosity of the fluid. 

(2.32b) 

( ∂u 

τ xy = τ yx = µ 

∂y + ∂v ) 

. (2.32c) 

∂x 

The heat fluxes are modeled according to Fourier’s law: 

q x = −λ ∂T 

∂x , (2.33) 

q y = −λ ∂T 

∂y , (2.34) 

where the thermal conductivity coefficient λ is related to the dynamic viscosity and 

the specific heat at constant pressure c p by the non-dimensional Prandtl number: 

P r = µc p 

λ . (2.35) 

2.2.1 Mathematical character of the Navier-Stokes equations 

The presence of viscosity and heat conduction effects transforms the momentum 

and energy equations into second-order partial differential equations. This results 

in a hybrid system being parabolic-hyperbolic in time and space, but becoming 

elliptic-parabolic in space for steady state problems.


2.3 Non-dimensional form of the Navier-Stokes 

equations 

In order to study flows under different conditions around the same geometry and, 

moreover, in order to reduce the errors due to the finite precision of computers, a 

non-dimensionalization of the flow variables is performed. In Table 2.1 the reference 

quantities used are given, which were taken from [118]. 

Variable Reference quantity 

ρ 

ρ ∞ 

u, v C ∞ 

E 

ρ ∞ C∞ 

2 

p 

ρ ∞ C∞ 

2 

x, y L 

T 

T ∞ 

µ µ ∞ 

λ 

λ ∞ 

t 

L/C ∞ 

Table 2.1: Reference quantities used for the non-dimensionalization of the Navier- 

Stokes equations. 

The dimensionless variables, denoted by an over-bar, are the following: 

ū, ¯v = u, v , ¯p = p , Ē = E , 

C ∞ ρ ∞ C∞ 

2 ρ ∞ C∞ 

2 

¯x, ȳ = x, y 

L , 

¯t = tC ∞ 

L , 

¯T = 

T 

T ∞ 

, ¯µ = µ 

µ ∞ 

. 

Through the non-dimensionalization of the equations the following reference numbers 

appear: 

ˆ Reynolds number Re = ρ∞C∞L 

µ ∞ 

ˆ Prandtl number P r = µ∞cp 

λ ∞


ˆ Mach number Ma = U∞ 

C ∞ 

Note that the Reynolds number Re uses c ∞ and therefore Re based on the free 

stream velocity (which is the case for experimentally given Reynolds numbers) must 

be scaled by the Mach number Ma. The equation of state Eq. (2.6) becomes: 

¯p = 

¯ρ ¯T 

γ , (2.36) 

and the viscous stresses and heat fluxes are altered, given by: 

¯τ xx = − 2 2¯µ ∂ū 

¯µ∇ · ū + 

3Re Re ∂¯x , 

(2.37a) 

¯τ yy = − 2 2¯µ ∂¯v 

¯µ∇ · ū + 

3Re Re ∂ȳ , 

¯τ xy = ¯τ yx = 

¯µ Re 

(2.37b) 

( ∂ū 

∂ȳ + ∂¯v ) 

, (2.37c) 

∂¯x 

¯µ ∂ 

¯q x = − 

¯T 

ReP r(γ − 1) ∂¯x , 

¯µ ∂ 

¯q y = − 

¯T 

ReP r(γ − 1) ∂ȳ . 

(2.37d) 

(2.37e) 

The dynamic viscosity of the fluid is related to the temperature of the flow 

using Sutherland’s law: 

where S = 100.55K. 

¯µ = µ ( ) 3 

T 

2 T∞ + S 

= 

µ ∞ T ∞ T + S , (2.38)


2.4 Integral form of the compressible flow equations 

The integral form of the Navier-Stokes equations is the basis of the FV methods. 

Considering an arbitrary fixed control volume Ω as it is shown in Fig. 2.1 and 

integrating Eq. (2.29) over it, follows: 

n 

Ω 

∂Ω 

Figure 2.1: Control Volume (CV) Ω. 

∫ ∫ 

∫ 

∂ 

UdΩ + ∇ · F i dΩ = ∇ · F v dΩ ⇒ 

∂t Ω 

Ω 

Ω 

∫ ∮ 

∮ 

∂ 

UdΩ + F i · ndS = F v · ndS, (2.39) 

∂t Ω 

∂Ω 

∂Ω 

where n is the outward normal vector at the boundary of the control volume. Neglecting 

the viscous flux tensor in Eq. (2.39), the integral form of the Euler equations 

is obtained: 

∫ ∮ 

∂ 

UdΩ + F i · ndS = 0. (2.40) 

∂t Ω 

∂Ω

Chapter 3 

Mesh quality measures for the FV 

method 

The discretization methods mentioned in the introduction of the thesis are all mesh 

based methods and their solution quality is affected by the underlying mesh over 

which they are applied, especially the FV and the FD method. Thus, field simulations 

that incorporate complex physics and geometries poses a formidable challenge 

to grid generation [77]. The distribution of points and elements can be quite non– 

uniform and coarse leading to inaccurate computations. The inaccuracy is often 

assessed after the simulation is performed causing repetitive grid generations and 

subsequent field simulations, so it would be desirable to have an assessment of the 

appropriateness (quality) of the mesh before performing the simulation. The presented 

study on the mesh quality focuses on the FV method as its use is currently 

widespread in CFD. 

Grid or mesh quality is affected by two primary factors; the local size of 

the computational elements, and the uniformity of the spatial distribution of the 

points/elements. A primary method for solving the issue of coarse mesh resolution 

has been adaptive grid refinement [11, 70, 71, 76, 102]. While substantial work 

has been performed on this type of mesh adaptation, relatively less work has been 

devoted to study and improvement of the local distribution of the points, which 

49

CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 50 

basically relates to the shape of the elements. Assessment of adequacy of local resolution 

and shape of the elements depends on the discretization error. It is imperative 

to derive measures of this error in the computation. 

Two broad categories of error indicators are: (i) a priori and (ii) a posteriori 

estimation. The two approaches are basically complimentary. A priori error 

evaluation can aid in mesh generation, while a posteriori can provide guidance to 

mesh adaptation techniques during the simulation. Up to now, works regarding 

the a priori grid quality assessment are based on geometric characteristics of the 

elements such as ratios of sizes of neighboring elements, as well as on element shape 

measures, such as angles and ratios of the radii of inscribed to prescribed circles 

[48, 52, 55, 80, 111]. In the FE method, the quality of a mesh is often given in terms 

of the element/mesh regularity. This type of approach has given measures that can 

be computed easily and are quite popular with practical applications. 

Truncation error (TE) analysis usually falls under the a posteriori estimation. 

The FD discretization method has offered a vehicle for calculating the TE by 

performing a Taylor series expansions of the solution at the points forming the discretization 

stencil [132]. The complexity of the expressions has led research works to 

focus on simplified model field equations [8, 127, 154]. Nevertheless, those works expressed 

the strong dependence of the solution and the stability of the computations 

on the grid and its quality. 

Direct relations between TE and mesh distortion parameters, such as departure 

from orthogonality and uniformity have been reported in [66, 133]. Significantly less 

work exists for unstructured meshes [57] and for the FV method [72, 141]. However, 

despite the large number of studies on the relation on the TE to the numerical 

solution, there have been very few derivations of mesh quality measures based on it. 

A posteriori error estimation methods include the Richardson extrapolation 

[25, 26, 67, 134]. Use is made of two or more grids of the same domain with the 

difference in the yielded solution offering a measure of the local error distribution. 

Generating and solving on multiple meshes can be quite difficult for practical applications.


Another a posteriori method evaluates the error indirectly via tracking of the 

numerical solution variation [78]. Local field features, such as boundary layers, shock 

waves and vortices are detected using sensors that are based on variations of the 

computed field parameters, such as the pressure and velocity. The assumption here 

is that the discretization error is large where the solution variations are large. The 

method has been primarily used for guiding grid adaptation. It is not a method 

that can directly yield practical assessment of grid quality, taking also into account 

that it necessitates expensive computations for large meshes and/or complex fields. 

Another school of a posteriori error estimation work employs FE discretization 

and derives analytic expressions of error bounds [3, 106, 116, 117]. Various model 

equations have been utilized in order to provide the estimates. Although, the error 

bounds do not yield grid quality measures directly, there is potential for such use. 

Again, knowledge of the solution field is needed. 

The goal of the mesh induced error study is the derivation of a priori quality 

measures for mixed-type element meshes for the FV method. The most fundamental 

computation is that of first-order derivatives. The first derivative is common not 

only to fluid flow governing equations, but also to other field equations. The relative 

reduced complexity of the related mathematics and its commonality led to its choice 

in the present work. The FV method will be the method for the mesh induced error 

study, as the FV method is currently the most wide spread method in computational 

fluid dynamics and moreover, it is the first-order accurate version of the DG 

discretization. A node-based FV method is chosen as the vehicle for derivation of 

quality measures, since it is known to be sensitive to mesh non-uniformity in terms 

of its accuracy. The complete expressions for the TE will be presented for meshes 

that consist of structured, unstructured, or both types of elements. The complexity 

of the analytic expressions lead to study of the two–dimensional case.


3.1 Gradient computation in the FV method 

A common node based FV evaluation of the first-order spatial derivatives employs 

the median dual area around a grid point as Fig. 3.1 depicts [15, 63]. This quite 

common FV evaluation is chosen to study quality measures. It is expected to reveal 

distortions of the mesh as it corresponds to a central differencing, and thus, deemed 

sufficient for this study. 

0 u e 

∆x e 

∆y e 

i-1 

e, 1 

i 

e, 2 

i+1 

Figure 3.1: Median dual surface for evaluating first-order derivatives at grid point 

0 via the FV method. 

The evaluation of the gradient ∇u is accomplished via the following contour 

integration: 

∇u ≈ 1 S 

∮ 

∂l 

udl ≈ 1 ∑ 

u e (∆y e î − ∆x e ĵ) ≡ ∇ h u, (3.1) 

S 

e 

where S is the dual area, u e is the field value at the middle of each edge of the dual 

contour , while ∆x e and ∆y e are the projections in the x and y direction of the 

edges of the dual contour joining the centroids of each element sharing point 0 with


the middle of the grid edges sharing this point . This contour is termed median 

dual [15]. Other choices of dual surfaces include the surface formed by joining the 

centroids of the elements sharing point 0 (centroid dual), as well as the surface of 

the union of those elements. 

Since the goal is deriving a quality measure the specific choice is not important. 

A node-based evaluation of the derivatives is considered, which will reveal the 

inaccuracy due to the distorted elements in a relatively straightforward manner. 

3.2 The general form of the truncation error for 

the node based FV method 

The TE in the computation of the gradient is defined as: 

E(x, y) = ∇ h u(x, y) − ∇u(x, y), (3.2) 

with ∇ h u and ∇u being the numerical and the analytical values of the gradient, 

respectively. The analysis is similar in the x and y directions and employs Taylor 

series expansions for u h at the required locations around point 0. The expansions are 

substituted into the TE definition of Eq. (3.2). The amount of operations involved 

is very large. This must have been one of the main reasons for not seeing complete 

TE analysis in previous works. In the present work the hurdle is overcome via use 

of the symbolic mathematics capability of Matlab [101]. 

The TE terms are grouped according to the following general form given for 

the case of evaluating the derivative in the x direction (u h x): 

E x =e x xu x + e x yu y + e x xyu xy + e x xxu xx + e x yyu yy + 

e x xxxu xxx + e x yyyu yyy + e x xyyu xyy + e x xxyu xxy + ... 

(3.3) 

The terms with the symbol e are functions of the metrics of the mesh and will be


the link for making the connection of the TE with mesh distortion. They will be 

termed error coefficients (EC). Their analytic expressions are grouped according to 

the order in terms of the local mesh size h. 

The first two terms are involved in the consistency checks of the numerical 

approximation: 

e x x = 1 

2S 

e x y = 1 

2S 

n∑ 

(∆x e,1 + ∆x e,2 )∆y e − 1, 

e=1 

n∑ 

(∆y e,1 + ∆y e,2 )∆y e . 

e=1 

(3.4a) 

(3.4b) 

The grid metrics involved in the above expressions are defined as follows: 

(∆x e,k ) j (∆y e,k ) l = 1 

N e,k 

N e,k 

∑ 

(x m | e,k − x 0 ) j (y m | e,k − y 0 ) l , 

m=1 

(3.5) 

∆y e = y e,2 − y e,1 , 

where x m | e,k , y m | e,k and N e,k are the coordinates and the number of nodes participating 

in the averaging of u at the dual vertex e, k (k = 1, 2). Generally, if a 

median dual vertex coincides with the middle point of an edge sharing point 0, then 

N e,k = 2, and if it coincides with the center of a quadrilateral or a triangle then 

N e,k = 4, or N e,k = 3, respectively. In Eq. (3.5) it is observed that the metric terms 

∆x e,1 , ∆x e,2 , ∆y e,1 and ∆y e,2 are O(h) each. 

In the above expressions, n is the number of the dual edges, S is the dual area 

and it is O(h 2 ), while e, 1 and e, 2 denote the end points of each edge of the dual 

contour, as shown in Fig. 3.1. 

The next three terms are first-order:


e x xy = 1 

2S 

e x xx = 1 

2S2! 

e x yy = 1 

2S2! 

n∑ 

(∆x e,1 ∆y e,1 + ∆x e,2 ∆y e,2 )∆y e , 

e=1 

n∑ 

[(∆x e,1 ) 2 + (∆x e,2 ) 2 ]∆y e , 

e=1 

n∑ 

[(∆y e,1 ) 2 + (∆y e,2 ) 2 ]∆y e . 

e=1 

(3.6a) 

(3.6b) 

(3.6c) 

For the purpose of deriving quality measures, the second-order derivative terms are 

retained as well: 

e x xxx = 1 

2S3! 

e x yyy = 1 

2S3! 

e x xyy = 1 

2S2! 

e x xxy = 1 

2S2! 

n∑ 

[(∆x e,1 ) 3 + (∆x e,2 ) 3 ]∆y e , 

e=1 

n∑ 

[(∆y e,1 ) 3 + (∆y e,2 ) 3 ]∆y e , 

e=1 

n∑ 

[∆x e,1 (∆y e,1 ) 2 + ∆x e,2 (∆y e,2 ) 2 ]∆y e , 

e=1 

n∑ 

[(∆x e,1 ) 2 ∆y e,1 + (∆x e,2 ) 2 ∆y e,2 ]∆y e . 

e=1 

(3.7a) 

(3.7b) 

(3.7c) 

(3.7d) 

When the projections ∆x e and ∆y e are symmetrical relative to point 0, the 

computation becomes of second-order and the TE expression reduces to the following:


E x = e x xxxu xxx + e x xyyu xyy , 

E y = e y yyyu yyy + e y xxyu xxy . 

(3.8) 

Meshes that give a TE form given in Eq. (3.8) will be termed ideal meshes. Orthogonal 

and equally spaced in each direction quadrilaterals form an ideal mesh. The 

equivalent with triangles is an unstructured mesh with equal-length projections of 

their dual edges in the x and y directions. This definition will be used subsequently 

to define a quality measure. 

Similar expressions for the EC regarding evaluation of the derivative in the y 

direction hold. The projection ∆y e is replaced with ∆x e and a minus sign is placed 

in front of the sums. The EC e y x and e y y are given below: 

e y x = − 1 

2S 

e y y = − 1 

2S 

n∑ 

(∆x e,1 + ∆x e,2 )∆x e , 

e=1 

n∑ 

(∆y e,1 + ∆y e,2 )∆x e − 1. 

e=1 

(3.9a) 

(3.9b) 

3.3 Consistency condition for the node based FV 

method 

A consistent discretization of the gradient requires the TE to vanish as the mesh 

size approaches zero. For zero mesh size all the EC become equal to zero, except 

for e x x, e x y in Eq. (3.4) and e y x, e y y in Eq. (3.9), and the TE in Eq. (3.2) becomes:


E x = e x xu x + e x yu y , 

E y = e y yu y + e y xu x . 

(3.10) 

It is noted that the EC e x x, e y x, e x y, e y y remain unchanged for fixed–shape mesh elements 

even if the local mesh size is changing. For a consistent discretization, it is required 

that: 

e x x = e x y = e y x = e y y = 0. (3.11) 

Following Eq. (3.11) the expressions of the EC e x x in Eq. (3.4) and e y y in Eq. 

(3.9) show that in order for them to vanish on a general mesh, the area evaluation 

of the following two expressions must yield the same result. 

S x = 1 n∑ 

(∆x e,1 + ∆x e,2 )∆y e , 

2 

S y = − 1 2 

e=1 

n∑ 

(∆y e,1 + ∆y e,2 )∆x e . 

e=1 

(3.12) 

Using the median dual surface, from Eq. (3.12) it results that the two forms 

for evaluating the dual area produce the same result. Specifically, their difference is 

the following: 

∑ 

[x e,2 y e,2 − x e,1 y e,1 − 2x 0 (y e,2 − y e,1 )] = 0, (3.13) 

e 

and so e x x = e y y = 0 on a mixed-type element mesh. Carrying out the algebra in the 

expression of the EC e x y in Eq. (3.4) using the median dual results:


e x y = 1 

2S 

Analogously, the EC e y x in Eq. 

∑ 

[ye,2 2 − ye,1 2 − 2y 0 (y e,2 − y e,1 )] = 0. (3.14) 

e 

(3.9) results to be always zero on a mixed-type 

element mesh. So, the median dual surface leads to a consistent approximation of 

∇u. 

Using the centroid dual, from Eq. 

(3.12) it results that the two forms for 

evaluating the dual area do not produce the same result on structured meshes. 

Their difference considering a structured mesh is: 

1 

8 

∑ 

[x i (y i+1 − y i−1 ) + y i (x i+1 − x i−1 )] ≠ 0, (3.15) 

i 

and so e x x ≠ 0, e y y ≠ 0 on a structured mesh. Carrying out the algebra in the 

expression of the EC e x y in Eq. (3.4) using the centroid dual it gives: 

e x y = 1 

8S 

∑ 

[y i (y i+1 − y i−1 )] ≠ 0. (3.16) 

i 

Analogously, the EC e y x in Eq. (3.9) is not always zero on structured meshes. So, 

the consistency of the centroid dual depends on the geometry of a structured mesh 

in the approximation of ∇u. 

Using the centroid dual, from Eq. 

(3.12) it results that the two forms for 

evaluating the dual area produce the same result on unstructured meshes. 

difference of the two forms in Eq. (3.12) considering an unstructured mesh is: 

1 

6 

The 

∑ 

[x i (y i+1 − y i−1 ) + x i (y i+1 − y i−1 )] = 0, (3.17) 

i 

and so e x x = e y y = 0 on unstructured meshes. Carrying out the algebra in the 

expression of the EC e x y in Eq. (3.4) using the centroid dual it gives: 

e x y = 1 

6S 

∑ 

[y i (y i+1 − y i−1 )] = 0. (3.18) 

i


Analogously, the EC e y x in Eq. (3.9) is always zero on unstructured meshes. So, the 

centroid dual gives a consistent approximation of ∇u on unstructured meshes. 

At mixed-type element mesh interfaces it is difficult to check analytically the 

consistency of the approximation of ∇u using the centroid dual. Despite this fact, 

it is anticipated that the computation will be inconsistent due to the inconsistency 

of the centroid dual expressions on structured meshes. 

3.4 Verification of the error coefficients analytic 

expressions 

It is of importance to check the complex expression of the TE using analytic field 

functions u(x, y). The TE is computed by substituting the analytic values of the 

spatial derivatives and the values of the EC in Eq. (3.3). This will be termed 

the analytic TE. The TE is also computed directly by subtracting the analytic 

expression for ∇u from the FV expression ∇ h u (Eq. (3.2)). This second approach 

will be termed numerical TE. 

Let us consider a field function u(x, y), which resembles a boundary layer type 

of flow field with a linear variation in the streamwise direction: 

u(x, y) = xy 2 . (3.19) 

The differences between the analytic and the numerical TE values are calculated for 

the mixed-type element mesh of Fig. 3.2 using both the median and the centroid 

dual. A feature of this mesh is a local lateral displacement in the vicinity of an 

interface point. 

For the median dual the distribution of the analytic and the numerical TE is shown 

in Fig. 3.3. It is observed that the derived expressions are coincident within plotting 

accuracy. 

Check of consistency “experimentally” via successive reductions of the size of


Figure 3.2: Mixed-type element channel mesh. 

the mesh elements may be impractical for large meshes. An indirect check of the 

consistency is proposed here by computing the difference between the analytic and 

numerical TE, without including in the analytic expression the error coefficients 

“responsible” for consistency (e x x and e x y). When including these terms, which are 

independent of the local mesh size, the difference will be zero within machine accuracy. 

If it is not, as the case is for the centroid dual discretization (see Fig. 3.4), a 

local inconsistency is revealed. The “spikes” in Fig. 3.4 correspond to points in the 

vicinity of the locally distorted mesh interface. The same inconsistency was revealed 

for the case of a distorted structured mesh. 

Similar results were observed using other field functions, as well. This provides 

assurance that the derived complex expressions are correct. A further check of their 

validity, even though indirect, will be provided when checking the subsequently 

derived quality measures. 

3.5 Elementary Types of Mesh Distortion 

The general mesh depicted in Fig. 3.1 exhibits skewness, stretching, offset of point 

0 from the center of the control surface, as well as interfaces between structured


0.01 

0.005 

Error 

0 

-0.005 

-0.01 

0 20 40 60 80 100 120 140 160 180 200 220 

Node 

Figure 3.3: Distribution of the analytic (✷) and the numerical (✸) TE for the field 

u(x, y) = xy 2 on the mesh of Fig. 3.2 using the median dual. 

and unstructured elements. It is important in the context of the present work to 

“isolate” each type of mesh “defect” and relate it directly to the TE. This is very 

important for future work on improving the mesh during or immediately after its 

generation. Distortion types are easier to define on a quadrilateral mesh, compared 

to a triangular for which the offset of point 0 from the dual centroid is a reasonable 

measure. Regarding the interfaces between quadrilateral and triangular elements, 

three distinct types were identified and studied. It should be noted that skewness, 

stretching and center point offset are distortions that are not independent from each 

other. Application of one of them to an ideal mesh causes appearance of the rest. 

