Development of a High-Order Finite-Volume Method for ... · Abstract Development of a High-Order Finite-Volume Method for Unstructured Meshes Sean McDonald Masters of Applied Science

Development of a High-Order Finite-VolumeMethod for Unstructured Meshes

by

Sean McDonald

A thesis submitted in conformity with the requirementsfor the degree of Masters of Applied Science

Graduate Department of Aerospace EngineeringUniversity of Toronto

Copyright © 2011 by Sean McDonald

Abstract

Development of a High-Order Finite-Volume Method forUnstructured Meshes

Sean McDonald

Masters of Applied Science

Graduate Department of Aerospace Engineering

University of Toronto

2011

The development of high-order methods remains a very active field of research in Computational

Fluid Dynamics (CFD). These types of schemes have the potential to reduce the computational

cost necessary to compute solutions to a desired level of accuracy. The goal of this thesis has

been to develop a high-order Central Essentially Non Oscillatory (CENO) finite-volume scheme

for multi-block unstructured meshes. In particular, solution to the compressible, inviscid Eu-

ler Equations are considered. The CENO method achieves a high-order spatial reconstruction

based on the k-exact method, combined with hybrid switching to limited piecewise linear recon-

struction in non-smooth regions to maintain monotonicity. Additionally, fourth-order Runge-

Kutta time marching is applied. The solver described has been validated through a combination

of high-order function reconstructions, and solutions to the Euler equations. Cases have been

selected to demonstrate high-orders of convergence, the application of the hybrid switching

method, comparisons with structured solver, and the multi-block techniques which have been

implemented.

ii

Acknowledgements

I would like to thank a number of people without whom the completion of this thesis would

not have been possible.

First, I would like to thank my family. My accomplishments with this work could not have

been possible without the loving support of my parents. Furthermore, any achievement that I

have made throughout my life is strong a reflection on them and the encouragement they have

given me, for that I am forever grateful. I would also like to express my deepest gratitude to

my brother James and sister Lindsay.

I have been very fortunate to be working under the insightful guidance of Professor C. P.

T. Groth throughout this project. His enthusiasm and expert advice has been a significant

contribution and driving force throughout this research.

I would like to thank the Natural Sciences and Engineering Research Council of Canada, the

University of Toronto, and Professor Groth for their financial support throughout this project.

I would also like to thank my colleagues at The University of Toronto Institute for Aerospace

Studies Computational Fluid Dynamics laboratory for their invaluable assistance and friendship

throughout my Masters’ degree.

Finally, I would like to express my profound appreciation to Jennifer DeWit for her encourage-

ment, understanding, and patience. She has given me the extra strength, motivation and love

necessary throughout my degree.

Toronto, December 2010 Sean McDonald

iii

Contents

Abstract ii

Acknowledgments iii

Contents iv

List of Tables viii

List of Figures ix

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Godunov-Type Finite-Volume Methods 6

2.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Navier-Stokes Equations in Cartesian Coordinate Frame . . . . . . . . . . 7

2.1.2 Euler Equations in Cartesian Coordinate Frame . . . . . . . . . . . . . . 8

2.1.3 Thermodynamic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Euler Equations in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 10

iv

2.3 Godunov-Type Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Concept of Finite-Volume Methods . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Flux Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.2 Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.3 Comparison of Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Second-Order Godunov Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.1 Godunov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.2 Piecewise Linear Reconstruction . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.3 Enforcing Monotonicity: Slope Limiters . . . . . . . . . . . . . . . . . . . 19

2.6 Time Marching Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6.1 Semi-Discrete Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.2 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.3 Courant Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Godunov-Type Finite-Volume Schemes in

Multi-Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7.1 Multi-Dimensional Example . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Unstructured Mesh 27

3.1 Unstructured Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Gmsh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.2 Geometry Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.3 Mesh Generation Module . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.4 Mesh Generation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.5 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Domain Decomposition and Multi-Block Partitioning . . . . . . . . . . . . . . . . 35

v

3.2.1 PMETIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Finite-Volume on Unstructured Mesh . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.1 Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.2 Finite-Volume Example on Unstructured Mesh . . . . . . . . . . . . . . . 38

4 High-Order CENO Scheme 40

4.1 Previous High-Order Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Overview of the CENO Scheme . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 k-Exact Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.1 Conditions on Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Determination of Reconstruction Stencil . . . . . . . . . . . . . . . . . . . 46

4.2.3 Evaluation of the Derivatives for the Reconstruction . . . . . . . . . . . . 48

4.3 CENO Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.1 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2 Smoothness Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.3 Monotonicity Enforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.4 Flux Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Multi-block Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 High-Order Treatment for Curved Boundaries . . . . . . . . . . . . . . . . . . . . 57

4.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Numerical Results 60

5.1 Theory for Evaluating High-Order Algorithms . . . . . . . . . . . . . . . . . . . . 61

5.1.1 Determining the Order of Accuracy . . . . . . . . . . . . . . . . . . . . . 61

5.1.2 Accuracy Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

vi

5.2 Reconstruction of Two-Dimensional Functions . . . . . . . . . . . . . . . . . . . . 62

5.2.1 Spherical Cosine Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.2 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.3 Abgrall’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Solution of Two-Dimensional Euler Equations . . . . . . . . . . . . . . . . . . . . 71

5.3.1 Unsteady Shock-Box Problem . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.2 Unsteady Cylindrical Expansion . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.3 Steady Supersonic Flow Over a Bump . . . . . . . . . . . . . . . . . . . . 75

6 Conclusions and Future Research 81

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

References 90

vii

List of Tables

3.1 Comparison of Meshing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Example of Edge Type Data Structure . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1 Orders of accuracy for the Spherical Cosine Function. . . . . . . . . . . . . . . . 65

5.2 Orders of accuracy for the Exponential Function. . . . . . . . . . . . . . . . . . . 66

5.3 Orders of accuracy for the Abgrall’s Function. . . . . . . . . . . . . . . . . . . . . 71

5.4 Orders of accuracy for the Cylindrical Expansion. . . . . . . . . . . . . . . . . . . 74

viii

List of Figures

2.1 Depiction of the Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Depiction of five possible solutions to the Riemann Problem - s = shock, c =

contact, r = rarefaction [42] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Depiction of integration loop for HLLE flux function . . . . . . . . . . . . . . . . 15