Nevertheless, this categorization offers a clear way to express the TE as a direct 

function of them and to come up with ways to reduce those distortions. It should 

be noted that the present work considers interior point configurations. Boundary 

“central” points (node 0 in Fig. 3.1) are not examined as the application of boundary 

conditions often “discards” the local discretization.


0 

Difference in Error 

-0.001 

-0.002 

0 20 40 60 80 100 120 140 160 180 200 220 

Node 

Figure 3.4: Difference between the analytic and the numerical TE for the field 

u(x, y) = xy 2 on the mesh of Fig. 3.2 using the centroid dual. 

3.6 Stretching 

Stretching is a common type of deviation from a uniformly–spaced mesh and it is 

readily defined for structured grids. In order to compare quadrilaterals and triangles 

in terms of sensitivity to stretching, the stretching type of mesh distortion is defined 

in a similar way as Fig. 3.5 depicts. In the quadrilateral case, the stretching is 

considered as the displacement of the edges sharing the center point 0 quantified 

by the stretching factors d x and d y . These edges are displaced in a structured 

way retaining their orientation, while stretching in the triangle case is defined via 

displacing the center node according to Fig. 3.5(b).


dx∆x 

• 

∆y 

0 

dy∆y 

∆y 

∆x 

(a) 

∆x 

d x h 

• 

d y h 

h 

0 

(b) 

Figure 3.5: Stretching for (a) structured and (b) unstructured meshes defined to 

facilitate comparison of accuracy degradation on them. 

3.6.1 Skewness and Shearing 

The four edges sharing the center point 0 in a quadrilateral grid usually meet forming 

angles that deviate from 180 ◦ as Fig. 3.6 illustrates. This type of distortion is 

typically called skewness. The angle ω quantifies the degree of skewness and ranges 

from 0 ◦ to 90 ◦ . Shearing is quite different from skewness. It preserves the 180 ◦ 

angle between the edges and deviates the mesh from being orthogonal (Fig. 3.7). 

The angle φ quantifies this deviation and ranges from 0 ◦ to 90 ◦ . It will be seen that 

shearing and skewness affect the accuracy very differently.


0 ω 

∆x 

Figure 3.6: Skewed mesh, depicting the deviation angle ω from the 180 ◦ angle 

between one of the two pairs of edges sharing point 0 (ω ∈ [0 ◦ , 90 ◦ ]). The lengths of 

the edges are equal. 

3.6.2 Mesh rotation 

Non–alignment of a structured mesh to a specific direction dictated by the form 

of the field is another type of distortion examined. The quadrilateral elements 

are rotated with respect to the x–axis of the coordinate system by an angle θ as 

illustrated in Fig. 3.8. This angle varies from 0 ◦ to 45 ◦ . 

0 

φ 

∆x 

Figure 3.7: Sheared mesh, depicting displacement of the edges causing deviation 

from orthogonality expressed by the angle φ (φ ∈ [0 ◦ , 90 ◦ ]). The lengths of the 

edges are equal.


∆y 

y 

∆y 

0 

θ 

x 

∆x 

∆x 

Figure 3.8: Rotated structured mesh depicting non alignment of the edges with the 

axes of the global system (∆x ≠ ∆y and θ ∈ [0 ◦ , 45 ◦ ]). 

3.6.3 Mixed-type element mesh interfaces 

Change in the topology of the elements creates special interfaces which require 

examination in terms of the local accuracy of the discretization [75]. The elements 

forming the interface (quadrilaterals and triangles) are considered to have the same 

size so that the effect of the changing topology is isolated and studied. Three cases 

of such interfaces are identified in two dimensions and are illustrated in Fig. 3.9. 

ψ 2 ψ 1 

h 

h 

h 

0 

0 

0 

h 

h 

h 

h 

(a) 

h 

(b) 

h 

(c) 

Figure 3.9: Common mixed-type element mesh interfaces.


3.7 Direct relation between truncation error and 

mesh distortion 

The general expressions of the error coefficients (EC) given in Eqs. (3.6) and (3.7) 

can yield simpler expressions directly related to each type of mesh distortion presented 

here. This is very important to subsequent work on improving the grid. 

Table 3.1 gives the expressions of the EC for a mesh exhibiting only one type 

of distortion for each case (a row of the table). The related parameters d x , d y , ω 

are present in the expressions. For clarity of the table, the terms related to the 

higher order EC are not given, and it is just noted if they are zero or not. It is 

observed that stretching and skewness reduce the formal second order of the FV 

discretization to first. The error is proportional to the stretching parameters d x and 

d y . In the case of skewness, the relationship with ω is not linear. Finally, for the 

case of shearing and non–alignment of a structured mesh, no reduction of the order 

of accuracy occurs. However, the structure of the TE changes (terms e x xxx, e x yyy, e x xyy 

and e x xxy) but the discretization remains second order. 

Mesh e x xy e x xx e x yy e x xxx e x yyy e x xyy e x xxy 

Fig. 3.5(a) 

1 

2 d y∆y d x ∆x 0 ≠ 0 0 ≠ 0 ≠ 0 

Fig. 3.5(b) d y h d x h 0 ≠ 0 0 ≠ 0 ≠ 0 

Fig. 3.6 

3 sin (2ω)∆x 

8 cos (ω)+8 

[cos (ω)−1]∆x 

2 

[1−cos (ω)]∆x 

4 

≠ 0 ≠ 0 ≠ 0 ≠ 0 

Fig. 3.7 0 0 0 ≠ 0 0 ≠ 0 ≠ 0 

Fig. 3.8 0 0 0 ≠ 0 ≠ 0 ≠ 0 ≠ 0 

Table 3.1: Expressions for the error coefficients for each type of mesh distortion. 

The EC corresponding to evaluation of the derivative u y of the three types of


mixed-type element mesh interfaces are presented in Table 3.2. For the first type 

of interface, it is considered that ψ 1 = ψ 2 . It is observed that mesh interfaces yield 

first order accuracy for the evaluation of the derivative u y . 

Mesh e y xy e y xx e y yy e y xxx e y yyy e y xyy e y xxy 

Fig. 3.9(a) 0 − h 48 

0 ≠ 0 ≠ 0 ≠ 0 ≠ 0 

Fig. 3.9(b) 0 0 − h 10 

0 ≠ 0 ≠ 0 ≠ 0 

Fig. 3.9(c) − h 6 

0 0 0 ≠ 0 ≠ 0 ≠ 0 

Table 3.2: Expressions for the error coefficients for the mixed-type element mesh 

interfaces. 

However, it is possible to relocate node 0, so that the order of accuracy of the u y 

evaluation is improved, but only for the mesh interfaces of Figs. 3.9(b),(c). In Table 

3.3 the displacements of point 0 for improving the accuracy of the u y evaluation are 

given. It should be noted that the results of Table 3.3 concern interfaces with equal 

“heights” (h) of the elements. 

Mesh ∆x ∆y 

Fig. 3.9(a) non existent non existent 

Fig. 3.9(b) 0 2(−10 + 3 √ 11)h 

Fig. 3.9(c) − h 6 

0 

Table 3.3: Central interface point displacements for improving the accuracy of the 

u y derivative evaluation for the mixed-type element mesh interfaces.


3.8 Definition of an appropriate index of mesh 

distortion 

The focus of the present work is on a priori evaluation of a grid with regards to 

the shape and topology of its elements. The other aspect of grid quality, namely 

the local resolution is not addressed here. Therefore, the defined measure (index) of 

mesh distortion should be independent of the local size of the mesh. Other properties 

that this index should have include: 

1. simple mathematical form, 

2. direct relation with the TE, as well as the mesh distortion parameters, 

3. ability to capture distortions in any direction, and 

4. ability to detect relatively small distortions in the presence of larger ones. 

3.8.1 Normalized error coefficients 

The EC of Eqs. (3.6) and (3.7) are divided by the appropriate power of a characteristic 

local length scale L. Care must be taken in order to account for directionally-sized 

(high aspect ratio) local mesh elements. This implies appropriate use of two length 

scales (L x ,L y ) expressing the local size in the x and y direction, respectively. 

The “normalized” EC of Eqs. (3.6) and (3.7) are defined as : 

e x xx = ex xx 

L x 

, 

e x yy = ex yy 

L y 

, 

e x xy = ex xy 

L y 

, (3.20) 

e x xxx = ex xxx 

L 2 x 

, e x yyy = ex yyy 

L 2 y 

, e x xxy = ex xxy 

L 2 y 

, e x xyy = ex xyy 

. (3.21) 

L 2 y 

The denumerators in the above definitions have been chosen based on the mesh 

metrics appearing in the EC expressions and the order of each EC.


Careful consideration is needed to define the characteristic local length. It is 

incorrect to define a single length for every direction, as then, on meshes with cells 

of high aspect ratio, the normalized EC would either be overestimated or underestimated. 

The present work uses an approach that resembles the FV evaluation of 

the derivative. It defines the following grid functions : 

f x = ∆x e,k |∆x e,k |, f y = ∆y e,k |∆y e,k |, k = 1, 2. (3.22) 

Then, the lengths L x ,L y are computed in a similar manner to Eq. (3.1) with u being 

replaced by f x for the x–component and by f y for the y–component: 

L x = 1 ∑ 

f x,e ∆y e , 

S 

e 

L y = − 1 ∑ 

f y,e ∆x e (3.23) 

S 

e 

It should be noted that for a uniform structured mesh with element sizes ∆x and 

∆y, the characteristic lengths are: 

L x = ∆x and L y = ∆y. 

For the case of a uniform (honeycomb shape) triangular mesh of edge size h, the 

local lengths are: 

L x = 35h 

48 

and 

L y = 7√ 3h 

16 . 

Also, for the examined mixed-type element mesh interfaces the characteristic local 

lengths are given in Table 3.4. 

3.8.2 Mesh quality index 

The normalized EC are grouped to yield a single number that characterizes the 

quality of the grid. Two such groupings are being presented and evaluated. The 

first is expressed via use of the first-order error coefficients:


Mesh L x L y 

Mesh interface of Fig. 3.9(a) 

11 

12 h h 

Mesh interface of Fig. 3.9(b) h h 

Mesh interface of Fig. 3.9(c) h h 

Table 3.4: Characteristic local lengths for the mixed-type element mesh interfaces. 

Q x = |e x xx| + |e x yy| + |e x xy|, 

(3.24a) 

Q y = |e y xx| + |e y yy| + |e y xy|, 

(3.24b) 

where the first expression regards the u x derivative evaluation, and the second concerns 

the u y computation. On meshes which yield second-order accuracy in the 

evaluation of the derivatives u x and u y , both Q x and Q y are equal to zero. 

The second index that is presented expresses the deviation of a mesh from 

being ideal via the ratio of a form of the EC appearing on a general distorted mesh 

to a form of those coefficients for an ideal mesh. Specifically: 

q x = |ex xx| + |e x yy| + |e x xy| + |e x xxx| + |e x yyy| + |e x xyy| + |e x xxy| 

, (3.25a) 

|e x xxx| + |e x xyy| 

q y = |ey xx| + |e y yy| + |e y xy| + |e y xxx| + |e y yyy| + |e y xyy| + |e y xxy| 

. (3.25b) 

|e y xxx| + |e y xyy| 

Index q expresses the deviation of the mesh locally from being ideal. It should be


noted that q x and q y are equal to one for an ideal mesh, and have values greater 

than one for a general distorted grid. 

It is interesting to examine the values of the indices for a structured mesh with 

high aspect ratio elements that is typical of boundary layer regions and illustrated 

in Fig. 3.10. 

0 

2∆y 

2∆x 

Figure 3.10: High aspect ratio quadrilaterals typical of a structured mesh in a 

boundary layer region. 

The relevant computation here is that of the derivative u y . The relevant EC 

in Eq. (3.8) for this case are: 

e y xxy = (∆x)2 

8 

, e y yyy = (∆y)2 , 

6 

which yields the mesh quality index Q y to be zero and the q y to be one, which are 

the best possible quality index values. 

The experiment is repeated employing the triangular mesh created by subdividing 

the quadrilaterals of Fig. 3.10 along their diagonals. The corresponding error 

coefficients are: 

e y yyy = (∆y)2 

6 

, e y xyy = − ∆x∆y 

6 

, e y xxy = (∆x)2 . 

6 

It is observed that the order of accuracy remains second (Q y = 0). However, more 

non–zero EC appear and the value of q y is not that of the ideal mesh, but is greater 

than one.


3.8.3 Analytic expressions of the quality index Q for each 

type of elementary mesh distortion 

The index Q has a simpler mathematical form and thus amenable to a direct relation 

to each type of mesh distortion. These expressions are given in Table 3.5. The 

equations for the stretched unstructured and the skewed structured mesh cases are 

given in Appendix A. 

Mesh 

Q 

Stretched structured [Fig. 3.5(a)] 

Stretched unstructured [Fig. 3.5(b)] 

Skewed [Fig. 3.6] 

Q x = | dx | + | 

1+d 2 x 

Q x (Eq. A-1) 

Q x (Eq. A-2) 

d y 

2+2d 2 y 

| 

Sheared [Fig. 3.7] Q x , Q y = 0 

Non–aligned structured [Fig. 3.8] Q x , Q y = 0 

Mesh interface [Fig. 3.9(a)] Q y = 1 

44 

Mesh interface [Fig. 3.9(b)] Q y = 144 

533 

Mesh interface [Fig. 3.9(c)] Q y = 1 6 

Table 3.5: 

distortion. 

Mesh quality index Q expressions for the elementary types of mesh 

It is observed that, the sheared and rotated (non–aligned) structured meshes do not 

exhibit reduction of accuracy and thus the indices are equal to zero. The unstructured 

mesh exhibits higher sensitivity in mesh distortion as it is depicted graphically 

in Fig. 3.11. 

The variation of the index with respect to the skewness angle is given in the Appendix 

A, and it is shown graphically in Fig. 3.12. A peak in the degradation of


0.5 

0.4 

0.3 

Q x 

0.2 

0.1 

Unstructured 

Structured 

0 

0 0.1 0.2 0.3 0.4 0.5 

d x 

Figure 3.11: Mesh quality index Q x for the stretched unstructured mesh (◦) and the 

stretched structured mesh (□) vs. displacement factor d x (d y = 0). 

accuracy is observed at about 80 ◦ . It is observed that the normalized error coefficient 

e x xy reduces as the skewness angle approaches 90 ◦ , which leads to the reduction of 

Q x in the same region in the graph. 

Finally, the influence of the mesh interface angle ψ 2 on degradation of accuracy 

is shown in the graph of Fig. 3.13. This concerns the mixed-type element 

mesh interface shown in Fig. 3.9(a). The curve is symmetric around the value of 

tan −1 (2) ≈ 63 ◦ for which Q y is minimum. 

3.8.4 Calibration of the mesh quality index Q 

Any practical use of the grid quality index to judge a mesh requires knowledge of the 

range of permissible values of Q. A calibration is needed which is based on upper 

values of stretching d x , d y and skewness ω that are considered acceptable for typical 

field solvers. The proposed calibration has a conservative character as the upper 

bound for index Q is chosen as max(Q x , Q y ). An upper value of 20% stretching 

1 

means a value of d x and d y of , which when substituted in the expression for Q 

11


0.8 

Q x 

Normalized EC and Q x 

0.6 

0.4 

0.2 

e x xy 

e x 2x 

e x 2y 

0 

0 20 40 60 80 100 

ω 

Figure 3.12: Mesh quality index Q x and its corresponding normalized error coefficients 

vs. the skewness angle ω. 

yields a value of approximately 0.13. Similarly for a maximum skewness angle ω of 

20 ◦ , a value of Q ≈ 0.17 is found. 

Considering the case of a stretched unstructured mesh and applying the same 

stretching (center node displacement factor) of 20% a value of Q ≈ 0.24 is found. 

Finally, for the mixed-type element mesh of Fig. 3.9(a), setting ψ 2 = tan −1 (2) ± 20 ◦ 

with ψ 1 = 2 · tan −1 (2) − ψ 2 , an upper bound of Q ≈ 0.16 is derived. Therefore, an 

upper bound value for Q of around 0.20 is a reasonable choice for all types of mesh 

distortion. 

3.9 Application of the mesh quality indices 

The quality index Q that is based on the first-order error terms appears to be quite 

simpler to use compared to the index q that uses both the first and the second 

order terms. However, a decision to adopt Q cannot be made until both of them 

are implemented with general distorted meshes. This section employs three types 

of grids; structured, unstructured, as well as hybrid. The hybrid grids consist of 

separate quadrilateral and triangular layers, which is quite typical in applications.


0.25 

0.2 

0.15 

0.1 

0.05 

Q y 

ψ 2 

0 

30 40 50 60 70 80 90 100 

Figure 3.13: Mesh quality index Q y for the mixed-type element mesh interface of 

Fig. 3.9(a) vs. angle ψ 2 . 

The geometries involved are channel, airfoil and cylinder. It is important that the 

employed grids exhibit (i) both small and large magnitude distortions, as well as (ii) 

local and more global ones. The contours of the functions Q ≡ max(Q x , Q y ) and 

q ≡ max(q x , q y ) are plotted for all cases. 

In order to compare the effectiveness—fidelity of the indices for the different 

cases, the same contour levels are plotted, that is the minimum, maximum and 

number of levels are the same for all grids for each index. Further, both indices 

(Q and q) are shown with the same number of contour levels, which is essential for 

comparing the two on the same mesh. 

3.9.1 Channel grids 

The structured channel grid shown in Fig. 3.14(a) is uniform everywhere except 

in the middle where a point is displaced. Both indices capture this distortion with 

index Q–contours being more focused. 

The same displacement is applied to an almost uniform triangular mesh shown


in Fig. 3.15(a). Similarly here, both indices capture the primary distortion along 

with smaller ones with q capturing a bit broader area than Q. This is due to the 

second order EC included in the definition of index q. 

The third mesh is hybrid with a straight line transitioning from the quadrilaterals 

to the triangles shown in Fig. 3.16(a). The index Q is more focused on the 

interface region than index q does. 

3.9.2 Airfoil grids 

An O–type structured mesh for the NACA 0012 airfoil is employed next. There is 

a sudden expansion of the quadrilaterals size in the region close to the surface as 

shown in Fig. 3.17. The same observation is made here, namely Q captures the 

areas of mesh stretching and skewness (leading and trailing edge regions) stronger 

compared to index q. 

The unstructured mesh around the NACA 0012 shown in Fig. 3.18 is a good 

quality mesh with small distortions everywhere. This case was devised to check the 

indices’ sensitivity to relatively small (“background”) distortions. It is observed that 

both indices show sensitivity to these non–uniformities with q being more sensitive 

to them. 

Finally, a hybrid mesh around the RAE 2822 airfoil is employed (Fig. 3.19). 

This is a viscous mesh with a very good preservation of the order of accuracy in 

the boundary layer region, except at the mesh interfaces, as indicated by index 

Q. However, index q shows a large deviation from the ideal computation for the 

boundary layer region due to the non alignment of the structured mesh in that area 

with the axes of the system of reference. 

3.9.3 Cylinder grids 

The cylinder mesh has mild distortions away from the body surface. It is an interesting 

case to check the performance of the quality indices at the farfield. Fig. 3.20


illustrates the structured mesh which exhibits skewness along lines ”diagonally” off 

the surface. The grid is of good quality close to the cylinder and this is indicated by 

both indices. It is observed that the four “skewness lines” are captured more clearly 

by index Q. 

The next case involves a triangular grid exhibiting a large stretching area close 

to the surface, as well as a mild mesh distortion of irregular shape away from the 

cylinder (Fig. 3.21). The two indices do recognize both types of distortion, and 

their contours “follow” the mild distortion pretty closely. 

Finally, the hybrid grid of Fig. 3.22 exhibits irregularity of the interface between 

the quadrilaterals and the triangles, as well as quite strong distortions in the 

farfield. It is observed that both indices recognize both types of strong distortions; 

the circular–shaped interface and the “random” ones in the triangles zone.


(a) 

(b) 

(c) 

Figure 3.14: Locally stretched structured channel mesh: (a) mesh geometry, (b) 

index q and (c) index Q.


(a) 

(b) 

(c) 

Figure 3.15: Locally stretched unstructured channel mesh: (a) mesh geometry, (b) 



(a) 

(b) 

(c) 

Figure 3.16: Mixed-type element channel mesh: (a) mesh geometry, (b) index q and 

(c) index Q.


(a) 

(b) 

(c) 

Figure 3.17: O–type structured mesh around a NACA 0012 airfoil: (a) mesh geometry, 

(b) index q and (c) index Q.


(a) 

(b) 

(c) 

Figure 3.18: Unstructured mesh around a NACA 0012 airfoil: (a) mesh geometry, 



(a) 

(b) 

(c) 

Figure 3.19: Mixed-type element mesh around a RAE 2822 airfoil: (a) mesh geometry, 



(a) 

(b) 

(c) 

Figure 3.20: Structured mesh around a cylinder: (a) mesh geometry, (b) index q 

and (c) index Q.


(a) 

(b) 

(c) 

Figure 3.21: Unstructured mesh around a cylinder: (a) mesh geometry, (b) index q 

and (c) index Q.


(a) 

(b) 

(c) 

Figure 3.22: Mixed-type element mesh around a cylinder: (a) mesh geometry, (b) 


Chapter 4 

Discontinuous Galerkin 

discretization 

The equations governing compressible flow (Eqs. (2.1) and (2.29)) are discretized 

with the DG method. The DG method is a FE method which was first introduced 

in 1973 by Reed and Hill [122] and further developed to the so called Runge–Kutta 

DG (RKDG) method by Cockburn ans Shu in a series of papers [28, 40–44]. It uses 

discontinuous piecewise polynomials as basis functions and relies on the choice of 

the numerical fluxes, which handle effectively the interactions across the element 

interfaces to achieve stable and highly accurate numerical solutions. In the last few 

years, the DG method attracted significant interest in the computational mechanics 

community, because it offers certain advantages compared to the FV, FD methods 

as well as to the classical continuous FE method. 

A remarkable advantage of this method is that the local support of the numerical 

solution at the element level, makes it particularly suitable for flow simulations 

with high-order of accuracy. Moreover, mesh distortion does not affect the quality 

of the solution considerably as the order of the scheme increases [139, 150, 151] to 

very high-orders. Furthermore, hp-adaptation is easy to implement permitting high 

resolution of the flow field at selected flow regions. 

Inspired from the DG method other methods with similar features such as 

87

CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 88 

the spectral difference (SD) [98] and the SV [5, 149] methods have been recently 

developed. In this work the DG method is used because it is based on the FE 

discretization framework and it is straightforward to extend in three dimensions. 

The locality of the method makes it well suited for parallel implementation. Due 

to the large amount of computations required for the evaluation of the terms appearing 

in the semi-discrete form of the equations, it is imperative to implement 

the algorithms of the DG method for parallel computation. The DG method can 

easily be implemented on mixed-type element meshes, with the use of edge based 

data structures. 

By way of summary the DG method has several potential advantages compared 

to other methods including: 

ˆ Spectral accuracy on arbitrary meshes. 

ˆ Local h-refinement with the use of hanging nodes. 

ˆ Local p or hp-refinement. 

ˆ Boundary conditions are imposed weakly through the numerical flux. 

ˆ Local conservation laws allow for different fidelity models on neighboring elements. 

ˆ Highly parallelizable. 

4.1 Spatial Discontinuous Galerkin discretization 

As a FE method the DG discretization relies upon certain definitions, which are 

the building blocks of all the FE methods. These definitions are provided here in 

accordance to Ciarlet [37]. 

Definition 1 (FE) A finite element is defined as a triplet {Ω e , P e , Σ e }, where : 

ˆ Ω e is a subset of R d .


ˆ P e is a vector space of functions p e : Ω → R m for some positive integer m. 

ˆ Σ e is a set of linearly independent linear forms U i , 1 ≤ i ≤ N defined over 

the space P e . The set Σ e is P -unisolvent, that is given any real scalar a there 

exists a unique function p e ∈ P e , such that U i (p e ) = a. 

Definition 2 (FE Mesh) The set T h = {Ω 1 , Ω 2 , . . . , Ω n } over a domain Ω with a 

piece wise polynomial boundary defines a tessellation of Ω: 

Ω = ⋃ e 

Ω e . 

Each element Ω e has a polynomial order N e . 

Definition 3 (FE interpolant) Given a unisolvent finite element {Ω e , P e , Σ e }, let 

B = {b 1 , b 2 , . . . , b n } be the set of basis that span the space P e . Let v ∈ V where 

P e ⊂ V , be a function for which the linear forms U i (p e ) are defined. The local FE 

interpolant is defined as: 

n∑ 

U e = c e i b i . 

i=1 

From the linearity of the forms b i , it follows the linearity of the interpolation operator 

U e . 

4.1.1 Discontinuous weak formulation for the Euler equations 

A FE mesh T h over an arbitrary domain Ω is considered, where the physical field 

is approximated by a function Ũ(x, t), constructed by a locally continuous approximation 

(FE interpolant) Ũe(x, t) over each Ω e . For a DG spatial discretization 

Ũ(x, t) is considered discontinuous at the inter element boundaries as depicted in 

Fig. 4.1. Symbolically, it may be written as:


Ũ(x, t) = ∑ e 

Ũ e (x, t). 

Considering a single element Ω e , the approximation Ũe(x, t) substituted in the system 

of Euler equations (2.1), results in an approximation error, or residual, of the 

field over the elemental region, equal to: 

R e (Ũe(x, t)) = L(Ũe(x, t)), (4.1) 

leading to the residual over domain Ω: 

R(Ũ(x, t)) = ∑ e 

R e (Ũe(x, t)). (4.2) 

Ũ h 1 

Ũ h 2 

Ω 1 

Ω 2 

Figure 4.1: Local reconstruction of the field at two adjacent elements. 

As a FE method, the DG discretization, minimizes the approximation error over 

the computational domain in the weighted residual sense. That is, forming the 

Legendre inner product of R(Ũ(x, t)) with an arbitrary test or weighting function 

v, and setting it equal to zero:


∫ 

Ω 

R(Ũ(x, t))vdΩ ≡ (R(Ũ(x, t)), v) = 0, 

leads to: 

∫ 

Ω 

[ 

v ∂Ũ 

∂t + v∇ · F (Ũ) ] 

dΩ = 0 ⇒ 

∫ 

Ω 

v ∂Ũ 

∫ 

∂t 

∫Ω 

dΩ − ∇v · F (Ũ)dΩ + ∇ · vF (Ũ)dΩ = 0. 

Ω 

Using Gauss’ theorem, the last equation results in: 

∫ 

Ω 

v ∂Ũ 

∮ 

∂t 

∫Ω 

dΩ − ∇v · F (Ũ)dΩ + vF (Ũ) · n∂l = 0, (4.3) 

∂Ω 

where n is the outward normal vector at the domain boundary. Alternatively, Eq. 

(4.3) may be written as: 

∑ 

e 

[ ∫ ∫ 

∮ 

] 

v ∂Ũe 

Ω e 

∂t dΩ e − ∇v · F (Ũe)dΩ e + vF (Ũe) · n e ∂l = 0, 

Ω e ∂Ω e 

with n e denoting the outward normal vector at the element boundaries. However, 

the discontinuous nature of the solution Ũ, permits to write the residual for every 

element as a separate equation: 

Eq. 

equations. 

∫ 

Ω e 

v ∂Ũe 

∂t dΩ e − 

∫ 

∮ 

∇v · F (Ũe)dΩ e + 

Ω e 

vF (Ũe) · n e ∂l = 0. 

∂Ω e 

(4.4) 

(4.4) represents the discontinuous weak formulation for the system of Euler 

The local approximation of the field is constructed by a linear combination of 

elemental basis functions b e k (x), according to the previous definitions:


Ũ e (x, t) = ∑ k 

c k e(t)b e k(x), (4.5) 

where the coefficient vector c k e(t) denotes the degrees of freedom (DOF) of the 

numerical solution to be advanced in time at every element. Substituting Ũe(x, t) 

in Eq. (4.4), yields: 

∫ 

Ω e 

vb e k 

∂ĉ k ∫ 

∮ 

e 

∂t dΩ e − ∇v · F (Ũe)dΩ e + vF (Ũe) · n e ∂l = 0, (4.6) 

Ω e ∂Ω e 

The test function v and the elemental expansion function b e k (x) belong to the same 

FE space: 

X = {f ∈ L 2 (Ω) : f e ∈ Pe m (Ω e ) ∀Ω e ∈ T h }, (4.7) 

where Pe 

m (Ω e ) denotes the local space of polynomial functions of degree at most m 

and L 2 (Ω) represents the space of functions, which are squared Lebesgue integrable 

over Ω e . Then a system of 4DOF × 4DOF equations results for the system of Euler 

equations in two dimensions. 

4.1.2 The Numerical flux 

An important component for a DG discretization of the system of Euler equations 

is the evaluation of the numerical flux. Generally, a numerical flux that satisfies the 

following conditions is suitable for a solution algorithm for conservation laws based 

on the DG discretization: 

ˆ Consistent. 

ˆ Conservative.


The condition of consistency requires that the numerical flux H should approximate 

the line integral in Eq. (4.21). Typically a requirement of Lipschitz continuity is 

imposed, so that if a constant D exists the following inequality is satisfied: 

) 

∣ 

∣H 

(Ũ+ e , Ũ− e 

(∣ ) 

∣∣ 

− F (U) ∣ ≤ Dmax Ũ + ∣ 

− U∣ , ∣Ũ− − U∣ 

. (4.8) 

The requirement for conservation in the formulation of the numerical flux is mandatory 

in order to obtain physically correct solutions for flows with discontinuities. 

The numerical fluxes may be chosen based on the exact or approximate Riemann 

solvers, developed for the FD and FV methodologies [135]. Qiu et al. in [119] 

examined several numerical fluxes starting from the exact Godunov flux up to more 

sophisticated fluxes from approximate Riemann solvers, such as the Roe [124] and 

the HLLC [136] flux and it was demonstrated that the local Lax-Friedrichs (LLF) 

flux is the most efficient in terms of CPU time, and that at orders of approximation 

higher than 3, the behavior of the resulting scheme with the LLF flux being almost 

the same with other less diffusive, but more computationally expensive numerical 

fluxes. This must have been the main reason that most RKDG works in the literature 

to be formulated with the LLF flux, as it has been done in the present work. 

The LLF flux is: 

) 

H 

(Ũ+ e , Ũ− e = 1 2 

[ ] + (fi + f − i )n x + (g + i + g − 1 

i )n y − 

2 k ( ) 

U + e − U − e , (4.9) 

where the superscripts (+) and (-) denote the exterior and interior elements respectively, 

sharing the same edge and k is the spectral radius (maximum eigenvalue) of 

the flux Jacobian given in Eq. (2.24) along the direction n normal to the edge and 

is equal to: 

k = u · n + c,


and it is obtained from the flux Jacobian evaluated at the Roe averaged state: 

ρu ˜ = 

√ 

ρ+ u + + √ ρ − u − 

√ 

ρ+ + √ ρ − , (4.10) 

ρv ˜ = 

√ 

ρ+ v + + √ ρ − v − 

√ 

ρ+ + √ ρ − , (4.11) 

˜h = 

√ 

ρ+ h + + √ ρ − h − 

√ 

ρ+ + √ ρ − , (4.12) 

˜c = (γ − 1) 

√ 

˜h − 1 2 ( ρu2 ˜ + ρv ˜ 

2 ), (4.13) 

with h = E + p. 

The Roe flux [124] has also been implemented. Roe’s flux is: 

) 

H 

(Ũ+ e , Ũ− e = 1 2 

[ ] + (fi + f − i )n x + (g + i + g − 1 

i )n y − 

2 LΛR ( ) 

U + e − U − e , (4.14) 

where L, R are the left and right eigenvectors (Eqs. (2.28) and (2.27)) and Λ is the 

matrix of eigenvalues given in Eq. (2.25) computed at the Roe average state. The 

Harten and Hyman entropy fix is used to avoid the expansion shock through a sonic 

point, alternating the eigenvalues in matrix Λ according to: 

The quantity ɛ is determined as follows: 

λ i = λ2 i + ɛ 2 

. (4.15) 

2ɛ 

ɛ = max ( 0.01, λ i − λ + i , λ− i − λ i 

) 

. (4.16) 

where λ i is the eigenvalue computed using the Roe averaged variables and λ + i , λ− i 

are the eigenvalues computed using the elements’ traces. The index i refers to the i 

characteristic field of the system.


4.1.3 Basis functions and standard elemental configuration 

The choice of the basis functions affects the behavior and computational efficiency of 

the DG discretization [83, 150]. Three types of basis functions may be constructed, 

while satisfying the unisolvency of the FE interpolation: modal, nodal and mixed 

modal-nodal. 

ˆ Modal bases are a set of approximating polynomials which have the property 

that by increasing the order of approximation, all the previous orders are still 

included in the approximation set. 

For example, the basis defined by the 

following set: {1, x, y, xy, x 2 , y 2 } is a modal hierarchical base. 

ˆ Nodal bases are a set of approximating polynomials, which have the property 

of being equal to unity at specific points and zero at other points in the approximation 

space. Lagrange interpolation polynomials is an example of nodal 

bases. 

ˆ Mixed modal-nodal bases are a combination of the above polynomial sets. 

When considering of applying a p-type refinement to the numerical solution, it 

is advantageous to use modal hierarchical expansion bases. Such bases may be 

constructed using the orthogonal set of Jacobi polynomials [83]. This approach was 

followed in the present work. The basis functions are defined over the standard 

square elemental configuration, which permits their systematic construction and 

implementation of any numerical operation involving them. 

Application of a Gauss type numerical integration, requires the definition of a 

standard elemental region spanning the domain [−1, 1] × [−1, 1]. In Fig. 4.2, the 

transformation of the physical quadrilateral to its standard elemental configuration 

Ω q st is depicted, and in Fig. 4.3, the transformation of the physical triangle to 

its standard elemental configuration Ωst t is shown. In this figures the Ω q st region, 

where the basis functions are to be constructed and all the numerical operations are 

performed is also shown.


n 2 

n 1 

C 

x = x(n 1 , n 2 ) 

(-1,1) (1,1) 

D 

C 

D 

B 

(0,0) 

y 

q 

A 

y = y(n 1 , n 2 ) 

A 

(-1,-1) 

B 

(1,-1) 

O 

x 

p 

Ω q e 

Ω q st 

Figure 4.2: Transformation of the physical quadrilateral to the standard quadrilateral 

element configuration (η 1 , η 2 ∈ [−1, 1]). 

For quadrilateral elements, the basis is constructed using the tensorial product 

of Legendre polynomials ˆψ p (n), which are subset of the Jacobi polynomials. 

according to a specific indexing, over Ω q st [83]: 

b k (η 1 , η 2 ) = ˆψ p (η 1 ) ˆψ q (η 2 ), −1 ≤ η 1 , η 2 ≤ 1, (4.17) 

with: 

k = p + q(N + 1), 0 ≤ p, q ≤ N. 

If a local approximation of order N is applied, then the number of bases in X is 

equal to (N + 1) 2 . 

For triangular elements, the basis is defined over Ωst t and constructed using 

the tensorial product of Jacobi polynomials over the region Ω q st (Fig. 4.3) [49]: 

b k (ξ 1 , ξ 2 ) = Pp 0,0 (η 1 )( 1 − η 2 

) p Pq 2p+1,0 (η 2 ), (4.18) 

2 

with the use of collapsed coordinates [83]:


(-1,1) 

C 

ξ 2 

(0,0) 

ξ 1 

A 

B 

(-1,-1) 

(1,-1) 

x = x(ξ 1 , ξ 2 ) 

Ω t st 

n 1 = 2 1+ξ 1 

1−ξ 2 

− 1 

y = y(ξ 1 , ξ 2 ) 

n 2 = ξ 2 

C 

B 

x = x(n 1 , n 2 ) 

(-1,1) (1,1) 

D 

C 

y 

A 

Ω t e 

y = y(n 1 , n 2 ) 

q 

A 

(-1,-1) 

(0,0) 

n 2 

n 1 

B 

(1,-1) 

O 

x 

p 

Ω q st 

Figure 4.3: Transformation of the physical domain triangle to the standard triangular 

element (ξ 1 , ξ 2 ∈ [−1, 1] with ξ 1 + ξ 2 ≤ 1) and standard quadrilateral element 

configuration (η 1 , η 2 ∈ [−1, 1]). 

ξ 1 = (1 + η 1)(1 − η 2 ) 

2 

and according to the following indexing [93]: 

− 1, ξ 2 = η 2 , 

k = p + (N + 1)q − q q − 1 , 0 ≤ p, q ≤ N with p + q ≤ N. 

2 

For a triangular element the number of bases for a local approximation of order N, 

is equal to (N+1)(N+2) 

2 

. 

In Tables 4.1 and 4.2 the analytical form of the basis functions, over the standard 

quadrilateral and triangular region, for a third order expansion are given, and


plotted in Figs. 4.4 and 4.5, respectively. 

(a) (b) (c) 

(d) (e) (f) 

(g) (h) (i) 

Figure 4.4: Basis functions over the standard square element: (a) (p, q) = (0, 0), 

(b) (p, q) = (1, 0), (c) (p, q) = (2, 0), (d) (p, q) = (0, 1), (e) (p, q) = (1, 1), (f) 

(p, q) = (2, 1), (g) (p, q) = (0, 2), (h) (p, q) = (1, 2), (i) (p, q) = (2, 2).


p, q b k p, q b k p, q b k 

(0, 0) 1 (1, 0) η 1 (2, 0) 

3n 2 1 −1 

2 

(0, 1) η 2 (1, 1) η 1 η 2 (2, 1) 

3n 2 1 +1 

2 

η 2 

(0, 2) 

3n 2 2 −1 

2 

(1, 2) η 1 

3n 2 2 +1 

2 

(2, 2) 

(3n 2 1 +1) (3n 2 2 +1) 

2 2 

Table 4.1: Basis functions for a third-order approximation over the standard quadrilateral 

region. 

p, q b k p, q b k p, q b k 

1−η 

(0, 0) 1 (1, 0) η 2 

3 

1 (2, 0) (η 2 2 1 − 1) 2 + 3η 1 − 2( 1−η 2 

) 2 

2 

(0, 1) 

3η 2 +1 

1−η 

(1, 1) η 2 5η 2 +3 

5 

2 1 (0, 2) (η 2 2 2 2 − 1) 2 + 6η 2 − 3 

Table 4.2: Basis functions for a third-order approximation over the standard triangular 

region. 

(a) (b) (c) 

(d) (e) (f) 

Figure 4.5: Basis functions over the standard triangular region: (a) (p, q) = (0, 0), 

(b) (p, q) = (1, 0), (c) (p, q) = (2, 0), (d) (p, q) = (0, 1), (e) (p, q) = (1, 1), (f) 

(p, q) = (0, 2).


4.1.4 Elemental operations 

In order to handle arbitrary geometries, it is necessary to numerically evaluate all 

the integrals appearing in Eq. (4.6). These are the mass matrix M kj , the volume 

integral V k and the surface (line) integral S k : 

M kj = 

∫ 

b e kb e jdΩ e , 

Ω e 

(4.19) 

∫ 

V k = ∇b e k · F (Ũe)dΩ e , 

Ω e 

(4.20) 

S k = 

∮ 

b e kF (Ũe) · n e ∂l, 

∂Ω e 

(4.21) 

where in place of the test function v the elemental basis function b e j has been substituted. 

In the present work, Gaussian quadrature rules are used for numerical 

integration. 

For quadrilateral elements, the integrals given in Eqs. (4.19) and (4.20) are 

then computed by the weighted summations (η = η 1,i , η 2,m ): 

∫ 1 ∫ 1 

M kj = b k b j |J|dη 1 dη 2 = 

−1 −1 

Q 

∑ 1 Q 

∑ 2 

w i w m b k (η)b j (η)|J(η)|, 

i=1 m=1 

(4.22) 

∫ 1 ∫ 1 

V k = J −1 · ∇b k · F (Ũ e )|J|dη 1 dη 2 = 

−1 −1 

Q 

∑ 1 Q 

∑ 2 

w i w m J −1 (n) · ∇b k (η) · F (Ũ e (η)|J(η)|, 

i=1 m=1 

(4.23)


and for triangular elements, the same integrals are computed by the following 

weighted summations: 

∫ 1 

M kj = 

∫ −ξ2 

−1 −1 

∫ 1 ∫ 1 

−1 −1 

Q 

∑ 1 Q 

∑ 2 

i=1 m=1 

b k b j |J|dξ 1 dξ 2 = 

b k b j |J|( 1 − η 2 

)dη 1 dη 2 = 

2 

( 1 − η2,m 

w i w m b k (η)b j (η)|J(η)| 

2 

) 

, 

(4.24) 

∫ 1 

V k = 

∫ −ξ2 

−1 −1 

∫ 1 ∫ 1 

−1 −1 

Q 

∑ 1 Q 2 

i=1 m=1 

J −1 · ∇b k · F (Ũ e )|J|dξ 1 dξ 2 = 

J −1 · ∇b k · F (Ũ e )|J|( 1 − η 2 

)dη 1 dη 2 

2 

∑ 

( 1 − 

w i w m J −1 η2,m 

(n) · ∇b k (η) · F (Ũ e (η))|J(n)| 

2 

) 

, 

(4.25) 

where |J| is the Jacobian of the transformation, J −1 is the inverse of the Jacobian 

matrix of the transformation from the physical to the computational space, w i , w m 

are the quadrature weights and Q 1 and Q 2 is the number of quadrature points 

along the η 1 and η 2 direction respectively. Matrix M is called the mass matrix and 

possesses a diagonal structure only for straight sided triangles and for straight sided 

quadrilaterals with parallel edges. 

The line integral in Eq. (4.21) is computed by the following weighted summation, 

both for triangles and quadrilaterals:


S k = 

Q m 

∑ 

m=1 

∫ 1 

−1 

b k F (Ũe) · n e |l e |dn = 

w m b k H(Ũ+ e , Ũ− e ) · n c e|l e |, 

(4.26) 

where Q m is the number of quadrature points, |l e | is the edge length, H the numerical 

flux at the element boundary, Ũ+ e and Ũ− e are the element traces on an edge shared 

by two elements and n c e is the normalized outward vector of the local element edge 

at the computational space. 

Clearly, in our implementation all numerical computations of the DG method 

are carried out in the canonical computational domain of the square element. This 

choice gives a significant advantage for use of mixed-type element meshes and the 

novel implementation of the limiters for arbitrary elements that will be shown later. 

4.1.5 Selection of Gauss type numerical integration 

Gauss numerical integration is classified according to the way the end points of the 

domain of integration are handled [83]. The classification is given in Table 4.3, with 

the corresponding order of accuracy using Q n quadrature points. 

Type End points Order of accuracy 

Gauss-Legendre (GL) no end points 2Q n − 1 

Gauss-Radau (GR) only one end point 2Q n − 2 

Gauss-Lobatto-Legendre (GLL) both end points 2Q n − 3 

Table 4.3: Types of Gauss numerical integration.