2.4 Shock tube problem: n=100 cells, CFL=0.5, t=0.007 s (a) Various Riemann

solvers (b) Close-up of contact wave . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Shock tube problem: n=100 cells, CFL=0.5, t=0.007 s - Order of Accuracy effect

on solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Shock tube problem: n=100 cells, CFL=0.5, t=0.007 s - Slope limiter effect on

solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Finite-volume Shock-Box calculation: (a) Density - Initial Conditions (b) Density

- Results at 7 ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Two-dimensional geometry created using Gmsh . . . . . . . . . . . . . . . . . . . 29

3.2 Two-dimensional mesh generated using Gmsh . . . . . . . . . . . . . . . . . . . . 30

3.3 Visualization of Mesh Modifications [53] . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Hierarchical structure of triangle types [52] . . . . . . . . . . . . . . . . . . . . . 33

3.5 Various meshing algorithms: (a) Geometry (b) MeshAdapt (c) Delaunay (d)

Frontal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 PMETIS Partitioned Mesh showing: (a) Blocks (b) Blocks and Mesh . . . . . . 37

ix

3.7 Finite-Volume Shock-Box Calculation on Unstructured Mesh: (a) Density - Ini-

tial Conditions (b) Density - Results 7 ms . . . . . . . . . . . . . . . . . . . . . . 39

4.1 (a) Nearest and next to nearest neighbors on structured grid (b) Nearest and

next to nearest neighbors difficult to determine on unstructured grid . . . . . . . 47

4.2 (a) Nearest neighbors (b) Nearest and next to nearest neighbors (c) All neighbors 47

4.3 Graph of f(α) = α1−α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Quadrature points on a typical high-order unstructured cell . . . . . . . . . . . . 56

4.5 Ghost cells required for message passing . . . . . . . . . . . . . . . . . . . . . . . 57

4.6 Finite-Volume Shock-Box Calculation on Unstructured Mesh: (a) 2nd-order (b)

4th-order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1 Reconstruction of spherical cosine function. . . . . . . . . . . . . . . . . . . . . . 64

5.2 Convergence studies for spherical cosine function reconstruction. . . . . . . . . . 65

5.3 Reconstruction of an Exponential Function. . . . . . . . . . . . . . . . . . . . . . 67

5.4 Convergence studies for Exponential Function Reconstruction. . . . . . . . . . . 68

5.5 Reconstruction of Abgrall’s function. . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.6 Results for Abgrall’s function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.7 Unsteady Shock Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.8 Demonstration of the Smoothness Indicator. . . . . . . . . . . . . . . . . . . . . . 77

5.9 Density variation on a structured mesh. . . . . . . . . . . . . . . . . . . . . . . . 77

5.10 Unsteady Cylindrical Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.11 L1 Error for the Cylindrical Expansion . . . . . . . . . . . . . . . . . . . . . . . . 79

5.12 Flow over a bump. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

x

Chapter 1

Introduction

Numerous engineering applications rely on a detailed understanding of fluid flows. These appli-

cations cross a broad range of industries including aerospace, energy, medical, and automotive.

Therefore, an understanding into the fundamentals of fluid dynamics has the potential to impact

a broad range of industries, having a tremendously positive impact on society.

One of the capabilities offering excellent potential for an increased understanding of fluid dy-

namics is the development of Computational Fluid Dynamics (CFD) tools to compute numerical

solutions to problems involving fluid flow [1, 2, 3, 4, 5]. These simulation tools can be very

useful by assisting to improve designs of engineering machines and systems that involve flu-

ids. The ultimate goal of CFD is to understand the underlying physical principals when a

fluid flows around and within a specified domain. This is done through systemized numerical

solution techniques applied to the partial differential equations governing the physics of fluid

flow [1, 2, 3, 4, 5].

The research outlined within this thesis serves as an advancement towards this goal of under-

standing the physics involved in fluid flow, by producing significantly more accurate simulations

tools for predicting fluid flow through complex geometry.

1.1 Motivation

CFD has proven to be an important enabling technology in many areas of science and engi-

neering. In spite of its relative maturity and widespread successes in aerospace engineering,

there remain a variety of physically-complex flows, which are still not well understood and have

proven to be very challenging to predict by numerical methods. Such flows would include but

1

Chapter 1. Introduction 2

are not limited to: (i) multiphase, turbulent, and combusting flows encountered in propulsion

systems (e.g., gas turbine engines and solid propellant rocket motors); (ii) compressible flows

of conducting fluids and plasmas; and (iii) micro-scale non-equilibrium flows, such as those

encountered in micro-electromechanical systems. These flows present numerical challenges for

they generally involve a wide range of complicated physical/chemical phenomena and scales.

Considering the physically-complex example of simulating turbulent reacting flows, typical of

many engineering applications including internal combustion engines. Tools for their accurate

simulation are of both practical and theoretical interest. Current tools for the simulation of

these flows include Direct Numerical Simulation (DNS) methods [6, 7, 8] and Large Eddie

Simulation (LES) methods [9, 10, 11]. These methods have proven to be computationally

expensive, and current low-order methods (i.e., methods of second-order accuracy or less) can

exhibit excessive numerical error [12]. The numerical error that these low-order methods exhibit

is both dispersive and dissipative.

The potential of high-order methods to reduce the cost of simulations for these physically-

complex flows has been recognized, but this potential has not always been fully realized. For

this reason, development of effective high-order methods remains an active area of research.

Improved numerical efficiency may be achieved by raising the order of accuracy of the spatial

discretization, thereby reducing the number of computational cells required to achieve the

desired accuracy. For hyperbolic conservation laws and/or compressible flow simulations, the

challenge has been to achieve accurate discretizations while coping in a reliable and robust

fashion with discontinuities and shocks [13]. However, the overall benefit of high-order methods

could be a significant decrease in the overall computational cost and memory requirements to

obtain a solution for a desired or specified level of accuracy.

A decrease in overall computational cost has the potential to allow large scale computations

to be carried out on a more routine basis. The computations targeted here could include, but

are not limited to DNS and LES of turbulent non-reactive and reactive flows, numerical com-

putations of complex unsteady aerodynamic flows, aeroacoustic modelling, and computational

electromagnetics [12]. As these computations become more routine, CFD codes will become

more useful in developing designs of machinery involving fluid flow. Computer simulations

can be conducted in place of building and testing prototypes, reducing cost and development

time. As an example, this could lead to improved combustor designs capable of meeting power

generation requirements while responding to increasing environmental concerns and regulations.