The GL and GLL type have been implemented for both the surface and the 

line integrations in the present work. Moreover, in Eqs. (4.22), (4.24) and (4.26), 

it is seen that the product of two polynomials of order N is integrated. Thus, the 

same number of quadrature points is used for volume and surface integrations and 

equal to: 

Q 1 = Q 2 = Q m = N + 1, (4.27) 

for a GL integration, and: 

Q 1 = Q 2 = Q m = N + 2, (4.28) 

for a GLL integration. 

4.1.6 Mapping between physical and computational space 

The computation of the integrals in Eqs. (4.22) to (4.25) requires the evaluation of 

the Jacobian matrix of the transformation at every quadrature point. This necessitates 

the construction of a mapping from the physical to the computational space. 

Usually, according to the polynomial degree N B of the physical space representation, 

curved elements are classified in sub-, iso- and super-parametric elements, by 

comparing the order N B with that of the order N of the discretization scheme. In 

detail: 

N B < N : sub-parametric element 

N B = N : iso-parametric element 

N B > N : super-parametric element 

For straight sided quadrilateral elements, the mapping from the physical configuration 

of the element Ω e to the standard elemental region (see Fig. 4.2), is given 

by the following relations:


(1 − η 1 ) (1 − η 2 ) 

x =x A 

2 2 

(1 − η 1 ) (1 + η 2 ) 

+ x D 

2 2 

(1 + η 1 ) (1 − η 2 ) 

+ x B 

2 2 

+ x C 

(1 + η 1 ) 

2 

(1 + η 2 ) 

, 

2 

(4.29) 

(1 − η 1 ) (1 − η 2 ) 

y =y A 

2 2 

(1 − η 1 ) (1 + η 2 ) 

+ y D 

2 2 

(1 + η 1 ) (1 − η 2 ) 

+ y B 

2 2 

+ y C 

(1 + η 1 ) 

2 

(1 + η 2 ) 

, 

2 

(4.30) 

and for straight sided triangular elements (see Fig. 4.3) by: 

(1 − η 1 ) (1 − η 2 ) 

x = x A 

2 2 

(1 + η 1 ) (1 − η 2 ) 

+ x B 

2 2 

1 + η 2 

+ x C , (4.31) 

2 

(1 − η 1 ) (1 − η 2 ) 

y = y A 

2 2 

(1 + η 1 ) (1 − η 2 ) 

+ y B 

2 2 

The Jacobian matrix of the transformation is defined as: 

1 + η 2 

+ y C . (4.32) 

2 

⎡ 

⎢ 

J = ⎣ 

∂x 

∂η 1 

∂y 

∂η 2 

⎤ 

∂x 

∂η 2 

⎥ 

∂y 

∂η 2 

⎦ , (4.33) 

with its inverse equal to: 

J −1 = 1 

|J| 

⎡ 

⎢ 

⎣ 

∂y 

∂η 2 

− ∂y 

∂η 1 

− ∂x 

∂η 2 

∂x 

∂η 1 

⎤ 

⎥ 

⎦ . (4.34) 

In the present work, in order to reduce the amount of operations required for 

the evaluation of the surface integrals, the product of the elements of the inverse


Jacobian of the transformation with the gradient of the weighting functions and 

the quadrature weights at each quadrature point, is computed and stored in a preprocessing 

step. 

4.2 DG discretization for the Navier-Stokes Equations 

Following the same procedure for the derivation of the weak form for the system 

of the Euler equations, Eq. (4.6), in the DG discretization, the weak form for the 

system of Navier-Stokes equations is the following: 

∫ 

vb e ∂ĉ k ∫ 

∮ 

e 

k 

Ω e 

∂t dΩ e − ∇v · F (Ũe)dΩ e + vF (Ũe) · n e ∂l = 

Ω e ∂Ω 

∫ 

∮ 

e 

− ∇v · F v (Ũe, ∇Ũe)dΩ e + vF v (Ũe, ∇Ũe) · n e ∂l. 

Ω e ∂Ω e 

(4.35) 

From the previous system it is observed that the inclusion of the viscous and thermal 

conduction effects has lead to a system including the solution gradient. This 

complicates the solution of the Navier-Stokes system in the DG framework, since, 

the solution gradient has to be computed and appropriate numerical fluxes for the 

viscous part must be chosen to provide a stable and accurate discretization scheme. 

4.2.1 Computation of the viscous terms in the DG method 

The computation of the viscous terms in the DG method requires a special attention 

as has been demonstrated in [155]. Specifically, considering the one dimensional heat 

equation: 

u t − u xx = 0, (4.36)


with periodic boundary conditions and following the same procedure for the derivation 

of the weak form in the DG method as for the purely hyperbolic system of the 

Euler equations, yields: 

∫ 

E 

v du h 

dt 

∮∂E 

dE + v du h 

dx 

∫E 

de − dv 

dx 

du h 

dE = 0. (4.37) 

dx 

where u h denotes the solution expansion of u. It is obvious that someone now has 

to deal with the derivative du h 

in the computation of the integrals appearing in 

dx 

Eq. (4.37). Due to the absence of an upwind mechanism in the heat equation, the 

natural choice is to define the numerical flux for u x as follows: 

( 

û x = 1 (duh ) + 

+ 

2 dx 

( ) ) − duh 

. (4.38) 

dx 

If a P 0 expansion of the solution was chosen then this would lead to the analytical 

value of u h x to be equal to zero and hence the discretization would be innately 

inconsistent. Furthermore, in [155] it was demonstrated that this holds for orders 

of approximation greater than 0. 

Several schemes for the discretization of the viscous terms have been proposed 

in the literature, such as the schemes developed by Bassi and Rebay [17, 19, 20], 

the Local DG method (LDG) of Cockburn and Shu [43], which is a generalization 

of the ideas of Bassi and Rebay, and the method developed by Baumann and Oden 

(BO) [22, 23] and [107]. In the BO DG method, extra terms were added in the 

numerical flux that accounted for the jump in the solution between the elements, 

but for a P 0 solution expansion the scheme is inconsistent. Moreover, for a P N order 

solution expansion, the order of accuracy of the BO scheme is N for even N, that 

is sub-optimal, and N + 1 for odd N. Furthermore, in [155] it was demonstrated 

that the BO method has larger errors than the LDG method. For those reasons, 

in this work the LDG method is used for discretizing the system of Navier-Stokes 

equations in a consistent, stable and accurate way.


4.2.2 The LDG method 

The basic idea of the LDG method is to perform an order reduction of the initial 

system into a degenerate first order system and then discretize it within the DG 

framework to compute the solution gradient. For doing so, the following auxiliary 

variable Θ e at each element is introduced: 

⎡ ⎤ 

∇ρ e 

Θ e = ∇Ũe = 

∇u e 

⎢ 

⎣∇v ⎥ 

e ⎦ , (4.39) 

∇T e 

which in essence is the solution gradient at each element. Multiplying the above 

equation by the same weighting function v appearing in Eq. (4.35) and integrating 

over the elemental space Ω e , obtain: 

∫ 

∫ 

vΘ e dΩ e − 

Ω e 

Ω e 

v 

⎡ ⎤ 

∇ρ e 

∇u e 

⎢ 

⎣∇v ⎥ 

e ⎦ dΩ e = 0, 

∇T e 

which after application of Gauss’ theorem, leads to: 

⎡ ⎤ 

ρ e 

∫ 

∫ 

∮ 

vΘ e dΩ e + ∇v 

u e 

⎢ 

Ω e Ω e ⎣v ⎥ 

e ⎦ dΩ e − 

T e 

∂Ω e 

v 

⎡ ⎤ 

ρ e 

u e 

⎢ 

⎣v ⎥ 

e ⎦ n e∂l = 0. (4.40) 

T e 

The evaluation of the line integral in Eq. (4.40) along with the numerical viscous 

flux computation has a pivotal role in the accuracy and stability of the resulting DG 

scheme. 

Arnold et. al in [6] performed an analysis of the resulting schemes depending 

on the choice of the line integral evaluation. Furthermore, in [6] it was proven


that two main requirements have to be met by the numerical fluxes in order to 

obtain a stable scheme, which furthermore achieves an optimal rate of convergence. 

These requirements are that the fluxes have to be consistent and conservative. The 

examined formulations are given in Table 4.4, where the following notation of jump 

[] and averaging {} operators for scalar and vector fields is used. 

No. Method û ˆΘ 

1 Bassi-Rebay (BR1) {u h } {Θ} 

2 Brezzi et. al 1 {u h } {Θ} − a r [u h ] 

3 LDG {u h } − β · [u h ] {Θ} + β[Θ] − a j [u h ] 

4 IP {u h } {∇ h u h } − a j [u h ] 

5 Bassi et al. (BR2) {u h } {∇ h u h } − a r [u h ] 

6 Baumann-Oden {u h } + n κ · [u h ] {∇ h u h } 

7 NIPG {u h } + n κ · [u h ] {∇ h u h } − a j [u h ] 

8 Babuska-Zlamal (u h | K )∂K −a j [u h ] 

9 Brezzi et. al 2 (u h | K )∂K −a j [u h ] 

Table 4.4: DG methods and their interiors fluxes taken for Arnold et. al [6]. 

ˆ for a scalar variable s: 

[s] = s + n + + s − n − = n(s + − s − ), 

{s} = 1 2 

( 

s + + s −) ,


ˆ for a vector variable g: 

[g] = g + · n + + g − · n − = n(g + − g − ), 

{g} = 1 2 

( 

g + + g −) . 

The functional operators a r and a j in Table 4.4 are penalty terms which their appearance 

in the gradient variable Θ is of significant importance for obtaining a stable 

scheme. The methods which do not contain any of these terms are only weakly stable 

and it is obvious that the current chosen LDG method may be or may not be 

weakly stable. All the methods with penalty terms may be interpreted as interior 

penalty (IP) methods. The penalty method generally imposes a penalty term in 

the formulation in order to enforce the solution and must automatically satisfy the 

boundary conditions. This is achieved by the presence of the penalty term which 

adds more artificial dissipation to the jump at the boundary so that to smear out 

the discontinuity that appears there. 

In the present work, the utilization of the LDG method permits, as it easily 

verified from Table 4.4, the application of the BR1 scheme which in the literature 

is known as the first Bassi-Rebay scheme. However, Cockburn and Shu in [43] 

demonstrated some deficiencies of the BR1 scheme: 

ˆ Convergence order of N only for odd weighting function. 

ˆ Spread stencil. 

Both problems may be remedied by using an upwind like fashion on the choice of 

fluxes in the gradient computation and in the viscous flux evaluation. In the present 

work this is achieved by the following use of the parameters β = ±0.5 and a j = 0.


Recently in the literature appeared the so called hybridizable DG (HDG) [38, 

39, 104] methods, which aim at alleviating the high computational cost of the DG 

discretization for the viscous terms. In HDG methods the final system is produced 

in terms of the DOF of the approximate traces of the field variables, that is the 

solution approximation at the inter element boundaries. Since the approximate 

traces are defined at the edges/faces only and are single valued, the HDG methods 

include significantly fewer coupled unknowns than other DG methods for higher 

order terms. 

The HDG methods are devised using a hybridization procedure of the DG 

methodology to discretize a system of equations. This procedure starts by introducing 

first the approximate traces and then define a total numerical flux in terms 

of them. A continuity requirement is weakly imposed for the normal component of 

the numerical flux and this leads to a large nonlinear system of equations for the 

approximate variables (including their spatial gradient) and the numerical fluxes. 

After linearization of the final system the local variables and their spatial gradient 

may be condensed in an element by element fashion and obtain a reduced global 

system involving only the approximate traces as the unknowns. 

4.3 The treatment of initial and boundary conditions 

According to the flow conditions that have to be met at specific regions of the 

computational domain, appropriate boundary conditions (BC) have to be applied 

to the system of the governing equations. In the present work, the implementation 

of Neumann and Dirichlet BC is performed in a weak manner. This follows from the 

flux formulation of the present DG discretization leading to a natural weak inclusion 

of the BC through the numerical fluxes. It is noted that the weak enforcement of 

Dirichlet BC leads to solutions that still ”feel” the influence of the boundary through 

the numerical fluxes, but in a manner that is consistent with the accuracy of the 

interior solution, thus, leading to improved solutions away from the boundaries [9,


45]. Also, the application of Neumann BC comes naturally in the DG discretization 

again through the flux formulation of the scheme. 

The technique for weak enforcement of BC requires the use of ghost elements, 

where a solution state U g is determined such that the appropriate boundary values 

at every quadrature point over a boundary edge are set. Then, the underlying 

Riemann problem is solved according to the main discretization scheme. 

It is noted, however, that a polyline representation of the wall geometry is 

not appropriate for the DG discretization [16, 18, 92, 93], as the DG method is 

highly sensitive to the accuracy of the boundary representation. This necessitates 

either use of curved elements or refinement of the mesh at the regions where the 

wall curvature is high. In the present work, the last approach is followed as only 

straight sided elements are used. 

The application of the initial conditions (IC) in the DG method may be performed 

in two ways that will be explained in the current chapter. These are the 

collocation and the discrete Galerkin projection. Both may be applied to solution 

expansions of nodal or modal hierarchical bases, but the first is more suitable for 

solution expansions with nodal bases which approximate the solution at specific 

points in the computational element, while the second is more suitable for modal 

hierarchical bases. 

4.4 Wall boundary conditions 

For a flow simulation over an arbitrary geometry, the BCs that have to be satisfied 

at the boundary segments defining the geometry depend on the physical modeling 

being performed for the flow field. Specifically, for inviscid flow numerical solutions 

it is required the flow to be tangent at the geometry, leading to the employment of 

the so called slip-wall BC, which is a Dirichlet type condition and simply expresses 

that the flow cannot penetrate through the segments defining the geometry. For the 

DG discretization this means that at the quadrature points on the edges defining the 

wall geometry the velocity field must be enforced to be tangent. This is expressed


by the following Dirichlet BC: 

v · n = 0, (4.41) 

The weak enforcement of the BC condition of tangency in the velocity field is implemented 

at every quadrature point at the wall edges by the solution of a Riemann 

problem. Specifically, a ghost state is created at the opposite side of the computational 

domain at a wall edge, where the density, pressure and the magnitude of the 

velocity field are extrapolated from the interior solution. This corresponds to the 

following Neumann BC for the pressure at the wall: 

∂p 

∂n = 0. 

Then the velocity components at the ghost state are determined so that the averaged 

velocity field at the quadrature points to be tangent to the edge. Analytically, the 

ghost state is the following: 

ρ g = ρ, 

u g = u t cos (a) + u n sin (a) , 

v g = −u n cos (a) + u t sin (a) , 

(4.42) 

P g = P, 

where u t , u n are the velocity components tangent and normal to the edge and a is 

the angle formed by the wall segment with the x coordinate axis. 

For viscous flow computations, the flow must adhere to the geometry resulting 

to the application of another Dirichlet condition, namely the no-slip wall condition. 

The weak enforcement of the no-slip wall condition is performed by constructing the


following ghost state: 

ρ g = ρ, 

u g = −u t cos (a) + u n sin (a) , 

v g = −u n cos (a) − u t sin (a) , 

(4.43) 

P g = P, 

and the underlying Riemann problem is solved. Furthermore, for viscous computations 

additional considerations have to be taken regarding the heat flux in the 

flow field at the boundaries of the computational domain. This coordinates the use 

either adiabatic, isothermal etc. conditions that are enforced to the Navier-Stokes 

system through Neumann or Dirichlet BC. For instance, an adiabatic wall requires 

that there is no heat flux normal to the wall and this is expressed as follows: 

dT 

dn 

= 0, (4.44) 

and implemented by requiring at the quadrature points on the wall edges the temperature 

gradient normal to the wall to be zero. For isothermal walls or for flows 

with specific temperature distribution on the wall boundary, the temperature field 

must be set at every quadrature point to the prescribed values. 

4.5 Far field boundary conditions 

At the regions of the computational domain where the inlet and outlet of the flow 

is discretized, BC depending on the local characteristic structure of the flow field 

must be applied [64, 113, 140]. In the current DG discretization, their application, 

also follows that of the weak enforcement, and they are based on the assumptions 

of a locally one dimensional isentropic flow. Also, the application of the far field BC


is performed on the primitive variables [ρ, u, v, p] T . 

Considering the system of Euler equations and their eigenvalues given in Eq. 

(2.25), it is observed that for a supersonic flow all the characteristic waves are 

traveling in one direction. This dictates the conditions at the inlet to be imposed 

by the conditions of the free stream flow and at the outlet all the conditions to be 

extrapolated from the interior of the domain. However, for the case of subsonic flow, 

the inlet and outlet regions are crossed by waves coming both from the interior and 

the exterior of the domain. This situation necessitates modeling of the waves crossing 

the far field boundaries from the exterior, in order to determine the appropriate BC 

at subsonic inlets and outlets. The Riemann invariants [64] are used for determining 

the far field BC according to the previously made assumptions: 

J + = u + + 2 

γ − 1 c+ , 

J − = u − − 2 

γ − 1 c− , 

(4.45) 

where the + and - superscript refer to the appropriate side of the inlet or outlet 

region. The primitive, and thus the conservative variables, are then calculated according 

to the following algorithm: 

1: u b = J + +J − 

2 

2: c b = g−1 

4 

(J + − J − ) 

3: p b = ρc2 

γ 

4: S b = p b 

ρ γ 

( ) 1 

c 3 γ 

5: ρ b = b 

γS


4.6 Application of Initial Conditions 

The use of a modal bases set requires determination of the initial values of the 

DOF according to the IC over the computational domain. Two methods may be 

used for the initialization of the DOF: collocation projection and discrete Galerkin 

projection. 

Considering a two dimensional function Ũ(x, t) which does not lie within the 

polynomial space of the expansion basis, then there will be an error R e (Ũ) in the 

approximation of the function over an arbitrary element where a solution expansion 

Ũ e (x, t) is employed: 

R e (Ũ) = Ũ(x, t) − Ũe(x, t). (4.46) 

Following the method of weighted residuals the Legendre inner product of Eq. 

(4.46) with an arbitrary function v is formed: 

∫ 

Ω 

∫ 

(Ũ(x, t) − Ũ e (x, t)) 

vdΩ = R e (U)vdΩ, (4.47) 

Ω 

and setting the right hand side equal to zero, the following system is obtained for 

the determination of the expansion coefficients: 

∫ 

Ω 

If the function v in Eq. 

∫ 

Ũ e (x, t)vdΩ = 

Ω 

Ũ(x, t)vdΩ. (4.48) 

(4.47) is set equal to the Dirac delta function at 

the quadrature points, it is then implied that ∫ R(u)dΩ = 0 and the expansion 

Ω 

coefficients are obtained by: 

Û k e = B −1 · U k e, (4.49) 

where the matrix B i,j = b i (η j ). This approach corresponds to the collocation projection. 

If the function v is set equal to the basis function b i then the discrete


Galerkin projection is obtained with B i,j = ∫ b Ω i(η)b j (η)dΩ. In the present work 

the Galerkin projection has been used for applying the IC due to the use of a modal 

bases set.

Chapter 5 

Time discretization 

For time advancement of the compressible flow equations either the method of lines 

or the space-time DG method could be used. For space-time DG discretizations 

the solution is not only discretized in space but in time as well. This leads to 

a discretization scheme where the DOFs are scalar but the basis functions of the 

solution expansion depend both in space and time. For more details in the spacetime 

DG discretization refer to [86, 143]. 

For the method of lines, the space discretization is performed first and the 

time is considered continuous. Then, a set of ODEs of size equal to the number of 

DOFs in Eq. (4.6) for the Euler equations and in Eqs. (4.35) and (4.40) for the 

Navier-Stokes equations in the current implementation is obtained, which can be 

advanced in time either with explicit or implicit methods for ODEs. 

Use of explicit time integration schemes has been widespread in DG discretizations. 

However, explicit schemes suffer from severe stability limitations. The allowable 

time step due to CFL-stability limits is very small and few cases may be solved 

at a reasonable computing time. On the other hand, the application of implicit time 

integration does not have limitations imposed by CFL-stability limits, at least for 

linear conservation laws, and thus, can allow larger time steps, which are often determined 

only by accuracy considerations. Moreover, implicit methods are suitable 

for stiff problems. 

117

CHAPTER 5. TIME DISCRETIZATION 118 

5.1 Runge-Kutta methods 

Runge-Kutta (RK) methods belong to a class of multistage single step methods. 

They apply numerical integration rules over a specified set of stages, where progressively 

more accurate approximation of the solution’s slope is performed. They are 

classified as explicit and implicit, with the solution at every stage being well defined 

for small enough step size. Although, their implementation is expensive in terms 

of storage and processing, they are widely applied in computational mechanics, due 

to their exceptional properties in terms of stability accompanied with high-order 

accuracy. 

The generic form of an initial value problem for a scalar equation is the following: 

dy 

=f (t, y (t)) , t ∈ [a, b] , 

dt 

y (a) = y 0 , 

(5.1) 

and the numerical solution sought is denoted as y : [a, b] → R. Next, defining the 

following constants: q ∈ N, τ i ∈ R, a ij ∈ R, b i ∈ R with i, j = 1, . . . , q, and usually 

0 ≤ τ i ≤ 1 for and arbitrary function ψ (s) : R → R the following integrals: 

∫ ti 

0 

ψ (s) ds, 

∫ 1 

0 

ψ (s) ds, 

are approximated by the following summations, respectively: 

q∑ 

a ij ψ (s) , 

j=1 

q∑ 

b i ψ (τ i ) , i = 1, . . . , q. 

j=1 

The constants a ij , τ i and b i define q + 1 quadrature rules, where τ i are the 

quadrature points and b i and a ij are the quadrature weights for the approximations 

of the integrals of ψ over the interval [0, 1] and [a, b], respectively. Each set of those 

constants define a RK method and it is customary to depict them in a tabular form 

known as Butcher’s table (Table 5.1). It is noted that the index i refers generally


to the stages of the RK method. 

a 11 a 12 . . . a 1q τ 1 

a 21 a 22 . . . a 2q τ 2 

. . . . 

a q1 a q2 . . . a qq τ q 

b 1 b 2 . . . b q 

Table 5.1: Butcher’s table 

Discretizing the interval [a, b] into N equally spaced elements of size h = b−a 

N , 

the nodal coordinates are equal to t n = a + nh, n = 0, . . . , N. Let y n be the solution 

given by a RK method as t n . Furthermore, as it has been mentioned that a RK 

method during each stage computes progressively a better approximation of the 

solution’s slope, the following constant is defined: 

t n,i = t n + τ 1 h, i = 1, . . . , q, (5.2) 

which simply expresses the parameterization of the sub interval [t n , t n+1 ] to the RK 

stages. 

Integrating Eq. (5.1) over [t n , t n+1 ], follows: 

y ( t n,i) − y (t n ) = 

∫ t n +τ i h 

which after some algebra may be written as: 

y ( t n,i) = y (t n ) + 

∫ τi 

0 

t n 

f (t, y (t)) dt, 

f (t + sh, y (t n + sh)) ds. (5.3) 

Applying the quadrature formulae with constants a ij and τ j , the approximation 

y n,i of y (t n,i ) results to be equal to: 

q∑ 

y n,i = y n + h a ij f ( t n,j , y n,j) , i = 1, . . . , q. (5.4) 

j=1


Integrating now Eq. (5.1) from t n to t n+1 , by applying the variable transformation 

t = t n + hs, and approximating the final integral over the interval [0, 1] 

with the quadrature rule defined by the points τ i and the weights b i , the following 

approximation y n+1 of y (t n+1 ) follows: 

q∑ 

y n+1 = y n + h b i f ( t n,i , y n,i) . (5.5) 

i=1 

As a result, the RK method described by Butcher’s table produces a sequence 

of approximations given by the following relations: 

y 0 = y n = y 0 , 

y n,i = y n + h ∑ qa ij f ( t n,j , y n,j) , 1 ≤ i ≤ q, 

j=1 

q∑ 

y n+1 = y n + h b i f ( t n,i , y n,i) . 

i=1 

(5.6) 

In the present implementation of the DG discretization of the system of the 

compressible flow equations, explicit time advancement of the solution is performed 

with high-order strong stability preserving (SSP) Runge-Kutta methods, that preserve 

the monotonicity of the spatial discretization in any norm or semi-norm coupled 

with first-order forward Euler time stepping [85]. That is, if the monotonicity 

requirement: 

||u n+1 || ≤ ||u n ||, (5.7) 

is met for a step size of ∆t F E with a first-order forward Euler time stepping, then a 

high-order Runge-Kutta method is SSP if there exists a number a ∈ (0, 1] for which 

the step size ∆t under ∆t ≤ a∆t F E satisfies the monotonicity requirement. 

Extensive use of the explicit third order accurate SSP Rung-Kutta method


proposed by Shu and Osher [130] is made: 

u (1) = u n + ∆tR(u n ), 

u (2) = 3 4 un + 1 4 u(1) + 1 4 ∆tR(u(1) ), 

(5.8) 

u n+1 = 1 3 un + 2 3 u(2) + 2 3 ∆tR(u(2) ), 

where R denotes the residual of the system after the DG discretization. Furthermore, 

SSP explicit Runge-Kutta methods provided by the PETSc library package are 

employed [58, 59]. 

5.2 Implicit time marching 

By grouping together the time dependent and spatial contributions resulting from 

the DG discretization of either the Euler or the Navier-Stokes equations given in 

Eqs. (4.6) and (4.35) respectively, the following system of ODEs results: 

M · dU dt 

+ R(U) = 0, (5.9) 

where M is the mass matrix given by Eqs. (4.22) and (4.24) and R is the residual 

including all the spatially discretized terms. 

Considering for simplicity the backward implicit Euler time discretization of 

Eq. (5.9), obtain: 

M · U n+1 − U n 

δt 

+ R(U n+1 ) = 0. (5.10) 

This is a system of nonlinear equations of the form: 

F (U) = 0, (5.11)


and must be solved numerically using a Newton like method that is described next. 

5.3 Newton’s method 

The Newton iteration for Eq. (5.11) is derived from a multidimensional Taylor series 

expansion about a point U k : 

F (U k+1 ) = F (U k ) + F ′ (U k ) · (U k+1 − U k ) + higher order terms. (5.12) 

Neglecting higher order and setting the right hand side equal to zero, yields Newton’s 

method, which basically is an iterative procedure over a sequence of linear systems: 

J (U k ) · ∆U k = −F (U k ) , U k+1 = U k + ∆U k , k = 0, 1, . . . , (5.13) 

given an initial guess for the solution U 0 , J ≡ F ′ 

= ∂F 

∂u 

being the approximate 

Jacobian of the system, U is the state vector to be found and k is the nonlinear 

iteration index. The Newton iteration is terminated based on the drop in the norm 

of the nonlinear function F under a specified tolerance or a sufficiently very small 

Newton correction ∆U. 

The sequence of linear systems is almost always solved 

using an iterative method belonging to the class of Krylov subspace methods. 

Forming each element of the Jacobian matrix requires the analytic or discrete 

derivatives of the system with respect to the state vector U. However, the analytic 

determination of the derivatives in order to form the Jacobian matrix is error-prone 

and even impossible, as the derivatives of an nondifferentiable function may need to 

be determined. Therefore in this work the Jacobian free method [36, 88] employing 

discrete evaluation of the Jacobian is used.


5.4 Krylov subspace methods 

Krylov subspace methods are approaches for solving large linear systems and first 

appeared in the 1950s and became very popular afterward since Reid [123] introduced 

them as iterative methods for the solution of large linear systems. They can 

be considered as projection or generalized projection methods for solving a linear 

system A · x = b using the Krylov subspace K j defined as: 

K j = span ( r 0 , A · r 0 , A 2 · r 0 , A 3 · r 0 , . . . , A j−1 · r 0 

) 

, 

where r 0 = b − A · x 0 is the system’s residual. These methods require only matrixvector 

products during the iteration and this is the key element that favors use of 

Newton’s method in a Jacobian free context as it will be shown. 

A wide spectrum of iterative methods belongs to the class of Krylov methods 

[13, 84, 125]. Two primary points of classification appear regarding the solution of 

symmetric and non-symmetric systems and the procedure followed for the construction 

of the bases of the Krylov subspace. The bases may be constructed using the 

long-recurrence Arnoldi orthogonalization procedure, which generates orthonormal 

bases of the Krylov subspace. Specifically, the Arnoldi iteration [7] uses the stabilized 

Gram-Schmidt process to produce a sequence of orthonormal vectors q 1 , q 2 , q 3 , . . ., 

known as the Arnoldi vectors such that the vectors q 1 , q 2 , q 3 , . . . , q j span the Krylov 

subspace K j . The algorithm for producing the Arnoldi vectors is the following: 

1: Start with an arbitrary vector q 1 with norm 1. 

2: Repeat for k = 2, 3, . . . 

3: q k ← Aq k−1 

4: for j to k − 1 do 

5: h j,k−1 ← q ∗ j q k 

6: q k ← −h j,k−1 q j 

7: end for 

8: h k,k−1 ← ||q k ||


9: q k ← 1 

h k,k−1 

q k 

The widely used Generalized Minimal RESidual method (GMRES) [126] is an 

Arnoldi-based method. In GMRES, the Arnoldi basis vectors form the trial subspace 

out of which the solution is constructed. One matrix-vector product per iteration 

is required to create a new trial vector, and the iterations are terminated based on 

a by-product estimate of the residual that does not require explicit computation of 

the constructed solution, which is a beneficial feature of the algorithm. The residual 

minimization property of the method is in the Euclidean norm, but requires the 

storage of all previous Arnoldi basis vectors, leading unfortunately to high requirements 

in storage. This necessitates the use of efficient preconditioning of the linear 

system. 

5.5 Jacobian-free Newton-Krylov Method 

The Jacobian-free Newton-Krylov (JFNK) method is a synergistic combination of 

the Newton method for solving a nonlinear system of equations and Krylov subspace 

methods for the solution of the linear system leading to the Newton correction. 

The specific advantage of the JFNK method is that the Jacobian of the system 

is not formed and stored, but only the Jacobian-vector product during the Krylov 

iteration is used, which is approximated numerically. This means that the gains 

both in processing time and memory requirements are extremely large. However, 

the application of the Krylov iteration necessitates adequate preconditiong of the 

linear system to be solved for the successful application of the JFNK method. 

The origins of the JFNK method can be found in efforts motivated by the 

solution of ODEs and PDEs [30, 31, 33, 56]. The primary goal in all cases has been 

the ability to perform a Newton iteration without the need to form the Jacobian. 

This has helped the application of higher-order implicit integration methods for 

ODEs as it is performed in the present work. 

In the JFNK approach a Krylov method is used to solve the linear system 

resulting from the linearization of the initial nonlinear system. An initial residual,


r 0 is defined, given an initial guess for, ∆u 0 , for the Newton correction: 

r 0 = −F (u) − J · ∆u 0 (5.14) 

Using the GMRES method for the solution of the linear system for determining 

the Newton correction and denoting by j the GMRES iteration index, the iteration 

for the solution of the linear system minimizes ‖J · ∆U j + F (U) ‖ 2 within a subspace 

of small dimensions compared to the total number of unknowns. The Newton 

correction ∆U j is drawn from the Krylov subspace spanned by the following vectors: 

{ 

r0 , J · r 0 , J 2 · r 0 , . . . , J j−1 · r 0 

} 

, 

and can be written as: 

j−1 

∑ 

∆U j = ∆U 0 + β i J i · r 0 , (5.15) 

where the scalars β i minimize the residual. Initially the Newton correction ∆U 0 is 

set to zero, which asymptotically is a reasonable guess as the approach corrections 

should converge to zero at late Newton iterations. Moreover, in practice ∆U j is 

determined as a linear combination of the orthonormal Arnoldi vectors produced by 

GMRES. 

Close examination of Eq. (5.15) reveals that during the GMRES iterations the 

action of the Jacobian is taking place in the form of matrix-vector products, which 

may be approximated as follows [31, 33]: 

i=0 

J · v ≈ 

F (U + ɛv) − F (U) 

, (5.16) 

ɛ 

where ɛ is a small perturbation. Eq. (5.16) is simply a first-order Taylor series 

expansion approximation to the Jacobian times a vector. For instance, considering 

a nonlinear system of two equations with two variables, the Jacobian of the system 

is the following:


J = 

⎡ 

⎢ 

⎣ 

∂F 1 

∂u 1 

∂F 1 

∂u 2 

∂F 2 

∂u 1 

∂F 2 

∂u 2 

JFNK does not require the formation of this matrix, instead a vector is formed 

that approximates the multiplication of this matrix by a vector. Specifically, approximating 

F (U + ɛv) − F (U) with a first-order Taylor series expansion about U, 

gives: 

⎤ 

⎥ 

⎦ . 

which simplifies to: 

F (U + ɛv) − F (U) 

ɛ 

≈ 

⎡ 

⎢ 

⎣ 

F 1 (u 1 ,u 2 )+ɛv 1 

∂F 1 

∂u 1 

+ɛv 2 

∂F 1 

∂u 2 

−F 1 (u 1 ,u 2 ) 

ɛ 

F 2 (u 1 ,u 2 )+ɛv 1 

∂F 2 

∂u 1 

+ɛv 2 

∂F 2 

∂u 2 

−F 2 (u 1 ,u 2 ) 

ɛ 

⎤ 

⎥ 

⎦ , 

⎡ 

⎢ 

⎣ 

v 1 

∂F 1 

∂u 1 

+ v 2 

∂F 1 

∂u 2 

v 1 

∂F 2 

∂u 1 

+ v 2 

∂F 2 

∂u 2 

⎤ 

⎥ 

⎦ = J · v. 

It is apparent that the error in the approximation is proportional to ɛ and its evaluation 

follows the approach proposed in [126] and [74]. 

5.6 Preconditioning of the JFNK method 

Preconditioning in the JFNK method [36, 46, 121, 146] has the purpose of reducing 

the number of GMRES iterations by efficiently clustering close to unity the eigenvalues 

of the iteration matrix during the solution of the linear system. The JFNK 

approach has as its main goal the avoidance of forming the Jacobian matrix of the 

system and an effective preconditioner for the linear solution part of the method 

is necessary. In the present work the preconditioner used is formed by numerically 

computing the elements of the Jacobian matrix corresponding only to perturbations 

of the elemental DOFs, using coloring techniques provided by PETSc [58, 59] and


applying an ILU(p) (for p = 0, . . . , 4) factorization procedure for computing the preconditiong 

matrix. However, it the experience gained in this work it was observed 

that this procedure did not work efficiently especially for large time step sizes. 

Unlike the matrix-based GMRES algorithm the Jacobian-free approach enjoys 

more flexibility as it allows to approximate the preconditioner matrix without affecting 

the quadratic convergence of the Newton algorithm. Preconditioning deals 

only with the solution of the linear system at every Newton iteration and in the 

present work advantage of this feature was taken by freezing the evaluation of the 

preconditioner for a number of Newton iterations (Jacobian lagging). Significant 

efficiency improvements have been reached by the lagging procedure because the 

matrix evaluation, for a DG code, is a rather computational expensive task [46].

Chapter 6 

Unified limiting 

The construction of the basis functions and the implementation of the DG method 

in the computational domain for the standard square element, allows straightforward 

numerical treatment of mixed-type meshes in a unified way. The expansion 

coefficients are defined in the computational space and are unique for every element. 

A Taylor series expansion [91] reveals that in the absence of discontinuities the solution 

coefficients are estimates of the solution derivatives. Thus, limiting of the 

solution coefficients amounts to limiting the solution derivatives. 

A significant innovation of this work is that a unified procedure was introduced 

that allows to preserve monotonicity of the solution in a TVB in the mean (TVBM) 

sense. A slope limiting procedure, which is transparent to the element type was 

developed and explained in this chapter. Specifically, the limiting operations are 

performed for every element over the standard square element configuration, where 

it is imposed that the slope of the solution at the element edges does not exceed the 

variation of the mean solution across them. 

Every element is checked for limiting, separately in each direction, by using 

essentially the same TVB limiter proposed by Cockburn and Shu [41, 129]: 

m(a 1 , a 2 , . . . , a n ) = 

{ 

a 1 

m(a 1 , a 2 , . . . , a n ) 

if |a 1 | ≤ ML 2 x,y, 

otherwise, 

(6.1) 

128

CHAPTER 6. UNIFIED LIMITING 129 

The significant modification of the present work is, however, that the parameter 

M ≥ 0 for each field is not set to a fixed value, but it is estimated as is shown 

bellow. Furthermore, L x,y in Eq. (6.1), is not the rectangular element length (∆x) 2 

in the physical space as in the original limiter, but represents the characteristic local 

lengths of the element in the physical space. The function m(a 1 , a 2 , · · · , a n ), is the 

usual minmod function: 

⎧ 

⎨ 

m(a 1 , a 2 , . . . , a n ) = 

⎩ 

s min 

1≤j≤n |a j| if sgn(a 1 ) = sgn(a 2 ) = . . . = sgn(a n ) = s, 

0 otherwise. 

(6.2) 

According to the analysis of Cockburn and Shu [40, 129] for scalar conservation 

law problems the parameter, M, is an estimate of the second derivative of the 

solution. It is furthermore required that for systems of conservation laws to perform 

limiting of the solution coefficients (solution moments), in the characteristic space. 

Evaluation of the parameter M is performed for each characteristic field in 

the regular computational domain. Limiting for each characteristic variable is performed 

separately along each direction. The directional Jacobian, F(u) · n on the 

element edges, the corresponding eigenvalues, and eigenvectors are evaluated at the 

Roe’s averaged state from the mean interior u − and the exterior u + states in order 

to transform the conservative space state variables to the characteristic space 

variables. An element is flagged for limiting, if the limiter in Eq. (6.1) returns an argument 

other than the first, essentially when the slope is smaller than an appropriate 

measure of the second derivative value. 

For p-adaptive computations of arbitrary order of approximation P p , p ≥ 1, 

only the P 1 expansion of the solution is used for elements flagged for limiting in order 

to preserve monotonicity. Then, if the solution at an element has been limited, the 

approximation order at that element is set to P = 1. 

For elements neighboring 

limited elements, a gradually higher order of approximation could be used until 

progressively the order of approximation increases to the preset order of accuracy 

P p . 

It is emphasized that for a multistage time marching method and for calcu-


lations with a preset order of approximation P p , if an element, e, with order of 

approximation p e > 1 is flagged for limiting during a stage, then the solution coefficients 

and the residual of the discretization corresponding to the higher order part 

of the approximation are set to zero. The rationale behind this is to keep the order 

of the approximation strictly equal to P 1 for all limited elements. 

For the effective use of the TVB limiter, it is crucial to have appropriate 

estimates of the solution’s second derivative for each field in the characteristic space, 

e.g. the parameters M wj , where w i , j = 1, 2, 3, 4 denotes the characteristic fields. 

The estimates for the second derivative are given bellow. Furthermore, because 

limiting is performed in the computational domain, an estimate of the local mesh 

size, L x,y , required. In this work L x,y was estimated as suggested by Eq. (3.23). 

The value of the parameter M (the estimate of second derivative of the solution 

in Eq. (6.1)) is evaluated at every element k, and for each field in the characteristic 

space, using central differences of the mean solution. Considering the mixed-type 

mesh patch given in Fig. 6.1, the parameter M is computed as follows: 

Figure 6.1: Mixed-element mesh patch for estimating parameter M.


∑ 

M = 

(u e − u k ) 

∣ 

∣ , (6.3) 

e 

where index e counts the neighboring elements of k and, which actually is an approximation 

of the sum of the second derivatives of the solution. Specifically, performing 

a Taylor series expansion [79] of the sum given in Eq. (6.3) relative to elements k, 

results in: 

M = e x u x + e y u y + e xy u xy + e xx u xx + e yy u yy + . . . , (6.4) 

where, e x , e y , e xy , . . . , etc are coefficients induced by the local mesh geometry. For a 

single type mesh e x = e y = e xy = 0, except for at mesh interfaces. It is observed, 

that: 

M ≈ u xx + u yy . 

In practical computations however, in order to account for the mesh distortion, 

truncation error and floating points arithmetic, the following slightly modified value 

for the parameter M is used: 

ˆM = M + 

b 

L 2 x,y 

, (6.5) 

where b is a constant. All results were obtained using the same value b = 0.1. 

Therefore in essence in Eq. (6.1), the first condition is replaced by |a 1 | ≤ ML 2 x,y+0.1, 

where M is the estimate of Eq. (6.3), and L x,y the element size estimate. 

6.0.1 Limiting for quadrilateral elements 

For a P 1 approximation over a quadrilateral element, the solution expansion is: 

u q h = cq 0b q 0 + c q 1b q 1 + c q 2b q 2 + c q 3b q 3. (6.6) 

For the standard square element, the bases functions b q j , j = 1, . . . , 4 in Eq. (6.6) 

relative to the local Cartesian system (η 1 , η 2 ) of the canonical computational domain


are: 

b q 0 = 1, 

b q 1 = η 1 , 

b q 2 = η 2 , 

b q 3 = η 1 η 2 . 

Modification of the c q j coefficients in Eq. (6.6) is carried out by the TVB limiter as 

shown below. It is observed that the base b q 3 is not linear (see Fig. 4.4). Therefore, 

the solution coefficient c q 3 could be set to zero if a quadrilateral element is flagged for 

limiting. Otherwise, the coefficient c q 3 could be modified with the same procedure 

used for the modification of the coefficients c q 1 c q 2 considering additional quadrature 

points. In this work the c q 3 was set to zero. The solution coefficient c q 1 corresponds to 

the u x solution derivative therefore the coefficient c q 1 is limited along the η 1 direction. 

For doing so, the P 1 part of the solution at the midpoints of the edges BC and AD 

(see Fig. 4.2) in Ω st is used. The midpoints of the edges are chosen for limiting the 

solution coefficients, for the following reasons: 

ˆ Any ambiguity and closure problems are completely avoided. 

ˆ It is computationally efficient. 

According to Eq. 

solution is equal to: 

(6.6), at the midpoints of edges BC and AD in Ω st the 

U BC = u q (1, 0) − c q 0 = c q 1, 

(6.7a) 

U AD = u q (−1, 0) − c q 0 = −c q 1, 

(6.7b) 

and the value for the coefficient c q 1 is expressed in terms of the midpoint values as: 

c q 1 = U BC − U AD 

. (6.8) 

2


Limiting of the solution along the η 1 direction is performed by limiting the midpoint 

values U BC and U AD , using Eq. (6.1), as follows: 

Ũ BC = m(U BC , c q 0,(i+1,j) − cq 0,(i,j) , cq 0,(i,j) − cq 0,(i−1,j) ), 

(6.9a) 

Ũ AD = −m(−U AD , c q 0,(i+1,j) − cq 0,(i,j) , cq 0,(i,j) − cq 0,(i−1,j) ), 

(6.9b) 

where c q 0,(i,j) 

denotes the mean solution for the element (i, j) in the computational 

domain Ω st . The new limited value of c q 1 is computed by substituting in Eq. (6.8) 

the limited values from Eqs. (6.9). 