1.2 Background

As high-order methods have been an active area of research now for several decades, various

CFD techniques have been applied to accomplish the goal of decreasing in the overall compu-

tational cost and memory requirements to obtain a solution for a desired or specified level of

accuracy. High-order finite-volumes schemes proposed in the past include the essentially non-

oscillatory (ENO) scheme by Harten et al. [13], the weighted essentially non-oscillatory (WENO)

schemes [14, 15, 16], central-essentially non-oscillatory (CENO) schemes [12, 17, 18, 19], Dis-

continuous Galerkin (DG) schemes [20], and Finite-Difference (FD) schemes [21, 22]. A major

challenge in high-order schemes is employing a method at the reconstruction stage to preserve

monotonicity.

Consider the ENO, WENO, and CENO schemes, all from a similar family of high-order schemes,

the main difference being they each employ a different technique for preserving monotonicity.

The ENO scheme avoids the use of computational stencils that contain discontinuities to achieve

solution monotonicity [13, 23, 24]. This is accomplished by reconstructing the solution on several

stencils and selecting the smoothest. The WENO scheme applies a solution-dependent weighting

to incorporate and appropriately select all of the stencils computed in the reconstruction [14, 15,

16]. Finally, the CENO scheme locally reduces the order of the reconstruction to a second-order

limited linear piecewise reconstruction where necessary [12, 17, 18, 19].

The application of the ENO scheme to unstructured mesh was first conducted by Abgrall [25],

and further investigations have also been considered [23, 26]. Additionally, WENO schemes

have also been applied on unstructured meshes [27]. Both ENO and WENO variants encounter

difficulties for general multi-dimensional unstructured meshes when selecting appropriate sten-

cils [25, 28, 29, 27] and they can result in poor conditioning of the linear systems that are

involved in the solution reconstruction [29, 27]. These, along with the associated computa-

tional cost and complexities involved in the ENO and WENO finite-volume schemes, have

limited the range of applications of these high-order methods.

The recently proposed CENO scheme of Ivan and Groth [17, 30] attempts to deal with the

somewhat restrictive computational issues associated with ENO and WENO schemes. The

CENO scheme avoids the complexity associated with the other schemes by using a fixed central

stencil. A central stencil in general provides the most accurate reconstruction due to cancella-

tion of truncation errors. This hybrid scheme first performs an unlimited, high-order, k-exact

least-squares reconstruction technique [31], and then automatically reverts to a monotonicity

preserving limited, piecewise, linear, least-squares reconstruction procedure in cells near shocks


or with under-resolved solution content. The switching is controlled by a solution smooth-

ness indicator. The CENO reconstruction is effective in eliminating the appearance of O(1)

numerical oscillations in under-resolved regions and in solution that contain strong discontinu-

ities and/or shocks. Although uniform accuracy is not achieved for non-smooth solutions, the

method would seem readily extendable to both multi-dimensions and unstructured mesh. The

ability to formulate a high-order scheme on unstructured computational mesh in a straightfor-

ward manner is particularly appealing due to the ready applicability of such mesh to practical,

complex, three-dimensional, flow-field geometries.

The CENO finite-volume scheme has been developed by Ivan and Groth [12, 17, 18, 19] for

application to solutions of the one- and two-dimensional Euler equations on structured meshes.

The extension to three-dimensional structured meshes and the Navier-Stokes equations has

been completed by Rashad [32]. An in-depth discussion of the CENO scheme which has been

implemented throughout this research can be found in Chapter 4.

1.3 Objectives

The primary objective of this research has been to develop and verify a higher-order CENO

finite-volume method capable of obtaining accurate solutions on unstructured meshes. In par-

ticular, cell-centered CENO schemes are examined for the solution of the conservation equations

of inviscid, compressible, gas dynamics on multi-block, unstructured meshes in two dimensions

using triangular computational cells. As noted above, the hybrid CENO approach avoids the

complexity associated with other ENO and WENO schemes that require reconstruction on mul-

tiple stencils, which in some cases can produce poorly conditioned coefficient matrices, and is

therefore well suited for solution reconstruction on unstructured mesh.

Verification of the proposed high-order CENO method for unstructured grids has been achieved

through comparison of results to other numerical solutions calculated using the existing high-

order structured mesh solution method. Additionally, the accuracy of reconstructed solutions

for arbitrary functions on unstructured mesh is examined and the influence of mesh resolution

on accuracy is assessed for several two-dimensional flow problems. The ability of the scheme

to accurately represent solutions with smooth extrema and yet robustly handle under-resolved

and/or non-smooth solution content (i.e., solutions with shocks and discontinuities) is demon-

strated.

The extension of the CENO finite-volume scheme to unstructured meshes is a logical step, as

experience with structured mesh solvers indicates that higher-order accurate approximations


can significantly improve the quality of numerical solutions [31]. The long-term application of

this research could be for the accurate solution of turbulent reactive flows using LES methods on

hybrid meshes (body-fitted structured and unstructured mesh) with adaptive mesh refinement.

In such an approach, quadrilateral elements would be used in the vicinity of solid boundaries,

and unstructured triangular elements elsewhere to facilitate greater automation in the mesh

generation procedure.

1.4 Overview of Thesis

This thesis provides a full description of the proposed high-order CENO finite-volume method

that has been implemented on two-dimensional general unstructured meshes. The computer

code developed has been thoroughly verified through comparison with existing numerical results

to good agreement.

Chapter 2 provides an outline of Godunov-type finite-volume methods which have been used

for this research. The purpose is to provide a background understanding of the methods used

in the high-order scheme. Additionally, the equations of interest for this research are presented.

Chapter 3 outlines information pertaining to unstructured meshes and general algorithms that

are used for their creation. Information is presented related to the specific tool that was used for

meshing, and the unstructured mesh data structure used within the code. Chapter 4 presents

information regarding the CENO finite-volume formulation, along with an in-depth description

of the k-exact reconstruction technique on which it is based. Chapter 5 shows the numerical

results that have been obtained through this work, and information regarding its validation

and discussion. Finally, chapter 6 outlines the overall conclusions of this thesis, and provides

details for continued work.

Chapter 2

Godunov-Type Finite-Volume

Methods

High-order extensions of Godunov-type finite-volume methods will be considered in this thesis

for solution of the Euler equations of compressible gas dynamics. This finite-volume method

discretizes the computational domain into many control volumes, then the integral form of

the conservation laws can be applied to each control volume. Finite-volume methods have

become very popular in CFD based on to two main advantages. First, they ensure that the

discretization is conservative, meaning that mass, momentum and energy will be conserved

discretely throughout the calculations. Second, a general finite-volume approach does not

require a coordinate transformation in order to be applied to irregular meshes, as is generally

required by other approaches, such as finite-difference methods [5]. These advantages make

finite-volume methods very suitable for the current research applying a high-order solution

technique to unstructured meshes.

Godunov-type methods are also particularly suitable to this research as they have been applied

successfully for the calculation of inviscid compressible flows described by the Euler equations

in numerous previous studies [33, 34, 35]. These methods are characterized by their robustness

and usefulness in computing flows with very complicated shock structures [36]. Godunov-type

finite-volume methods rely on the solution to the non-linear Riemann problem discussed in

section 2.4.1. It is worth noting that the exact solution to the Riemann problem requires an

iterative procedure, which can add unnecessary computational expense, particularly for systems

of equations beyond those of compressible gas dynamics. This expense can be overcome through

the use of approximate Riemann solvers which are explained in section 2.4.2.

6

Chapter 2. Godunov-Type Finite-Volume Methods 7

2.1 Conservation Equations

The system of equations conventionally used to describe fluid flow govern the conservation of

mass, momentum and energy of the fluid through a domain [5]. These equations are a coupled

set of non-linear partial differential equations (PDE’s), that describe the motion of a fluid

in space and time. This section first presents a mathematical description of this system of

equations, known as the Navier-Stokes equations. This is followed by the the description of

the Euler equations which govern inviscid fluid flow and are a simplification of their viscous

counterpart. It should be noted that while the Navier-Stokes equations are presented, it is the

Euler equations that are of particular interest to this work.

2.1.1 Navier-Stokes Equations in Cartesian Coordinate Frame

Using matrix-vector notation, which is the applicable form for use with finite-volume algorithms,

the integral form of the Navier-Stokes equations may be written as follows:

∫∫∫V

[∂U∂t

+∇ · (−→F −

−→Fv)

]dV = 0 , (2.1)

where U is the vector of conserved solution variables

U = [ρ ρu ρv ρw E]T , (2.2)

−→F = (F,G,H) is the inviscid flux dyad, with components

F =

ρu

ρu2 + p

ρuv

ρuw

u(E + p)

, G =

ρv

ρuv

ρv2 + p

ρvw

v(E + p)

, H =

ρw

ρuw

ρvw

ρw2 + p

w(E + p)

, (2.3)


and−→Fv = (Fv,Gv,Hv) is the viscous flux dyad, with components

Fv =

0

τxx

τxy

τxz

uτxx + vτxy + wτxz − qx

, Gv =

0

τxy

τyy

τyz

uτxy + vτyy + wτyz − qy

,

Hv =

0

τxz

τyz

τzz

uτxz + vτyz + wτzz − qz

.

(2.4)

In the above equations, u, v, and w are the velocity components in the x, y, and z directions,

respectively, p is the pressure, ρ is the mass density, E is the total specific energy given by

E = e + 12 |~u|

2, and e is the specific internal energy. Introducing the fluid viscosity, µ, for

an isotropic fluid and using Stokes’ hypothesis, the stress tensor terms (i.e., components of ~~τ)

appearing in the viscous flux dyad can be related by the constitutive equations, as follows [37]:

τxx =µ

3

(4∂u

∂x− 2

∂v

∂y− 2

∂w

∂z

), τxy = µ

(∂v

∂x+∂u

∂y

),

τyy =µ

3

(4∂v

∂y− 2

∂u

∂x− 2

∂w

∂z

), τxz = µ

(∂w

∂x+∂u

∂z

),

τzz =µ

3

(4∂w

∂z− 2

∂u

∂x− 2

∂v

∂y

), τyz = µ

(∂w

∂y+∂v

∂z

).

(2.5)

Finally, using Fourier’s law of heat conduction, and introducing the coefficient of thermal ex-

pansion, k, and the temperature, T , we define the components of the heat flux vector, q, as

qx = −k∂T∂x

, qy = −k∂T∂y

, qz = −k∂T∂z

. (2.6)

2.1.2 Euler Equations in Cartesian Coordinate Frame

The Navier-Stokes equations presented in Section 2.1.1 can be simplified by neglecting viscosity

and heat conduction. This simplification leads to the Euler equations which govern compressible

inviscid fluid flow. The equations are shown below in integral form, suitable for solution using

a finite-volume method, as equation 2.7:


∫∫∫V

[∂U∂t

+∇ ·−→F]dV = 0 , (2.7)

where the vector of conserved variables, U, and the inviscid flux dyad,−→F , are as given by

equations (2.2) and (2.3) above, respectively.

This thesis will focus on high-order solutions to the two-dimensional form of the Euler equations

on arbitrary unstructured meshes composed of triangles. The proposed high-order solution

methods are discussed further in Sections 3 and 4.

2.1.3 Thermodynamic Relations

The Euler equations discussed above must be closed by an equation of state that relates pressure,

density, and temperature. This work considers thermally and calorically perfect gases, and can

therefore be assumed they follow an ideal-gas equation of state given by equation 2.8,

p = ρRT , (2.8)

where R is the gas constant, p is the pressure, ρ is the density, and T is the temperature. The

specific energy e, the enthalpy h, and the speed of sound a can then be defined as

e =p

ρ(γ − 1), h =

γ

(γ − 1)p

ρ, a2 =

γp

ρ, (2.9)

where γ is the specific heat ratio for the gas of interest, taken to be 7/5 = 1.4 for air. These

equations makes the assumption that the average distances between molecules is so large that

the intermolecular potential energy may effectively be neglected. Therefore, the particles will

be independent from one another, and the situation obtained is referred to as an ideal gas [38].

Focusing on the two-dimensional Euler equations, as they are of particular interest to this

work, this thermodynamic relation allows for the mathematical description to be a system of

equations, with three independent variables (x, y and t). Additionally, in two-dimensions, there

are four unknown, dependent variables (ρ, u, v, and p or T ). Therefore, the Euler equations,

along with the thermodynamic closing equation provide a complete mathematical description

of the fluid dynamics that are considered for this work.