Along the η 2 direction the solution coefficient c q 2 is limited using a similar 

procedure as for the c q 1 coefficient. Specifically, at the midpoints of the edges AB 

and CD the solution expansion is equal to: 

U AB = u q (0, −1) − c q 0 = −c q 2, 

(6.10a) 

U CD = u q (0, 1) − c q 0 = c q 2, 

(6.10b) 

and the value of coefficient c q 2 is expressed in terms of the midpoint values as: 

c q 2 = U CD − U AB 

. (6.11) 

2 

Limiting of the values U AB and U CD using Eq. (6.1) is performed as follows: 

Ũ AB = −m(−U AB , c q 0,(i,j) − cq 0,(i,j−1) , cq 0,(i,j+1) − cq 0,(i,j) ), 

(6.12a) 

Ũ CD = m(U CD , c q 0,(i,j) − cq 0,(i,j−1) , cq 0,(i,j+1) − cq 0,(i,j) ). 

(6.12b)


The new limited value for c q 2 is computed by substituting in Eq. (6.11) the limited 

values from Eqs. (6.12). 

6.0.2 Limiting for triangular elements 

For a P 1 approximation over a triangular element, the solution expansion: 

u t h = c t 0b t 0 + c t 1b t 1 + c t 2b t 2, (6.13) 

where the basis functions relative to the local Cartesian system (η 1 , η 2 ) of Ω st (see 

Fig. 4.3 ) are: 

b t 0 = 1, 

b t 1 − η 2 

1 = η 1 , 

2 

b t 2 = 3η 2 + 1 

. 

2 

For triangular elements, the coefficients c t 1 and c t 2 correspond to u x and u y , 

respectively. The midpoints of the edges are chosen for limiting the solution, and a 

similar procedure to that applied for the case of quadrilaterals is followed. Specifically, 

along the η 1 direction, coefficient c t 1 is limited, by limiting the P 1 part of 

the solution. Then, from the expansion given in Eq. (6.13), the solution at the 

midpoints of the edges AD and BC is equal to: 

U BC = u t (1, 0) − c t 0 = ct 1 

2 + ct 2 

2 , (6.14a) 

U AD = u t (−1, 0) − c t 0 = − ct 1 

2 + ct 2 

2 , (6.14b) 

and coefficient c t 1 is expressed in terms of the midpoint values as:


c t 1 = U BC − U AD . (6.15) 

Limiting of the values U AD and U BC is performed using Eq. (6.1) as follows: 

Ũ AD = m(U AD , c t 0,(i,j) − c t 0,(i,j−1), c t 0,(i,j+1) − c t 0,(i,j)), 

(6.16a) 

Ũ BC = m(U BC , c t 0,(i,j) − c t 0,(i,j−1), c t 0,(i,j+1) − c t 0,(i,j)), 

(6.16b) 

and the new value for c t 1 is computed by substituting in Eq. (6.15) the limited values 

from Eqs. (6.16). 

Along the η 2 direction the coefficient c t 2 is limited, but the P 1 part of the 

solution across the edge AB is limited only with the mean solution variation across 

that edge. The final value of the c t 2 coefficient is: 

c t 2 = m(U AB,(i,j) , c t 0,(i,j) − c t 0,(i,j−1), c t 0,(i,j) − c t 0,(i,j−1)). (6.17) 

Clearly, the approach described for quadrilateral and triangular elements could 

be extended for hexahedral and tetrahedral elements. Again, in three dimensions, 

use of collapsed coordinates must be employed and limiting will be performed in a 

dimension per dimension fashion for the standard cubical element. 

Remark 1. As has been noted the parameter M is computed in the characteristic 

space. For doing so, first the difference of the mean solution is formed at 

every element edge and then transformed to the characteristic space. 

Remark 2. If the solution at an element is limited then the order of accuracy 

does not drop to P 0 , since a modified slope is preserved for the approximation.


6.1 Positivity preserving limiters for the DG method 

The assumption of a thermally perfect gas for the equations of gas dynamics lead to 

the relation between the total energy, pressure and kinetic energy in a flow field given 

by Eq. (2.21). It is easily verified that the pressure p is a concave function of the state 

variables, e.g. defining two states W 1 = [ρ 1 , u 1 , v 1 , E 1 ] T and W 2 = [ρ 2 , u 2 , v 2 , E 2 ] T 

it is implied from Jensen’s inequality for 0 ≤ s ≤ 1, that: 

p (sW 1 + (1 − s) W 2 ) sp (W 1 ) + (1 − s) p (W 2 ) if ρ 1 , ρ 2 0, (6.18) 

and the set of admissible states is defined by: 

G = {W | ρ, p > 0}, (6.19) 

which is convex and this is the key observation for the positivity preserving limiters 

for the density and pressure fields developed by Zhang and Shu in [156] where the 

solutions coefficients are limited in such a way, so that: 

ˆ For smooth solutions accuracy is preserved. 

ˆ The scheme remains conservative. 

ˆ Limiting for positivity of density and pressure is performed locally at every 

element. 

As in [156] the TVB limiter is applied before the positivity limiters, which will 

be described hereafter, but taking into consideration that for the overall success of a 

positivity preserving scheme, a numerical flux that is preserving the positivity of the 

variables must be employed, such as the Lax-Friedrichs or the local-Lax-Friedrichs 

(Eq. (4.9)). Furthermore, in [112] the positivity of an Euler explicit scheme was 

analyzed and a sufficient condition was derived for the mean solution of the state 

variables to be in the set G:


∆t 

∆x || (|u| + c) || ∞ ≤ aα, (6.20) 

where a ∈ (0, 1]. SSP high order Runge-Kutta methods then are suitable for the 

positiveness of high order time accurate discretizations. 

The first step is to limit the coefficients for the density field. This is accomplished by 

first computing the minimum value of the density ρ min looping over the quadrature 

points on the edges. Then the following parameter is defined: 

( c 

ρ ) 

0 − ε 

θ 1 = min 

c ρ , 1 , (6.21) 

0 − ρ min 

where ε = min (1.0 −13 , c ρ 0, p, ) and p is the mean value of the computed pressure. 

Then, the final coefficients for the density expansion are modified as follows: 

˜c ρ i = θ 1c ρ i , (6.22) 

where the index i counts all the bases except for the one corresponding to the mean 

solution. 

The second step is to limit the pressure. This requires the alteration of all the 

solution coefficients for every conservative variable. This is accomplished as follows. 

Define: 

s = (1 − t)W + tq, (6.23) 

where W is the mean solution and q is the conservative variables state vector with 

the limited density solution. First, calculate: 

t = the solution of p(s) = ε, if p(q) < ε, (6.24) 

then, modify the solution coefficients for each field except for the ones corresponding


to the mean solution as: 

˜c i = θ 2 c i , (6.25) 

where θ 2 is the limiting function given by the following equation: 

θ 2 = min (1, t). (6.26) 

This limiter was applied only for quadrilateral elements by Zhang and Shu 

[156]. In this work, applications for unstructured triangular meshes are demonstrated 

with excellent results.

Chapter 7 

Numerical results 

In the present work a CFD solver based on the previously described DG discretization 

was developed and named HoAc as an abbreviation for High Order ACcuracy. 

The unified limiting procedure has been implemented in HoAc . Many cases of high 

speed flows with strong shocks have been analyzed and are presented in this chapter 

to demonstrate the improvements obtained with the proposed limiting approach. 

The results confirm the outstanding shock capturing and positivity preserving capabilities 

of the novel limiting approach. Moreover, applications for supersonic inviscid 

and subsonic viscous cases will be presented. 

In the computations with the new limiting approach, the value M in Eq. (6.1) 

was computed at each time step for each characteristic field. The variation of the 

computed maximum value of the Laplacian for each field in the characteristic space: 

W 1 ≡ δw 1 = δρ−δp/c 2 , W 2 ≡ δw 2 = −k x δu−k y δv, W 3 ≡ δw 3 = k x δu+k y δv+δp/ρc, 

and W 4 ≡ δw 4 = −k x δu − k y δv + δp/ρc, where k = [k x , k y ] is an arbitrary direction 

was recorded and will be presented in many cases. 

First, verification tests of the spatial DG discretization implemented in the 

standard space Ω st in HoAc will be presented followed by verification and validation 

tests for the limiting procedure, which are performed by comparing the analytical 

solution for the standard Sod’s shock tube problem and again for the Sod’s problem 

with an extreme pressure ratio. The supersonic flow over a cylinder is considered 

139

CHAPTER 7. NUMERICAL RESULTS 140 

next employing a P 2 and a P 3 expansion of the solution, followed by standard test 

cases involving flows with strong shocks proposed in [152]. These standard test cases 

for shock capturing schemes include the double mach reflection of a Mach 10 strong 

shock and a supersonic flow at Mach number of 3 in a wind tunnel with a forward 

facing step. The Shardin’s problem [60] is solved next, which also confirms the very 

good resolution of the developed limiting procedure, followed by a test case of an 

impinging shock at Mach Mach number of 5 around the same geometry. Finally, the 

diffraction of a strong shock over a backward facing step at Mach, Mach number of 

5 is considered for quadrilateral, rectangular and triangular meshes using P 2 and P 3 

expansions of the solution. Furthermore, it is noted that in all test cases the LLF 

flux has been applied and the GLL quadrature points are used for the computation 

of the integrals. Also, for the application of the IC Galerkin projection is employed. 

7.1 Solution output 

Most present CFD solvers store the solution at the element center or at the nodes in 

the mesh. However, a DG method computes the solution in a more general fashion, 

as for modal expansions especially the solution coefficients do not have a specific 

physical point of relevance. 

As the solution is discontinuous across the element edges two approaches have 

been followed in the present work for outputting and post processing of the computed 

results: (i) averaging of the solution at the nodes of the computational mesh 

and (ii) a discontinuous output of the solution, which is based on outputting the solution 

at points over the computational element, whose coordinates are determined 

by the inverse transformation of equally spaced points in the standard element configuration. 

Both approaches give results that are very similar on fine meshes, and for 

that, in test cases where fine meshes have been used averaged print of the solution 

is used. 

The discontinuous output of the solution leads to a visualization mesh that 

is finer than the computational mesh. In Fig. 7.1 a portion of a mixed-type ele-


ment computational mesh is shown together with the visualization mesh for a P 4 

expansion of the solution over every computational element. 

Figure 7.1: Computational mesh (blue lines) and visualization mesh (black lines) 

for a P 4 expansion of the solution over triangular and quadrilateral elements. 

In Appendix B the pseudo code for constructing the visualization mesh over a 

quadrilateral and a triangular element is provided. 

7.2 Code Verification 

In this section the verification of the implemented DG discretization for the Euler 

and the Navier-Stokes equations will be presented, which confirm the correctness of 

the solution algorithms employed in the HoAc solver. 

7.2.1 Convergence study for the Euler equations 

For verifying the correctness of the spatial DG discretization of the Euler equations, 

a stationary isentropic vortex is considered described by the following IC in the flow 

variables:


ρ = 

[1 − 

] 1 

γ−1 

(γ − 1) β2 

e (1−r2 ) , (7.1) 

8γπ 

(δu, δv) = β “ 

2π e 1−r 

2 ” 

2 

[− (y − y 0 ) , (x − x 0 )] , (7.2) 

p = ργ 

γ . (7.3) 

As the above IC satisfy exactly the Euler equations the residual of the system in 

every field must drop accordingly to the order of discretization using a refined mesh. 

In Fig. 7.2 the convergence rates for the residual of the system corresponding 

to the density field is given starting from a P 1 and going up to a P 4 expansion of the 

solution using the LLF flux. It is observed that the design order of the discretization 

is achieved. 

1 

0.01 

2 

L 2 

Error 

0.0001 

3 

P1 

P2 

P3 

P4 

1e-06 

4 

5 

1e-08 

0.1 1 

∆x,∆y 

Figure 7.2: Convergence rate for the DG discretization of the Euler equations.


7.2.2 Convergence study for the Navier-Stokes equations 

The correctness of the results for viscous computations is verified by considering the 

test case corresponding to the Poiseuille flow, which simply refers to a flow between 

two parallel plates under a pressure gradient. An analytical solution of this problem 

exists under the assumptions of incompressible flow and constant viscosity and in 

conservative variables is the following: 

⎡ 

U exact = 

⎢ 

⎣ 

ρ 0 

γ−1 

⎤ 

ρ 

y (y − b) 

0 

. (7.4) 

( 

p0 + dp x) ⎥ 

+ ρ dx 2 u2 ⎦ 

1 dp 

2µ 0 dx 

As HoAc solves the compressible Navier-Stokes equations, the introduction of a 

source term S to the right hand side of the system given in Eq. (2.29) is necessary 

in order to retrieve numerical results from the compressible Navier-Stokes equations 

corresponding to the incompressible solution given in Eq. (7.4). The source term S 

is equal to: 

⎡ 

S = 

⎢ 

⎣ 

1 

2µ 

( dp 

dx 

) 2 

[ 

1 

k−1 

0 

0 

0 

y (y − b) − 

(2y−b)2 

2 

⎤ 

. (7.5) 

⎥ 

] ⎦ 

. 

In Fig. 7.3 the convergence rates for the DG discretization of the compressible 

Navier-Stokes equations using the LDG method are given for a P 1 up to P 4 

expansion of the solution, with respect to the element length. It is observed that 

the design order of the method is achieved.


0.1 

0.01 

L 2 

Error 

0.001 

0.0001 

1e-05 

1e-06 

1e-07 

2 

3 

4 

5 

P1 

P2 

P3 

P4 

1e-08 

0.1 1 

∆x,∆y 

Figure 7.3: Convergence rate for the DG discretization of the Navier-Stokes equations 

with the LDG method. 

7.2.3 Inviscid flow over a cylinder 

The potential of the discontinuous output of the solution is depicted for the computation 

of the flow over a cylinder at a low Mach number (M ∞ = 0.14). Solutions are 

computed with a fixed mixed-type linear element mesh with quadrilateral elements 

in the vicinity of the cylinder and starting from a P 1 and going up to a P 3 expansion 

of the solution. It is noted that the curvature based boundary conditions proposed 

in [92] have been applied, due to the use of straight sided elements, otherwise an 

unsteady wake is formed and a steady state solution is not reached.


(a) Computational mesh 

(b) P 1 expansion 

(c) P 2 expansion 

(d) P 3 expansion 

Figure 7.4: Mach contours for the inviscid flow over a cylinder using discontinuous 

output of the solution and the curvature based boundary conditions in [92]. 

It is observed in Fig. 7.4 that as the order of the solution expansion increases so does 

the quality of the numerical results. Furthermore, in Fig. 7.5 the computed pressure 

coefficient at every quadrature point along the edges defining the cylinder geometry 

is compared with the analytical solution for the pressure distribution around the 

cylinder: C p = 1 − 3 sin 2 (θ). It is clear that as the order of the approximation 

increases, the numerically evaluated pressure distribution approaches that given by 

the analytical solution.


Figure 7.5: Comparison between the numerically evaluated pressure coefficient and 

the analytical solution for the inviscid flow around a cylinder. 

This test case was solved using an implicit first order backward Euler scheme, 

with a step size of dt = 1.0 using an unpreconditioned matrix-free approach. The 

large number of iterations observed in the convergence history plot in Fig. 7.6 for 

the density field is attributed to the placement of the far field boundary at a distance 

of 20 unit lengths from the cylinder center.


Figure 7.6: Convergence history for the residual in the density field for the inviscid 

flow over a cylinder. 

7.2.4 Standard Sod’s shock tube problem 

The standard Sod’s shock tube problem with IC [ρ, u, v, p] L = [1, 0, 0, 1] and [ρ, u, v, p] R = 

[0.125, 0, 0, 1], which is an exact solution of the 1D Euler equations is solved in two 

dimensional triangular and quadrilateral meshes, which are shown in Fig. 7.7. This 

problem is the first validation test for the developed limiting procedure.


(a) Quadrilateral mesh 

(b) Triangular mesh 

Figure 7.7: Meshes for the two dimensional Sod’s shock tube problem. 

The evolution of the limiting procedure for a P 1 numerical solution is depicted 

in Fig. 7.8 for the computations performed on the quadrilateral mesh. 

0.25 

0.2 

Time 

0.15 

0.1 

0.05 

0 

-1 -0.5 0 0.5 1 

Length 

Figure 7.8: Limited elements in time for the Sod’s shock tube problem using the 

quadrilateral mesh of Fig. 7.7a. 

It is observed that the limiting procedure follows the discontinuities appearing in 

this Riemann problem. The contact discontinuity is smeared as time advances and 

limiting is performed at the start and end of the contact. Similarly, at the start 

and end of the rarefaction wave the elements are limited. For the smooth parts of 

the flow limiting has been applied in few elements. The final density and entropy 

profiles at time t = 0.25 are shown in Fig. 7.9 and compared with the exact result


for the computations performed on the triangular channel mesh. The numerical 

solution was recorded along the center line of the channel mesh shown in Fig. 7.7b. 

The P 1 computed density distribution of Fig. 7.9a agrees quite well with the exact 

result. The comparison of the most sensitive entropy distribution given in Fig. 7.9b 

shows that only small oscillations exist at the location of the contact and that the 

shock is essentially captured within two elements. 

2 

1 

Exact Solution 

Numerical Solution P1 

1.8 

Numerical Solution P1 


1.6 

Density 

0.5 

Entropy 

1.4 

1.2 

1 

0 

-1 -0.5 0 0.5 1 

Length 

(a) Density 

-1 -0.5 0 0.5 1 

Length 

(b) Entropy 

Figure 7.9: Comparison of the exact density and entropy variation at t = 0.25 with 

the two-dimensional numerical solution for the Sod’s shock tube problem in the 

triangular channel mesh of Fig. 7.7b. 

7.2.5 Extreme Sod’s Riemann problem 

In order to demonstrate the proper implementation of the positivity preserving 

limiters a one-dimensional Riemann problem with an exact solution for an extreme 

pressure ratio was considered. This problem was solved with a two dimensional 

rectangular mesh and it is noted the TVB limiter is applied before the application 

of the positivity limiters. For this problem, as the solution was advanced in time 

it was verified that the positivity preserving limiters were activated. The Sod’s 

shock tube problem is considered with the following IC [ρ, r, v, p] l = [1.0, 0, 0, 100], 

[ρ, u, v, p] r = [0.125, 0, 0, 0.001], and a specific heat ratio γ = 1.4. For this shock 

tube problem an extreme pressure ratio p l /p r = 10 5 exists. In Figs. 7.10, 7.11 and 

7.12 the exact and the numerical solution computed with a P 1 expansion and 800


equally sized elements are compared. The computed density, pressure, and entropy 

are in good agreement with the exact result. 

1 

Numeical Solution 


Density 

0.5 

0 

-1 -0.5 0 0.5 1 

Length 

Figure 7.10: Density plot for the Sod’s problem with a pressure ratio of 100000. 

100 

Numerical Solution 


Pressure 

50 

0 

-1 -0.5 0 0.5 1 

Length 

Figure 7.11: Pressure plot for the Sod’s problem with a pressure ratio of 100000. 

100 

Entropy 

50 

Numerical Solution 


0 

-1 -0.5 0 0.5 1 

Length 

Figure 7.12: Entropy plot for the Sod’s problem with a pressure ratio of 100000.


For this problem an explicit third-order accurate SSP Runge-Kutta method 

was employed with 25 stages provided by the PETSc library package. The time 

step used in the computations was equal to dt = 2.0 · 10 −7 and the final time of the 

simulation was equal to t = 5.0 · 10 −2 . This very low step size was needed as the 

maximum Mach number developed in the flow approached the value of 12. 

7.3 Supersonic flow over a cylinder 

The inviscid, supersonic flow at Mach number of 2 over a cylinder is considered 

with a P 2 and P 3 expansion of the solution in order to demonstrate the developed 

p-adaptive limiting procedure for arbitrary shaped elements, which is noted again 

that it is being performed in the characteristic variable space and over the standard 

square element of the computational space. Three types of meshes quadrilateral, 

triangular and a mixed-type element mesh were used for the computations which 

are shown in Figs. 7.13, 7.14 and 7.15. An impulsive start of the numerical solution 

is performed with the following IC: [ρ, u, v, p] = [1.0, 2.0, 0.0, 0.714285714] 

Figure 7.13: Quadrilateral mesh for supersonic flow at Mach 2 around a cylinder.


Figure 7.14: Triangular mesh for supersonic flow at Mach 2 around a cylinder. 

Figure 7.15: Mixed type mesh for supersonic flow at Mach 2 around a cylinder. 

The computed pressure field, which is the same for the numerical solutions obtained 

with different meshes, is shown in Fig. 7.16 for a P 3 solution expansion over the 

quadrilateral mesh.


Figure 7.16: Pressure contour lines using 30 equally spaced intervals for flow at Mach 

2 over a cylinder for the numerical solution obtained with a P 3 approximation. 

It is emphasized that once an element is flagged for limiting based on the value 

of the estimate of the Laplacian (second derivative) M, this element is marked as 

limited. For a P 2 global expansion the elements next to the limited elements are 

P 1 and for the rest of the domain are P 2 . For computations with preset higher 

order P n expansions and for elements neighboring the P 2 elements the expansion 

will be raised to P 3 and progressively to P n . This is demonstrated in Fig. 7.17 for 

a computation with P 3 global approximation for a quadrilateral mesh.


Figure 7.17: Limited elements for the flow at Mach 2 over a cylinder for the numerical 

solution obtained with a quadrilateral mesh and a P 3 approximation; the blue 

elements are limited the adjacent green elements are P 1 expansions, the adjacent 

to green red elements are P 2 , and for the rest of the domain P 3 expansions are 

employed. 

The limited elements for a P 2 computation with the triangular and quadrilateral 

are shown in Fig. 7.18.


(a) Mixed-type mesh (b) Quadrilateral mesh (c) Triangular mesh 

Figure 7.18: Limited elements for the flow at Mach 2 over a cylinder for the numerical 

solution obtained with a P 2 approximation; the blue elements are limited 

the adjacent red elements are P 1 expansions and for the rest of the domain P 2 

expansions are employed. 