2.2 Euler Equations in One Dimension

In an effort to best explain the proposed Godunov-type finite-volume method used within this

work, we will first consider the Euler equations in one spatial dimension given by 2.11,

∂U∂t

+∂F∂x

= 0, (2.10)

U =

ρ

ρu

e

, F =

ρu

ρu2 + p

u(E + p)

, (2.11)

where U is the vector of conserved variables, and F is the inviscid flux dyad for one-dimensional

fluid flow as given above. The simplification to one-dimensional geometry is useful for under-

standing the Godunov-type method; however, is important to recognize that Godunov-type

finite-volume methods are easily extended to multiple-dimensions and this extension will be

discussed in Section 2.7.

2.3 Godunov-Type Methods

As mentioned previously, Godunov-type finite-volume methods discretize the solution domain

into control volumes (or cells) and apply the integral form of the conservation equations to each

of the discrete control volumes [39]. They make use of the solution to a Riemann problem, in

order to evaluate the inviscid fluxes into and out of each control volume. For one-dimensional

planar flow problems this procedure results in the following expression for the update of the

solution in cell i

Un+1i = Un

i −∆t∆x

[Fni+ 1

2

−Fni− 1

2

], (2.12)

where the control volume of interest is identified by the index i, the time level is identified by n,

and the inviscid numerical solution-flux, F , at the cell interfaces are found from the solution to

the Riemann problem. This can be applied throughout the computational domain to advance

the solution forward in time. It is worth noting that this form of the equation uses a first-order

explicit Euler time-marching scheme. Applications to different time marching schemes will be

discussed in Section 2.6.


2.3.1 Concept of Finite-Volume Methods

The basic concept of the finite volume method includes three operations. These operations

are: reconstruction in each cell; flux evaluation on each face or edge; and evolution in time of

the solution in each cell [40]. The reconstruction step is performed in each cell. Knowing the

cell-averaged values of the conserved solution variables at a given instance in time, a piecewise

polynomial is constructed that allows a specific value to be determined for each conserved

solution variable at any point within that cell. This research will focus on this step of the

finite-volume method, making use of the high-order CENO reconstruction technique to achieve

accurate spatial resolution on two-dimensional unstructured mesh. The CENO reconstruction

technique will be discussed fully in Chapter 4.

The flux evaluation at each cell face involves the application of a strategy to resolve the solution

discontinuities that can arise between neighboring cells. This requires a solution to the Riemann

problem as discussed in Section 2.4. The flux can be calculated, if necessary through performing

a high-order accurate flux quadrature. Finally, the solution in each cell can be marched forward

in time through a variety of time marching schemes. This research makes use of either second- or

fourth-order Runge-Kutta methods that will be explained in Section 2.6. The order of the time

marching method will be selected to at minimum match the order of the spatial reconstruction.

2.4 Flux Evaluation

The flux evaluation step requires the solution to the Riemann problem. It is at this step where

the flux of mass, momentum, and energy from adjacent cells is calculated. It is important to

point out that whatever solution flux exits one cell, must enter the adjacent cell. In this way the

finite-volume method is conservative at the discrete level. The following sections will describe

the Riemann problem and discuss appropriate solution techniques.

2.4.1 The Riemann Problem

The solution to the Riemann problem is of both practical and theoretical interest. The problem

is a special form of an initial-value problem such that the initial data is given by two piecewise

constant states described below by equation 2.13 and illustrated in Figure 2.1


Figure 2.1: Depiction of the Riemann Problem

Uo(x) = U(x, t = 0) =

{Uleft if x ≤ 0,

Uright if x > 0.(2.13)

The figure shows two adjacent cells that have a sharp discontinuity between the two solution

states. This discontinuity must be resolved in order to compute the flux of mass, momentum,

and energy from one cell to the other.

2.4.2 Riemann Solvers

Provided that viscous effects are neglected, the exact solution to the Euler equations can be

obtained on the basis of simple waves separating regions of uniform conditions [3]. These waves

can be rarefaction waves, contact waves, or shock waves. For gasdyamics (Euler equations),

the solution to the Riemann problem will involve one of five patterns that are depicted in

Figure 2.2. The first four situations are various combinations of left and right moving shock or

rarefaction waves separated by a contact wave. The fifth situation that can be encountered is

when a vacuum state appears in the pattern center. The occurrence of the latter will not be

dealt with herein.

Riemann solvers can be categorized as either exact or approximate. Approximate solution


Figure 2.2: Depiction of five possible solutions to the Riemann Problem - s = shock, c = contact,

r = rarefaction [42]

techniques are acceptable in many cases as it can be argued that in existing Godunov-type

finite-volume schemes much of the solution information is degraded by the discrete nature of the

solution method, and only certain features of the exact solution are worth striving for [41]. The

following sections discuss in further detail the exact Riemann solver of Gottlieb and Groth [42],

as well as Roe’s approximate Riemann solver [41], and the HLLE Riemann solver [36], and the

HLLL (Linde) [43] flux function.

Exact Riemann Solver of Gottlieb and Groth

The exact Riemann solver of Gottlieb and Groth [42] uses a standard Newton iterative method

in conjunction with an efficient initial guess to solve for the exact solution to the Riemann

problem. Additionally, this solver uses the state variables (p,a,u,γ,R) instead of (p,ρ,u,γ,R).

This is convenient because the speed of sound is a dependent variable which appears more

frequently than the density in the equations considered. The solver iterates by calculating the

intermediate velocity, and converges once the intermediate pressure difference across the contact

becomes smaller than a preset tolerance. The exact Riemann solver uses the Rankine-Hugoniot

relations and Riemann invariants to calculate the intermediate states. Once the solver has

converged, all other solution variables can be calculated.