It can be seen that limiting has been applied only for elements neighboring the 

shock. For a P 2 global expansion the elements next to the limited elements and for 

the rest of the domain are P 2 and only the limited elements are P 1 . It is noted 

that on the mixed-type element mesh the limited elements are not visible as they 

are concentrated in the region where the mesh is very much refined. 

A comparison of the computed pressure distribution along the stagnation line 

obtained with different meshes and polynomial expansions is shown in Fig. 7.19. 

The agreement of the computations is very good. The computation on the refined 

mixed-type element mesh has the sharpest capturing of the shock.


Figure 7.19: Comparison of the pressure distribution along the stagnation line obtained 

from P 2 and P 3 approximations of the numerical solutions with different 

meshes. 

The variation of the Laplacian estimate, in essence the parameter M in the 

TVB limiter in the current test case, for each field in the characteristic space W i , is 

shown in Fig. 7.20 for the P 2 computation using quadrilateral elements.


Figure 7.20: Variation of the maximum value of parameter M for each characteristic 

field obtained during convergence to steady state of the numerical solution for the 

flow at Mach 2 over a cylinder with a quadrilateral mesh and P 2 solution expansion. 

It must be emphasized that until now numerical solutions with the DG method 

and the TVB limiters were presented only for rectangular elements and for triangular 

elements following the elaborate procedure of Cockburn [82], but from the current 

test case it is observed that the developed limiting approach works quite well for 

elements of arbitrary shape, mixed-type element meshes and p-adaptive solutions. 

Furthermore, a crisp resolution of the shocks is obtained since M < 3 (in contrast to 

the suggested value of M = 50 by [82]) and fewer elements are limited as depicted 

in Fig. 7.18. 

7.4 Double Mach Reflection of a strong shock 

Next, the complex shock pattern generated by the reflection of a moving shock at 

Mach number of 10 inclined at an angle of θ = 60 ◦ with respect to a horizontal wall 

[152] is solved numerically in order to investigate the potential of the p-adaptive


limiting procedure for a flow with a strong moving shock. This is a standard test 

problem for shock capturing schemes. It originated by experimental and numerical 

studies of reflections of planar shock waves from wedges. The computational domain 

is a rectangle with a width of four units and a height of one unit. The shock is initially 

set up to be inclined at an angle of 60 ◦ to the x-axis, and has a Mach number of 

10. Therefore, the IC are: [ρ, u, v, p] = [8.0, 8.25 cos (30 ◦ ) , −8.25 cos (30 ◦ ) , 116.5] for 

x < 1 6 + y √ 

3 

and [ρ, u, v, p] = [1.4, 0, 0, 1.0] for x 1 6 + y √ 

3 

. At x = 0 the boundary 

conditions are specified to the values corresponding to the post-shock state, and at 

x = 4 extrapolation of the computed variables is performed. The lower y boundary 

for x 1 is a reflecting wall and for x < 1 the fluid values are set by the initial 

6 6 

post-shock conditions. The upper y boundary is constructed to follow the flow of the 

diagonal shock such that there is no interaction between the shock and this boundary. 

Given the IC here, the intersection of the diagonal shock and the upper boundary at 

time t occurs at x s (t) = 1 + 1+20t √ 

6 3 

. For x x s (t), the values at the upper y boundary 

are set by the initial pre-shock conditions. For x < x s (t), the state variable values 

are set by the initial post-shock conditions. In [82, 91, 119] this problem was solved 

using a DG discretization of the Euler equations on rectangular meshes, but in the 

present work is solved using quadrilateral, rectangular and triangular meshes. Both 

P 1 and P 2 approximations were used for the numerical solution that was run to a 

final time of t = 0.2 using the third order explicit SSP Runge-Kutta developed in 

[130]. For demonstration, a solution obtained in a relatively coarse quadrilateral 

mesh is shown first. The limited elements are shown in Fig. 7.21. 

Figure 7.21: Limited elements on the coarse quadrilateral mesh for the double mach 

reflection problem at t = 0.2.


It is seen that even for a coarse mesh limiting is applied in a relatively narrow region 

around the discontinuities. The final density at t = 0.2 obtained from the numerical 

solution with the coarse mesh is shown in Fig. 7.22. 

Figure 7.22: Density contours on the coarse quadrilateral mesh for the double mach 

reflection problem at t = 0.2. 

The numerical solution obtained on a mesh with rectangular elements of size 

h = 1/240, with a P 1 expansion of the solution, for the density field is shown in Fig. 

7.23 

Figure 7.23: Density contours for the numerical solution computed on the rectangular 

mesh, h = 1/240, and a P 1 approximation for the double mach reflection 

problem. 

and the elements where limiting of the solution was performed are shown for time 

t = 0.1 and t = 0.2 in Fig. 7.24,


(a) Limited elements at t = 0.1 

(b) Limited elements at t = 0.2 

Figure 7.24: Limited elements for the numerical solution computed on the rectangular 

mesh, h = 1/240, and a P 1 approximation for the double mach reflection. The 

blue elements are limited and for the rest of the domain a P 1 approximation of the 

solution is employed. 

where the domain decomposition employed for parallel computation is highlighted. 

Clearly, seamless limiting is achieved for parallel processing. The computed density 

of Fig. 7.23 agrees well with other numerical solutions in the literature [82, 91, 119] 

obtained with high resolution shock capturing schemes of equivalent order. 

A numerical solution for an isotropic mesh with triangular elements whose 

edge length is equal to h = 1/240 was also obtained for a P 1 solution expansion. 

The limited elements at time t = 0.1 and t = 0.2, which are marked in Fig. 7.25, 

show that limiting is performed at narrow zones around the discontinuities. The 

numerical solution for the density field computed on the triangular mesh is shown 

in Fig. 7.26.


(a) Limited elements at t = 0.1 

(b) Limited elements at t = 0.2 

Figure 7.25: Limited elements for the numerical solution computed on the triangular 

mesh, h = 1/240, and a P 1 approximation for the double mach reflection. The blue 

elements are limited and for the rest of the domain a P 1 approximation of the 

solution is employed. 

Figure 7.26: Density contours for the numerical solution computed on the triangular 

mesh, h = 1/240, and a P 1 approximation for the double mach reflection problem. 

This test case was also run on the same triangular mesh using a P 2 expansion 

of the solution. The density field at t = 0.2 is shown in Fig. 7.27 which agrees very 

well with the P 1 solution obtained with the rectangular mesh and with the results 

given in [119, 152].


Figure 7.27: Density contours for the numerical solution computed on the triangular 

mesh, h = 1/240, and a P 2 approximation for the double mach reflection problem. 

The variation of the parameter M in the application of the TVB limiter for 

the P 2 solution on the isotropic triangular mesh is depicted in Fig. 7.28, and it is 

observed a wide variation of its values for each characteristic field.


Figure 7.28: Variation of the maximum value of parameter M for each characteristic 

field obtained during the time advancement of the numerical solution for the double 

mach reflection problem obtained with a uniform triangular element mesh with a 

P 2 approximation and h = 1/240. 

7.5 Inviscid flow at Mach number of 3 in a tunnel 

with a step 

The inviscid supersonic flow at Mach number of 3 in a tunnel with a step has been 

computed with rectangular, triangular and mixed-type element meshes in order to 

demonstrate the flexibility and potential of the proposed limiting approach. This 

is another standard test case for shock capturing schemes and was introduced by 

Emery in [51]. The wind tunnel is one unit length wide and three unit lengths


long. The step is 0.2 unit lengths high and is located 0.6 length units from the 

left-hand end of the tunnel. The corner of the step is the center of a rarefaction fan 

and hence is a singular point of the flow field which is seriously affected by large 

numerical errors generated just in the neighborhood of this singular point. This 

may be alleviated in two ways: (i) by using a fine mesh in the region close to the 

corner or (ii) by applying special boundary conditions near the corner of the step as 

demonstrated by Woodward and Collela in [152]. Specifically, the density is reset 

so that the entropy has the same value in the zone just to the left and below the 

corner of the step and the magnitudes of the velocities are reset also, but not their 

direction, so that the sum of enthalpy and kinetic energy per mass has the same 

value as in the same zone used to set the entropy. In the present work the first 

method is followed. 

The numerical solution with the following IC [ρ, u, v, p] = [1.4, 3, 0, 1.0] for the 

tunnel with a step was obtained with a P 1 and P 2 expansions, because for computations 

with higher order expansions severe time step limitations are encountered 

due to the very small elements that had to be used at the corner in order to reduce 

the artificial entropy layers generated by the corner singularity. 

The first computation was performed with a P 1 expansion of the solution for 

a mesh with rectangular elements with the edge length equal to h = 1/80. In Figs. 

7.29 and 7.30 the limited elements and the numerical solution for the density field 

are shown at t = 2.0, respectively. 

Figure 7.29: Limited elements for the numerical solution obtained on a rectangular 

mesh and P 1 approximation and h = 1/80 for the flow at Mach number of 3 in a 

wind tunnel with a forward facing step at t = 2.0.


Figure 7.30: Density field for the numerical solution obtained on a rectangular mesh 

and P 1 approximation and h = 1/80 for the flow at Mach number of 3 in a wind 

tunnel with a forward facing step at t = 2.0. 

The limited elements of Fig. 7.29 closely match the captured discontinuities of Fig. 

7.30. At later times of the simulation the artificial entropy layer generated by the 

singularity at the corner creates a Mach stem. In an effort to alleviate the error by 

the singularity at the corner, a numerical solution on a isotropic fine triangular mesh 

with an average edge length of h = 1/100 and for a P 2 expansion of the solution 

was performed. The limited elements for this numerical solution along with the 

computed density field at t = 4 are shown in Figs. 7.31 and 7.32: 

Figure 7.31: Limited elements for the numerical solution obtained on a fine triangular 

mesh and a P 2 approximation and h = 1/100 for the flow at M = 3.0 in a 

wind tunnel with a forward facing step at t = 4.0. Blue elements: limited solution. 

Green elements P 1 expansion. Red elements P 2 expansion.


Figure 7.32: Density field for the numerical solution obtained on a fine triangular 

mesh and P 2 approximation and h = 1/100 for the flow at Mach number of 3 in a 

wind tunnel with a forward facing step at t = 4.0. 

It is observed that the Mach stem still appears in the flow field. This lead to a third 

effort for a numerical solution on a mixed-type element mesh refined at the corner 

with triangular elements as shown in Fig. 7.33. 

Figure 7.33: Mixed-type element mesh for the inviscid flow at Mach number of 3 in 

a tunnel with a step. 

The average edge length size in the quadrilateral region of the mesh is h = 1/40 and 

in the triangular region the edge length is equal to h = 3.0 · 10 −3 . The computed 

density field at t = 4 is shown in Fig. 7.34. It is observed that the Mach stem has 

diminished considerably.


Figure 7.34: Density field for the numerical solution obtained on a mixed-type 

element mesh and a P 1 approximation for the flow at Mach number of 3 in a wind 

tunnel with a forward facing step at t = 4.0. 

Furthermore, in Fig. 7.35 the limited elements on the mixed-type element 

mesh computation are shown at t = 0.1, where the TVB limiter has smoothly ”fireup” 

at the narrow region of the developing shock wave in the part of the mesh 

covered by quadrilateral and triangular elements. 

Figure 7.35: Limited elements at t = 0.1 using a mixed-type element mesh for the 

inviscid flow at Mach number of 3 in a wind tunnel with a forward facing step. 

The variation of the parameter M versus the number of time steps is shown 

in Fig. 7.36 where it is observed that there is a wide variation of its value for each


characteristic field and differs considerably from the constant value of M = 50 as 

suggested in [82]. 

Figure 7.36: Variation of the parameter M for the TVB limiter applied on the mixedtype 

element mesh for the inviscid flow at Mach number of 3 in a wind tunnel with 

a forward facing step. 

7.6 Diffraction of a strong shock over a backward 

facing step 

The effectiveness of the positivity preserving limiters combined with the TVB limiter 

for shock capturing is tested for the diffraction of a Mach number of 5.09 right 

moving shock over a backward facing step. This problem is solved on meshes with 

rectangular, quadrilateral and triangular elements. The IC for this problem is a 

right moving shock wave. The element size for the solutions was h = 0.025 for the 

rectangular and triangular meshes and of comparable size for the region close to the


corner for the mesh with quadrilateral elements. Sample meshes along with contour 

lines for the density field at t = 0.8 of the diffraction of the shock wave are given in 

Fig. 7.37. 

(a) Quadrilateral elements mesh 

(b) Rectangular elements mesh 

(c) Triangular elements mesh 

Figure 7.37: Sample meshes with density contours at t = 0.8 for the diffraction 

of a strong shock over a backward facing step using a P 2 solution expansion. The 

black square shows where the computational domain was clipped for showing the 

underlying computational mesh.


A P 2 expansion of the solution has been used in all cases. For demonstration, 

a solution with P 3 expansion was also computed. The elements for which limiting 

was applied at time t = 2.0 for the computations performed with different meshes 

are shown in Fig. 7.38. 

(a) Quadrilateral elements 

(b) Rectangular elements 

(c) Triangular elements 

Figure 7.38: Limited elements for the diffraction of strong shock over a backward 

facing step at t = 2.0. Blue elements: limited solution. Green elements P 1 expansion. 

Red elements P 2 expansion. 

Comparing Fig. 7.38 with the computed density at the same time in Fig. 7.39:





Figure 7.39: Density contours for the diffraction of strong shock over a backward 

facing step at t = 2.0 with 30 equally spaced intervals from 0.06 to 7.16. 

it is seen that limited elements were obtained at the shock locations. Comparing 

Fig. 7.39 with the pressure distribution in Fig. 7.40:





Figure 7.40: Pressure contours for the diffraction of strong shock over a backward 

facing step at t = 2.0 with 30 equally spaced intervals from 0.09 to 31.9. 

the location of the contact discontinuity can be identified. It appears no limiting was 

performed in the regions of the contact discontinuities. The line plot of Fig. 7.41 for 

the density and pressure along a line starting from the left side of the computational 

domain, ending at its right side and being above the step, also demonstrates the 

existence and quite good capturing of the contact discontinuity, which is further 

justified by the numerically computed gradient of pressure and density in Figs. 7.42 

and 7.43.


30 

25 

Density 

Pressure 

Density, Pressure 

20 

15 

10 

5 

0 

0 2 4 6 8 10 12 

Length 

Figure 7.41: Plot of density and pressure along the line defined by points [0, 7] and 

[13, 7]. 

Figure 7.42: Numerically computed pressure (left) and density (right) gradient field 

for the diffraction of strong shock over a backward facing step at t = 2.0, using a 

rectangular elements mesh.


Figure 7.43: Numerically computed pressure (left) and density (right) gradient field 

for the diffraction of strong shock over a backward facing step at t = 2.0, using a 

triangular elements mesh. 

The variation of the maximum value of the parameter M is shown in Fig. 

7.44 versus the iteration number for the simulation on the mesh with rectangular 

elements and a P 2 expansion. 

30 

W 1 

25 

W 3 

W 3 

W 4 

20 

M 15 

10 

5 

0 

0 5000 10000 

Iteration 

Figure 7.44: Variation of the second derivative estimate for the diffraction of a strong 

shock over a backward facing step using a P 2 expansion on a rectangular elements 

mesh.


It is observed that there is a wide variation of the maximum value of the parameter 

M for each field as the solution advances in time. In contrast to [156] where fixed a 

high value of the parameter M was prescribed (M = 100), with the present limiting 

approach these values were evaluated during the time advancement of the solution 

and limiting was performed only where the variation of the computed solution was 

very large. These regions are most of the time shock waves. 

Additional computations on the rectangular mesh were performed with P 1 and 

P 3 expansions of the solution, in order to examine the effect on the results of the 

combined application of the shock capturing and positivity preserving limiters. In 

Fig. 7.45 a plot over a line defined by the points [2, 5.5] and [5, 4] at time t = 0.9 of 

density and pressure is shown. 

14 

12 

Pressure P2 

Density P1 

Denstiy P2 

Density P3 

Density, Pressure 

10 

8 

6 

4 

2 

0 

0.5 1 1.5 2 2.5 

Length 

Figure 7.45: Plot of density and pressure along a line defined by the points [2, 5.5] 

and [5, 4] at time t = 0.9. 

Due to the fine mesh and the non-limiting of the solution at the slip line as it is 

evident for the P 2 computation from Fig. 7.38, the resolution of the computations 

is the same. However, the location of the shock has been captured in a different 

position for the P 1 computation in comparison with the P 2 and P 3 computations, 

which actually for the last two is the same. 

The large-scale wiggles at time t = 2.0 of Fig. 7.39(a) are due to the irregular 

and coarse mesh since the quadrilaterals expand away from the corner. The enlarged


plots of Fig. 7.37 also show that some wiggles could be introduced from irregular 

meshes (see Fig. 7.39(a)). For the finer and regular rectangular and triangular 

meshes of Figs. 7.39(b),(c) the wiggles are not caused by the mesh or the TVB 

limiter. Note that in Fig. 7.37 the slope limiter and the positivity preserving 

limiter have been applied already many times until t = 0.9, but no wiggles in 

the computed solution appear. The wiggles at the end of the simulation at time 

t = 2.0 are caused by small scale disturbances that are generated in the region 

between the shock and the contact discontinuity, and they are not well resolved. 

In the absence of physical viscosity, these disturbances cannot diffuse. Depending 

on the numerical viscosity introduced by the numerical scheme, and the resolution 

of the mesh, some disturbances may dissipate and the rest appear as wiggles in 

the computed density and pressure plots. The computed vorticity field at time 

t = 2.0, shown bellow in Fig. 7.46, for the simulations performed on the rectangular 

and triangular meshes, demonstrate that for the rectangular element mesh these 

disturbances are concentrated in the region 1.5 < x < 9 marked by a red line 

in Fig. 7.46(a). For the computation with the triangular mesh, which provides 

higher resolution per unit area, the onset of a second contact discontinuity after 

the Mach stem is evident. Further away, this contact could not be resolved in the 

computations. For the part of the domain were the contact remains unresolved (the 

region 5.5 < x < 11 also marked by a red line in Fig. 7.46(b)) wiggles are found in 

the computed solution.


(a) Rectangular elements 

(b) Triangular elements 

Figure 7.46: Vorticity field for the diffraction of a Mach 5 shock over a backward 

facing step.


intervals 

7.7 Shardin’s problem 

The effectiveness of the novel p-adaptive limiting procedure is further examined for 

the standard Schardin’s problem [60] in order to validate the use of the limiting 

procedure for shock capturing. For this problem a right moving shock at a Mach 

number of 1.34 impinges on a triangle. An unsteady flow is produced accompanied 

by the generation of vortices and complex wave patterns resulting from the diffraction 

of the shock at the triangle’s corners and multiple interactions between the 

reflected shock waves and the generated vortices. The complete problem setup and 

the experimental results are described in detail in [34]. 

For this problem two types of meshes have been used with different orders of 

expansion for the solution. In order to reduce the problem size only half of the 

triangle was considered and symmetry conditions were imposed. An unstructured 

triangular mesh with a P 1 expansion and approximately 200000 elements and a 

quadrilateral mesh with a P 2 expansion and approximately 285000 elements. Both 

meshes have an average mesh length of h = 0.002. The reason for employing a P 1 

expansion on the triangular mesh is that triangles provide almost double resolution 

over the same area in the computational domain compared to quadrilateral elements. 

The computed density and pressure fields at time t = 128µs are given in 

Figs. 7.47 and 7.48. Indeed the computed density field by both triangular and 

quadrilateral mesh numerical solutions with P 1 and P 2 expansions respectively are 

of comparable resolution.


(a) P 1 expansion. Triangular elements 

(b) P 2 expansion. Quadrilateral elements 

Figure 7.47: Density contours for Schardin’s problem with 30 equal spaced intervals 

from 0.6 to 2.4 using a P 2 solution expansion




Figure 7.48: Pressure contours for Schardin’s problem with 30 equal spaced intervals 

from 0.4 to 2.14 using a P 2 expansion. 

It is observed that both expansion sets and mesh types reveal the same structure of 

the flow field behind the triangle, with the exception, that the results produced by 

the P 2 expansion capture each discontinuity in a narrower region. Furthermore, in 

Fig. 7.49 where the numerical Schlieren for both polynomial expansions is shown:




Figure 7.49: Numerical Schlieren for Schardin’s problem using a P 2 solution expansion. 

it can be observed that the complex wave pattern produced by the interaction of 

the vortices and the impinging and reflected shock waves is basically the same, 

but that of the P 2 expansion has resolved better the vortices as also the diffracted 

waves and the slip lines. However, in the P 2 computation wiggles appear at the slip


lines locations, but not for the P 1 computation. This is due to the fact that the 

combination of a fine mesh and higher order expansion lead to smaller numerical 

viscosity. This observation is consistent with the trends found in high resolution 

inviscid flow computations carried out with increasing order of accuracy and mesh 

resolution using DG [82] and WENO discretizations [12]. The numerical results 

presented are in very good agreement with the experimental results given in [34], 

where the shadowgraph at t = 128µs shown in Fig. 7.50: 

Figure 7.50: Shadowgraph for the Schardin’s problem at t = 128µs. 

compares very well with the numerical results depicted previously in Fig. 7.49, 

and the numerical results obtained by a high fidelity numerical simulation for this 

problem performed by Chaudhuri et al. [35] obtained using a WENO scheme on a 8.4 

million points mesh. However, the P 2 results in Fig. 7.49 which were obtained with 

the 285000 element mesh required approximately 20 CPU hours on 50 processors 

in contrast to the results of [35] that required approximately 500 CPU hours using 

20-40 processors. 

In order to demonstrate that the accuracy is preserved in smooth regions of the 

flow field for the Schardin’s problem with the present shock capturing approach, a


grid and p-refinement study was performed on the triangular element mesh. Specifically, 

at the region behind the triangle where the vortex is formed, a refinement of 

the mesh was performed. The highly refined mesh and the computed Schlieren with 

P 2 expansion is shown in Fig. 7.51. Clearly, the shear layer is captured with 4 cells, 

while the shocks are captured with two cells. 

Figure 7.51: Numerical Schlieren for the Shardin’s problem on the finest mesh and 

using P 2 solution expansion. 

The first mesh refinement performed using an element size equal to h 1 = 0.001 

in the refined mesh region, while for the rest of the domain the size of the elements 

was h = 0.002 as before. Numerical solutions were obtained using P 1 and P 2 

solution expansions. The numerical Schlieren at time t = 128µs is shown in Fig. 

7.52, for both expansions, and it is apparent that a comparison between Figs. 7.52 

and 7.49(a) reveals that the solution on the refined mesh has resolved better the 

vortex formation behind the triangle. Furthermore, a solution was obtained with a 

more refined mesh having an element size of h 2 = 0.0005 in the region behind the 

triangle. As in the previous mesh refinement study, two numerical solutions were 

obtained using P 1 and P 2 solution expansion. The numerical Schlieren at a time 

t = 128µs is shown in Fig. 7.53 where it is observed that both solution expansions 

have resolved even finer and more complex flow structures.


(a) P 1 expansion. 

(b) P 2 expansion. 

Figure 7.52: Numerical Schlieren for Schardin’s problem on the refined triangular 

mesh with mesh element size in the vortex region equal to h 1 = 0.001.


(a) P 1 expansion. 

(b) P 2 expansion. 

Figure 7.53: Numerical Schlieren for Schardin’s problem on the refined triangular 

mesh with mesh element size in the vortex region equal to h 2 = 0.0005. 

7.7.1 Strong Mach 5 shock impingement on a triangle 

In this test case the same geometry employed in the computations for the standard 

Schardin’s problem is used, but the IC correspond to a right moving shock at a Mach


number of 5.09. The triangular mesh has been used but with a P 2 expansion of the 

solution as higher gradients in the unsteady flow behind the triangle were expected. 

Furthermore, the positivity limiters proposed in [156] were employed, otherwise the 

computations were not able to advance in time long enough due to the appearance 

of negative density and pressure. 

The computed density field is shown in Fig. 7.54. In Fig. 7.55, the elements 

where the solution was limited along with the computed pressure gradient gray scale 

contours are shown for three characteristic non dimensional times. The limited 

elements are marked with blue color, for elements neighboring limited elements P 1 

approximation is used and these elements are marked with green color. For the rest 

of the domain, marked with red color, P 2 approximation is used. It is observed that 

limiting has been performed mostly at the elements where the impinging shocks 

(detected by pressure gradients) as well as through elements, which the reflected 

shocks are crossing. Moreover, in Fig. 7.55 the numerical Schlieren is shown where 

it is evident that limiting has not been performed over the elements where the 

contact discontinuities are formed.


(a) t = 0.06 

(b) t = 0.08 

(c) t = 0.1 

Figure 7.54: Density contours using a P 2 expansion for the flow of a strong shock 

moving at Mach number of 5 around a triangle with 30 equal spaced intervals from 

0.28 to 13.28. Results are depicted at non-dimensional time.


(a) t = 0.06 

(b) t = 0.08 

(c) t = 0.1 

Figure 7.55: Limited elements for the flow of a strong shock moving at Mach number 

of 5 around a triangle along with the gradient of pressure over 30 equally spaced intervals 

from 0 to 1000. Results are depicted at non-dimensional time. Blue elements: 

limited solution. Green elements P 1 expansion. Red elements P 2 expansion.


7.8 Flat plate boundary layer 

The viscous flow over a flat plate with no pressure gradient at M = 0.2 and Re = 10 4 

is considered to further validate the viscous calculation with HoAc by comparing the 

velocity profiles for a P 1 , P 2 and a P 3 solution expansion. The free stream conditions 

are: [ρ, u, v, p] = [1, 0.2, 0, 0.714285714] with γ = 1 and the computational domain 

4 

has dimensions of [−2, 2] × [0, 10] with half of the lower horizontal line representing 

a symmetry boundary and the other half an adiabatic wall. At x = −2 the flow 

velocities and pressure are specified, while the density is extrapolated from the 

interior. At x = 2 and y = 10 all the computed variables are extrapolated except 

for the pressure which is specified at its mean stream value. A discontinuous output 

of the solution is employed as a relatively coarse mesh is used for the computations. 

Specifically, quadrilateral elements are used with a height of h = 10 −3 for the first 

layer of elements and a stretching factor of 1.2 in the direction normal to the wall. 

In Figs. 7.56 and 7.57 the computed velocity profiles are shown with comparison 

to the Blasius solution. It is observed that the P 1 computation hardly agrees with 

the Blasius solution especially for the V-velocity profile. But, the P 2 computation 

is close to analytical solution and the P 3 computation achieves the best resolution 

for the boundary layer.


Figure 7.56: Comparison of the U-velocity profile with the Blasius solution for the 

flow over a flat plate. 

Figure 7.57: Comparison of the V-velocity profile with the Blasius solution for the 

flow over a flat plate.


The velocity field vectors along with the vorticity field for the P 3 computation 

is depicted in Fig. 7.58, where it is observed that the boundary layer thickness 

grows along the plate and the velocity profiles are nicely resolved by the numerical 

solution. 

Figure 7.58: Velocity field vectors and vorticity field for the flat plate boundary 

layer solution using a P 3 solution expansion. 

7.9 Unsteady viscous flow over tandem airfoils 

This problem was a test case that was solved and presented in the first International 

Workshop on High Order CFD Methods [89]. The geometry of the problem consists 

of two tandem NACA 0012 airfoils with the forward airfoil at 10 ◦ angle of attack. 

The free stream flow conditions correspond to a Mach number M = 0.2 and a 

Reynolds number Re = 10 4 . The IC applied to this test case correspond to a C 1 

initial solution for the velocity field: 

⎧ 

√ γp∞ 

⎨ 

V = (u, v) = M ∞ (1, 0) ( 

ρ ∞ ⎩ sin 

) 

πd 

2δ 1 

1, if d > δ 1 , 

, if d ≤ δ 1 , 

(7.6) 

with δ 1 = 0.05 and d being the distance to the closest wall with the density and 

pressure initialed to their free stream values. A mixed-type linear element mesh 

was generated around the geometry of the airfoils with quadrilateral elements in the 

region close to the airfoils in order to achieve better resolution for the boundary layer.


In Fig. 7.59 the visualization mesh for a P 4 solution expansion over quadrilateral 

and triangular elements is shown. 

Figure 7.59: Visualization mesh for a P 4 computation for the two tandem NACA 

0012 airfoils. 

For this specific problem a sensitivity study regarding the far field boundary 

distance was carried out first in order to resolve the lift and drag coefficients within 

0.01 counts using a P 2 solution expansion over each element. The far field boundary 

was formed by a circle enclosing the airfoils. Three simulations were performed in 

order to examine the effect of the far field distance from the airfoils. These corresponded 

to the following radii of the circles: R = 25, R = 50, R = 100 times 

the chord length of the airfoil, which was assumed equal to unity. The sensitivity 

study was performed using an explicit third-order SSP Runge-Kutta scheme with 16 

stages up to t = 100, using a time step size of dt = 1.0e − 3, corresponding to nondimensional 

units. The cost of the computations is expressed in work units defined 

by comparing the execution time T 2 , excluding the initialization, post-processing 

data preparation time and file I/O time, to TauBench time T 1 , which is an unstructured 

grid benchmark. The respective kernels are derived from Tau - a Navier-Stokes 

solver, which has been developed at DLR in Germany.


Work units = T 2 

T 1 

. 

In Table 7.1 the work units, number of elements, number of DOF and the L 2 error 

for the whole time of the simulation for the aerodynamic coefficients for both airfoils 

are given. All the simulations were performed using 8 cores. From the results given 

in Table 7.1, the distance of the far field for the time accurate simulations was chosen 

to be R = 50. 

R Elem Work Units DOF C l forward C d forward C l aft C d aft 

25 2708 1163.49 146496 – – – – 

50 2788 1312.3 150336 5.4880e-7 1.0240e-7 2.1443e-6 1.6780e-7 

100 2914 1649.78 156384 6.8725e-8 1.0516e-8 1.2237e-7 3.0579e-8 

Table 7.1: Results for the sensitivity study regarding the far field boundary distance. 

Furthermore, a p-refinement study was performed using a P 2 , P 3 and P 4 

solution expansion for a time accurate computation using an explicit third order 

SSP Runge–Kutta method with 16 stages, provided by PETSc, and a small step 

size equal to dt = 5.0e − 4 for a simulation time equal to t = 100. The time history 

for the lift and drag coefficients for the forward and the aft airfoil are given in Figs. 

7.60 and 7.61, respectively.


(a) Lift coefficient t = 0 − 100 (b) Lift coefficient t = 40 − 100 

(c) Drag coefficient t = 0 − 100 (d) Drag coefficient t = 40 − 100 

Figure 7.60: Lift and drag coefficients for the forward airfoil.


(a) Lift coefficient t = 0 − 100 (b) Lift coefficient t = 80 − 100 

(c) Drag coefficient t = 0 − 100 (d) Drag coefficient t = 80 − 100 

Figure 7.61: Lift and drag coefficients for the aft airfoil. 

It is observed that a time periodic response appears to develop for both airfoils after 

time t = 80. In Figs. 7.62 and 7.63 the velocity and vorticity contours at t = 80 are 

shown for a P 2 and a P 4 solution expansion using a discontinuous output, where 

the difference between the resolved flow fields is apparent.


(a) P 2 Solution 

(b) P 4 Solution 

Figure 7.62: Velocity contours for the flow over NACA0012 tandem airfoils. 

(a) P 2 Solution 

(b) P 4 Solution 

Figure 7.63: Vorticity contours for the flow over NACA0012 tandem airfoils.

Chapter 8 

Conclusions and Future Work 

The present work developed and evaluated a priori indicators of mesh distortion. 

This was accomplished by deriving the complete truncation error expression for the 

FV discretization of the field gradient in two dimensions. The complexity of the 

mathematical operations involved was overcome via use of symbolic mathematics 

software. 

The presented work analyzed a generally-distorted mesh (structured, unstructured 

and mixed-type element) into elementary distortions (stretching, skewness, 

shearing, rotation), as well as three common types of interfaces, and those were directly 

related to the TE. The derived analytic expressions are relatively simple and 

amenable to future work on improving the mesh. This is especially true for index 

Q for which analytic expressions were presented directly relating it to the degree of 

stretching and skewness, as well as to mixed-type element mesh interfaces. 

The two indices (Q and q) “followed” the grid distortion quite faithfully with 

Q being more focused on these local areas. Index q marks broader areas of the mesh 

due to the second order error terms included in its definition. It also picks-up small 

(background) distortions which may not be of interest. Given the fact that Q is 

quite simpler to calculate, it is the recommended quality index. A disadvantage of 

Q is that it cannot indicate the quality of distorted meshes that still yield second 

order accurate computation of the gradient. Shearing and rotation do change the 

197

CHAPTER 8. CONCLUSIONS AND FUTURE WORK 198 

TE (i.e. only index q “picks them” up) but they do not reduce the order of accuracy. 

However, the goal is to fix meshes that degrade the accuracy to a lower order. 

The following observations were also made for both indices: (i) irregularlyshaped 

small distortions were captured faithfully, (ii) small distortions were detected 

even though much larger ones existed in the mesh, and (iii) distortions were captured 

in any direction. 

Employment of the centroid dual discretization must be avoided on general 

structured and mixed-type element meshes due to its inconsistency in the evaluation 

of the field gradient. 

Future work involves two directions: (i) working the analytical expressions for 

the three-dimensional case for mixed-type element (structured/unstructured) grids, 

and (ii) using the analytic expressions of the present work for a priori improvement 

of the mesh. 

In the present work a CFD solver was developed and named HoAc (an abbreviation 

for High order Accuracy) based on the DG discretization. The code was 

verified and its ability to achieve the design order of the DG discretization for up 

to fifth order accurate expansion of the solution for the Euler and Navier-Stokes 

equations was demonstrated. Furthermore, in the examined cases the ability of the 

HoAc solver to provide very accurate solutions on problems with complex flow 

features was demonstrated. 

The HoAc solver has the ability to perform parallel computations via the 

domain decomposition method on super computers and massively parallel systems 

using the MPI programming paradigm. It can handle arbitrary mixed-type element 

meshes over arbitrary geometries in two dimensions and solve unsteady problems 

for the Euler and Navier-Stokes equations with the aid of high order explicit and 

implicit time marching schemes. This was mainly achieved by the fact that all the 

operations for the DG discretization in HoAc are being performed in the standard 

square element configuration where the expansion bases are defined and constructed. 

Also, the capability to produce a high order discontinuous output of the solution has 

been implemented, which appeared to be fairly adequate for coarse meshes when a


high order approximation of the solution is employed. 

Furthermore, in the present work the shock capturing capability of the DG 

discretization was addressed and improved with a new limiting approach which 

is suitable for mixed-type meshes. A shock capturing p-adaptive DG scheme was 

developed and implemented in HoAc. In the new limiting approach, limiting is performed 

with a slight but essential modification of a TVB limiter extensively tested 

for DG discretizations. Limiting is carried out in the standard computational space 

where all types of elements transform to standard squares. Application of the new 

limiting approach for standard test cases for shock capturing schemes demonstrated 

that the new approach is robust, accurate, and it does not require adjustment of 

parameters. Moreover, it provides capabilities for p-adaptation because the solution 

accuracy could be increased away from regions of discontinuities that are clearly 

marked since accurate estimates of the second derivative are obtained. In addition, 

the developed limiting algorithm was combined with new positivity preserving limiters 

especially developed for the DG method and test cases involving high speed 

flows with strong shocks were examined with exceptional results. Specifically, these 

cases were solved using unstructured triangular meshes and up to the time of the 

present work nowhere in the literature the positivity preserving limiters appeared 

to be applied on triangular meshes. 

Future work in the DG discretization and implementation of the current thesis 

will involve: (i) the extension of the developed limiting approach in three dimensions, 

both for shock capturing and positivity preservation for density and pressure, (ii) 

the ability of the HoAc solver to work on meshes with curved elements, (iii) the 

extension of the HoAc solver to handle three dimensional flows around complex 

geometries using mixed-type element meshes with curved elements, (iv) the ability 

to perform high order DG simulations for turbulent flows and (v) the extension 

of the current implementation of HoAc for its use with Graphics Processor Units 

(GPUs).

APPENDIX A 

Mesh quality index Q x for the stretched unstructured mesh of Fig. 3.5(b): 

Q x = 96 

7 

dx 

∣|1 + 2dx| (1 + 2dx) + 4(1 + dx 2 ) + |2dx − 1| (1 − 2dx) ∣ + 

∣ 32√ ∣∣∣ 

dy 

3 

7 (2dy − √ 3)[|2dy − √ 3| + | √ 3 + 2dy|] + 2dy 

∣ . 

(A-1) 

Mesh quality index Q x for the skewed mesh of Fig. 3.6: 

Q x = 1 − cos2 (ω) 

2[cos 2 (ω) + 1] + cos (ω) − 1 

cos 2 (ω) − cos (ω) − 4 − 

3 sin (2ω) 

2[cos (ω) + 1][cos 2 (ω) − cos (ω) − 4] . 

(A-2) 

200

APPENDIX B 

Discontinuous Output for Quadrilateral Elements 

For an approximation order N, a quadrilateral element is subdivided into (N + 1) 2 

quadrilateral sub-elements. This element division is based on a listing Q q of equally 

spaced points according to Eq. (B-1): 

Q q = p + q(N + 2), 0 ≤ p, q ≤ N + 1, (B-1) 

over the standard square element configuration. Each point then defines a node 

whose coordinates in the physical space are computed by Eq. (4.29). Then the 

connectivity of the quadrilateral sub-elements is defined according to a structured 

mesh. Finally, the solution is projected according to Eq. (4.5) and printed at every 

new node. This leads to a print out of the solution that is discontinuous across the 

inter-element boundaries. 

Discontinuous Output for Triangular elements 

As for a quadrilateral element a triangular element is subdivided into (N + 1) 2 

triangular sub-elements whose nodes are based on a listing Q t of equally spaced 

points over the standard triangular element configuration given in the following 

equation: 

201


Q t = p + (N + 2)q − 

q(q − 1) 

, 0 ≤ p, q ≤ N with p + q ≤ N. (B-2) 

2 

The physical coordinates of the nodes are computed from Eq. (4.31) and the following 

pseudo code is provided for the connectivity list of the triangular sub-elements: 

m=0; 

for (i=0; i< # elements; i++) { // loop elements 

for (q = 0; q < N+1; q++) 

for (p = 0; p < N+1; p++) 

if (p + q < N + 1) { // if in tria 

print(Q_t(p, q) + m, Q_t(p+1, q, sys) + m, Q_t(p,q + 1, sys) + m); 

if (p + q + 2 < N + 2) 

print(Q_t(p + 1, q) + m, Q_t(p + 1, q + 1) + m, Q_t(p, q + 1) + m); 

} // if in tria 

m += (N + 2) * (N + 3) / 2; 

} // loop elments 

As for the quadrilateral elements, the solution is projected according to Eq. (4.5) 

and printed at every new node resulting in a discontinuous solution print.

APPENDIX C 

Convergence analysis 

The discretization error is defined as: 

Du = u n − u = O (h p ) , 

(C-1) 

where p is the order of the discretization and u n and u is the numerical solution 

obtained with a discretization size h and the exact solution respectively. 

The exact solution is necessary in order to compute the order of the discretization, 

but this is limited to cases that can be solved analytically. The formal order p may 

be determined by comparing two discretization errors with differenet sizes h 1 and 

h 2 , as follows: 

D 1 

which after applying logarithms yields: 

= hp 1 

, (C-2) 

D 2 

h p 2 

and p may be calculated as: 

ln D 1 

= ln hp 1 

, (C-3) 

D 2 

h p 2 

p = − ln D 1 − ln D 2 

ln h 1 − ln h 2 

. 

(C-4) 

203


In Eq. (C-4) scalar error values are needed, which are introduced as integral 

values of a norm ||∆u|| p of the solution error over each element: 

||∆u|| p = 1 ∑ 

(∫ 

N e 

n 

Ω e 

[Du] p dΩ e 

) 1/p 

, 1 ≤ n ≤ N e . (C-5) 

where N e is the total number of elements used in the mesh.

APPENDIX D 

HoAc run time options 

HoAc relies on PETSc for parallel computations and time marching of the solution. 

So, any run time option referring to PETSc is readily applied to HoAc as well. A 

sample input file for HoAc is given below: 

*CONTROL_EQUATION 

/*[0,1]=[Euler,NS]*/ 

1 

*END 

*CONTROL_FLOWCOND 

/*gamma*/ /*P_inf*/ /*U_inf*/ /*V_inf*/ /*rho_inf*/ /*C_inf*/ 

1.400000e+0 7.142857e-01 0.2 0.000000e+00 1.000000e+00 

1.000000e+00 

*END 

*CONTROL_SOLUTION 

/*n_out*/ /*PN*/ /*[0,1]=[start,restart]*/ /*[0,1]=[Roe,LLF]*/ /* # parts*/ 

5000 1 0 1 16 

*END 

*CONTROL_TIME 

/*End time */ /*Problem type */ /* Output type 

[0,1]=[continuous,discontinuous]*/ 

1.0 0 1 

*END 

205


*CONTROL_BOUNDARY_WALL 

/*[0,1]=[Straight,CBC]*/ 

0 

*END 

*CONTROL_NAVIER_STOKES 

/*Re*/ /*Pr*/ /*T_inf*/ /*LDG coefficient*/ 

100.000000 7.200000e-01 1.0 0.5 

*END 

*CONTROL_BOUNDARY_IN_SPEC 

/*rho_spec_in*/ /*u_spec_in*/ /*v_spec_in*/ /*p_spec_in*/ 

0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 

*END 

*CONTROL_BOUNDARY_OUT_SPEC 

/*rho_spec_out*/ /*u_spec_out*/ /*v_spec_out*/ /*p_spec_out*/ 

0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 

*END 

where form the comments provided in, the user is easy to check what a value means. 

Furthermore, in order to give more flexibility to the user additional run time options 

are provided which handle the DG discretization in HoAc . These are given in the 

following sample options file along with other PETSc commands for an implicit or 

explicit run, where it is noted that the comments should not exist when a simulation 

is to be performed. 

-llf_flux %% for Local Lax Friedrichs 

-roe_flux %% for Roe flux 

-explicit %% for an expicit run 

-explicit_type 0 %% For using the SSP RK develop by Shu and Osher 

-explicit_type 1 %% For using the SSP RK provided by PETSc 

-gl %% for Gauss-Legendre quadrature 

-gll %% for Gauss-Lobatto-Legendre quadrature 

-ts_type ssp


-ts_ssp_type rks3 

-ts_ssp_nstages 9 

-ts_dt 1.0e-4 

-dt 1.0e-4 %% if explicit_type is 0 

-order 1 %% Order of approximation 

-parts 16 %% Number of partitions 

-end_time 2000.0 %% End time of simulation 

-n_out 50000 %% Output frequency 

-shock_lim %% Limiter for shock capturing 

-pos_lim %% Limiters for positivity preservation 

-restart %% Switch for restarting a solution 

-snes_monitor 

-snes_atol 1.0e-8 

-snes_rtol 1.0e-10 

-snes_mf 

-ksp_type fgmres 

-pc_type none 

-mat_mffd_compute_normu 

-ksp_max_it 30 

-ksp_gmres_restart 200 

HoAc : 

At the command line the user then types the following command for running 

mpiexec -n {# parts} ./hoac {input file} -options_file {file_name}

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