Roe’s Approximate Riemann Solver

In the approximate Riemann solver of Roe, a solution is sought for the linearized form of

∂U∗∂t

+A∗∂U∗∂x

= 0, (2.14)

where A∗ is a local linearization of the flux Jacobian. In this local linearization, A∗ is a

function of intermediate states U∗. These intermediate solution values are calculated based

on the left and right states in such a way as to satisfy satisfy several properties known as′′Property U′′ which are described by Roe [41]. The flux can be evaluated based on characteristic

decomposition of the flux differences as follows

F =12

(FL + FR)− 12

n=3∑k=1

|λ∗,k|α∗,krc∗,k, (2.15)

where λk are the eigenvalues, αk are the left eigenvectors multiplied by the solution change

across the disontinuity, and rc,k are the right eigenvectors. All quantities with a subscript ∗ are

evaluated using the Roe-averaged state quantities that will not be defined herein, but can be

found within [41]. It should be noted, that a situation can occurs where a sonic point exists

inside a rarefaction wave. In this scenario, Roe’s approximate Riemann solver will produce an

entropy violating solution to the Euler equations, referred to as a rarefaction shock, In order to

resolve this issue, an entropy correction is required [44].

The HLLE Flux Functions

The HLLE flux function assumes a three state solution to the Riemann initial value problem [36].

These three states are separated by two waves that represent the maximum and minimum signal

velocities. Taking the closed loop integral of half the initial Riemann problem, with vertical

boundaries of x/t = 0 and the cell edge, the flux through the cell boundary can be determined

as follows

F =λ+FL − λ−FR

λ+ − λ−+λ+λ−(UR −UL)

λ+ − λ−, (2.16)

where λ+ and λ− represent the largest and smallest signal velocities respectively. The subscripts

L and R denote quantities determined based on the left and right states. The integration loop

used to derive the flux function is depicted in Figure 2.3.


Figure 2.3: Depiction of integration loop for HLLE flux function

The Linde Flux Functions

The HLLL flux function is a modification of the HLLE flux function. The modification is

to introduce a third wave when necessary, moving at the intermediate velocity. This third

wave is necessary for the recognition of intermediate waves such as contacts, entropy waves,

and shear waves. This flux function does not involve extensive characteristic analysis of the

governing equations but can improve significantly the solution resolution in the vicinity of

contact discontinuities and shear waves [43].

2.4.3 Comparison of Riemann Solvers

The solution to an initial-value problem has been calculated for the purposes of comparing the

various Riemann solvers that have been discussed above for evaluating the numerical flux. The

initial conditions have been chosen to represent a shock tube problem. These conditions are

ρl = 4.696 kg/m3, ul = 0 m/s, pl = 404.4 kPa, ρr = 1.408 kg/m3, ur = 0 m/s, pr = 101.1 kPa,

and 0 ≤ t ≤ 7 ms, and have been taken from the report by Groth and Gottlieb [45]. The results

that have been obtained for the different flux functions can be seen in Figure 2.4(a).

As shown in Figure 2.4(a), the solution obtained with a first-order accurate Godunov scheme

approximates the exact solution with each of the Riemann solvers. In all cases, the Godunov

scheme computes the general shape of the solution, however it has difficulties accurately cap-

turing the expansion wave, the contact surface, and the shock. Additionally, it should be

pointed out that the HLLL approximate Riemann solver is better able to resolve the contact

surface than the HLLE approximate Riemann solver. This result can be seen more clearly in

Figure 2.4(b), which shows a closeup of the contact. This result comes as no surprise as the

HLLL flux function accounts for the intermediate contact wave, while the HLLE flux function


(a)

(b)

Figure 2.4: Shock tube problem: n=100 cells, CFL=0.5, t=0.007 s (a) Various Riemann solvers

(b) Close-up of contact wave


does not.

2.5 Second-Order Godunov Methods

Given cell average values of the conserved solution vector, U, the purpose of reconstruction

is to determine the spatial distribution of the solution in each cell. Therefore, the solution

variables U becomes a function of the spatial coordinates. In one-dimension as we are currently

considering, we obtain U(x). Considering two-dimensions, which is the focus of this thesis, the

solution variables are then a function of both the x and y directions and we obtain U(x, y).

This reconstruction is computed by using information from nearby cells, referred to as the cell

stencil. In practice, this reconstruction is often computed using an over-determined system,

solving with a least-squares or Green-Gauss technique, and the reader is encouraged to consult

the paper by Barth [31] for further information on these techniques.

When considering second-order, or even high-order reconstruction techniques, one important

consideration is ensuring that solution monotonicity is preserved. A detailed description of

techniques used to ensure that no new extrema are created by the reconstruction procedure is

detailed in the following sections.

2.5.1 Godunov’s Theorem

Godunov’s Theorem states that among the class of constant coefficient schemes, there are

no schemes that are higher than first-order accurate and monotone [46]. Therefore in order

to implement higher-order schemes in a manner that preserves monotonicity, it is necessary

for the scheme to be non-linear, even for the solution of linear partial differential equations.

Monotonicity can be enforced through the use of slope limiters [40, 47, 48]. The strategy for

this method will be to revert to a first-order accurate monotone scheme when necessary, and

use a high-order scheme everywhere else. This method is essential for accurately capturing

discontinuities such as shocks or combustion flame fronts.

2.5.2 Piecewise Linear Reconstruction

For second-order reconstruction in one dimension, the derivative is calculated and the recon-

structed solution for cell i has the following linear form:


Figure 2.5: Shock tube problem: n=100 cells, CFL=0.5, t=0.007 s - Order of Accuracy effect

on solution

Ui(x) = Ui +∂U∂x

∣∣∣i(x− xi) . (2.17)

Here U can be computed within any cell i in terms of the solution average value, slope, and

distance from the cell center. If this reconstruction is known in every cell, a piecewise linear

reconstruction of the entire computational domain is achieved.

The initial-value problem that has been described previously in Section 2.4.3 has also been

calculated using a second-order implementation of Godunov’s method. The results of the higher-

order algorithm are shown in Figure 2.5 along with the first-order accurate solution, and the

exact solution for comparison purposes. The second-order solution uses the slope limiter of

Barth-Jesperson [31] that will be discussed in detail in Section 2.5.3. The higher-order solution

is much more capable of accurately resolving the exact solution as would be expected.

While this equation is useful for obtaining a second-order accurate scheme, it does not address


the issues related to Godunov’s Theorem. Therefore, the problem of maintaining monotonicity

with a higher-order scheme must still be addressed.

2.5.3 Enforcing Monotonicity: Slope Limiters

For one-dimensional second-order linear reconstruction, the solution states must be calculated

at the cell interface. In order to accomplish this goal, while still maintaining monotonicity, the

concept of a slope limiter is introduced. Therefore, the previously discussed piecewise linear

reconstruction of cell i, given in equation 2.17, is reformatted and re-written as

Ui(x) = Ui + φi∂U∂x

∣∣∣i(x− xi) , (2.18)

where the slope limiter, φ has been introduced. Notice that φ = 0 corresponds to a first-order

reconstruction, and φ = 1 corresponds to a second-order reconstruction. With the cell boundary

solution values calculated using equation 2.18, a Riemann problem can be solved to calculate

the flux between cells.

The slope limiters of Barth-Jesperson [31] and Venkatakrishnan [47] are considered here for

evaluation of φi. These limiters are calculated in such a way as to ensure the solution state

at the cell face is a value between the maximum and minimum average solution value of all

cells in the reconstruction stencil. In this way, monotonicity is preserved and the scheme can

accurately capture sharp discontinuities.

Barth-Jesperson Limiter

The Barth-Jesperson Limiter calculates the slope limiter, φ, using the expression

φik =

min(1, umax−uiuik−ui ) uik > ui

min(1, umin−uiuik−ui ) uik < ui

1 uik = ui

(2.19)

for which

φi = min(φik) (2.20)

and where u represents the primitive solution variable of interest, and the various terms of u

are defined as follows:

� u = cell average value (for cell i),


� umax = max(u, uneighbours) = maximum cell average value amongst all cells used in the

reconstruction of cell i,

� umin = min(u, uneighbours) = minimum cell average value amongst all cells used in the

reconstruction of cell i, and

� uk = unlimited reconstructed value at the kth quadrature point.

The Barth-Jesperson slop limiter only limits the solution data when required, and as such is a

low-dissipation limiter. This limiting procedure has been shown to be very effective in removing

spurious solution oscillations [40].

Venkatakrishnan Limiter

The Venkatakrishnan limiter is a modification to the well known Van Albada limiter [49]. The

modification to the Van Albada limiter has been to improve two draw backs. The first, is that

in practice, second-order accurate schemes exhibited orders of accuracy less than second-order.

The modification assists in obtaining a higher order of accuracy. Secondly, the modification

improve convergence to steady state solutions [47]. The Venkatakrishnan limiter calculates the

slope limiter, φ, using the expression

φik =

y2+2yy2+y+2

uik > ui y = umax−uiuik−ui

y2+2yy2+y+2

uik < ui y = umin−uiuik−ui

1 uik = ui

(2.21)

where again

φi = min(φik) (2.22)

and where the various values of u are the same as in the Barth-Jesperson limiter. It is worth not-

ing here that the Venkatakrishnan limiter is slightly more dissipative than the Barth-Jesperson

limiter.

Effect of Slope Limiters

The effect of slope limiters on the second-order solution have been investigated for the same

initial-value problem as before, and these results are shown in Figure 2.6. Note that when there

is no slope limiter implemented the computed solution exhibits oscillations near the expansion

wave, the contact, and the shock. However, when the two slope limiters are implemented,


Figure 2.6: Shock tube problem: n=100 cells, CFL=0.5, t=0.007 s - Slope limiter effect on

solution

monotonicity is preserved, and the solution oscillations are eliminated. Both slope limiters

shown here behave very similarly for this case.

2.6 Time Marching Schemes

Time-accurate solutions to systems of ordinary differential equations (ODE’s) can be achieved

by integrating the the solution in time by means of a time-marching method [5]. Section 2.6.1

discusses how to transform our system of conservation equations into a set of ODE’s for appli-

cation of a time-marching method.

Time-marching methods can be either explicit or implicit. Explicit time marching schemes

use the information from a previous time step to solve for the solution at a current time step.

Implicit methods use information from the current time step to solve for a solution at that time-

step. The latter will require the solution to a coupled system of nonlinear algebraic equations.


The CENO approach to high-order reconstruction is applicable for use with both explicit and

implicit time marching schemes. However, due to the desire to compare results for the high-

order unstructured solver, with results which have been obtained using the CENO high-order

structured solver, the time-marching schemes that have been implemented for this work are

limited to explicit schemes. The schemes that have been used are further described in Sec-

tion 2.6.2. For this work, the time-marching schemes used have been selected to at least match

the order of accuracy of the spatial reconstruction.

2.6.1 Semi-Discrete Form

Application of Godunov-type methods, with the intention of achieving a high-order accurate

solution requires the implementation of a high-order time-marching scheme, in addition to

high-order spatial reconstruction. In order to implement a variety of time-marching schemes

the Godunov method shown in equation 2.12, can be recast in to the following semi-discrete

form:

dUdt

= − 1∆x

{[Fx,(i+ 1

2) −Fx,(i− 1

2)

]}. (2.23)

This is a set of ordinary differential equations to which a variety of different time-marching

scheme can then be applied to advance the solution forward in time.

2.6.2 Runge-Kutta Methods

In numerical analysis, Runge-Kutta methods are an important family of explicit and implicit

time marching methods. They are a special subset of predictor-corrector methods [5] that can

be derived for different orders of accuracy. The two methods that have been used for this work

will simply be presented here, but the reader is encouraged to consult numerical textbooks such

as [5, 50] for further reading.

The Runge-Kutta methods that have been implemented in this work are second- and fourth-

order Runge-Kutta techniques. The second-order method is a two-stage method and has the

form

Un+1 = Un + (k2, )

k1 = ∆tf(tn,Un),

k2 = ∆tf(tn + 12∆t,Un + 1

2k1),

(2.24)


where Un+1 is the updated solution, Un is the solution, ∆t is the time step, and f is a function

of U and t for which the time marching procedure is being applied. Additionally, k1 and k2 are

required for the time marching scheme and are calculated as shown.

The fourth-order Runge-Kutta method is a four-stage scheme given by

Un+1 = Un + 16(k1 + 2k2 + 2k3 + k4),

k1 = ∆tf(tn,Un),

k2 = ∆tf(tn + 12∆t,Un + 1

2k1),

k3 = ∆tf(tn + 12∆t,Un + 1

2k2),

k4 = ∆tf(tn + ∆t,Un + k3),

(2.25)

where the variables are the same as the second-order method, with additional ki as required by

the scheme.

Within this work, the order of the time marching scheme has been selected to at minimum

match the order of the spatial resolution. Therefore, for a second-order spatial resolution,

a second-order time marching scheme has been implement. When the spatial resolution is

increased to third- or fourth-order, then a fourth-order time marching scheme was used.

Finally, it should be pointed out that the time-marching scheme can have an effect on the overall

solution monotonicity. While there has been work with slope limiters to ensure no new local

extrema will be introduced, when the solution state is updated in time, there is no guarantee

that the overall monotonicity of the scheme will be formally preserved. This is a consideration

that should be kept in mind when interpreting results from the high-order solutions, but will

not be discussed in detail here.

2.6.3 Courant Number

The stability of the explicit time-marching schemes that have been used for this work are

controlled by the appropriate selection of the time step, ∆t. An appropriate time step can be

calculated by making use of the Courant-Friedrichs-Lewy number, CFL, such that

∆t = CFL∆x

(|u|+ a)max. (2.26)

This ensures that the waves from adjacent Riemann problems do not travel far enough to

interact with one another. It should be noted that for time-accurate problems the time-step is

chosen by calculating the minimum time-step in each computational cell, and using the smallest


computed. The value of the CFL for this work has typically been chosen as equal or less than

unity, in order to maintain the numerical stability of the scheme.

2.7 Godunov-Type Finite-Volume Schemes in

Multi-Dimensions

While to this point the description of the Godunov-type finite-volume method has been limited

to one-dimensional solution techniques, the extension to multiple-dimensions is possible. As

this work is primarily interested in the solution to the two-dimensional Euler equations, we will

consider the extension of the Godunov-type finite-volume method to two dimensions, noting

that an extension to three-dimensions follows in a very similar manner.

If we consider the two-dimensional integral form of the Euler equations given below

∫∫A

[∂U∂t

+∇ ·−→F]dA = 0, (2.27)

where, for the 2D Euler equations in Cartesian coordinates, U is the vector of conserved solution

variables given by

U = [ρ ρu ρv E]T , (2.28)

and−→F = (F,G) is the inviscid flux dyad, with

F =

ρu

ρu2 + p

ρuv

u(E + p)

, G =

ρv

ρuv

ρv2 + p

v(E + p)

, (2.29)

and where u and v are again the velocity components in the x and y directions respectively, p

is the pressure, ρ is the mass density, E is the total specific energy given by E = e + 12 |−→u |2,

and e is the specific internal energy.

The semi-discrete form of the Euler equations on a two-dimensional Cartesian mesh can be

reduced to the following expression


dUdt

= − 1∆x

{[F(i+ 1

2),j −F(i− 1

2),j

]− 1

∆y

{[Gi,(j+ 1

2) − Gi,(j− 1

2)

]}, (2.30)

where we note that the only variation is the requirement for additional flux evaluations at the

y-direction interfaces. This can then be further extended to an arbitrary shaped mesh as

dUi

dt= − 1

Ai

Nf∑l=1

(−→F · −→n∆`)i,l, (2.31)

where−→F = Fx + Gy which is the flux dyad, Nf is the number of faces, Ai is the area, ∆` is

the edge length, and −→n is the unit vector normal to a given edge. The standard time-marching

schemes described above can then be applied to compute solutions to the resulting coupled

system of ODEs.

2.7.1 Multi-Dimensional Example

For the purposes of understanding, a simple shock box cases has been considered and solutions

computed on a Cartesian mesh with 200 cells in both the x and y directions. The results are

computed using the second-order Godunov-type finite-volume scheme described in the previous

sections. Fluxes are computed using Roe’s approximate Riemann solver [41] and monotonicity

is maintained with the slope limiter of Venkatakrishnan [47]. Second-order Runge-Kutta time

marching is used to advance the solution. The results are shown in Figure 2.7. The initial

conditions are shown in Figure 2.7(a), and are an area of high pressure and density, and an

area of low pressure and density. The two states begin to interact with one another as the high

density area expands into the low density area. This interaction produces a very unique wave

structure that is shown in the figure at 7 ms. At this time, the results will not be presented in

depth, but the reader should take from this that a multi-dimensional finite-volume flow solver

has produced these results.


(a) (b)

Figure 2.7: Finite-volume Shock-Box calculation: (a) Density - Initial Conditions (b) Density

- Results at 7 ms

Chapter 3

Unstructured Mesh

There are two main classes of grids: structured and unstructured. A structured mesh is charac-

terized by regular connectivity that can be expressed as a two- or three-dimensional array. This

generally restricts the element choices to quadrilaterals in two-dimensions (2D) or hexahedra in

three-dimensions (3D). The regularity of the connectivity implies a neighborhood relationship

defined by the storage arrangement. An unstructured mesh is, in general, triangles or hexagons

in two-dimensions and tetrahedra, pyramids, or extruded triangles in three-dimensions. These

meshes require a connectivity matrix to explicitly define the connections between points in the

computational mesh [51].

The greater geometrical flexibility offered by unstructured grids can be very effective for dealing

with complex geometries that are more typical of practical engineering applications. Therefore,

unstructured grids are now commonly used in computational fluid dynamics [52]. Additionally,

there has more recently been an interest in the application of high-order methods to unstruc-

tured mesh. With the widespread use of unstructured mesh, an effort has been directed towards

developing efficient and robust algorithms for their automatic generation.

3.1 Unstructured Mesh Generation

There are a wide variety of different unstructured mesh generation tools which use a great

variety of different algorithms to compute the node and element locations of the mesh. The

volume of literature on the subject has grown enormously in recent years [33]. The mesh

generation tool that has been used for this work is Gmsh, which has been developed by Geuzaine

and Remacle [53]. Gmsh has a wide variety of tools, and several different unstructured mesh

27

Chapter 3. Unstructured Mesh 28

generation algorithms implemented that will be discussed in the following sections.

3.1.1 Gmsh

This work has relied entirely on meshing created using the widely available meshing tool Gmsh

developed by Geuzaine and Remacle [53]. Gmsh is a three-dimensional finite-element grid

generator that has a build in CAD engine and post-processor. Gmsh has been designed in

such a way as to provide a fast, light, and user friendly meshing tool. It’s capabilities include

parametric input as well as advanced visualization capabilities. Instructions can be provided

to Gmsh either through an interactive Graphical User Interface (GUI), or through text files,

written using Gmsh’s scripting language that can be read in by Gmsh.

Gmsh contains four separate modules:

1. a geometry module;

2. a mesh generation module;

3. a finite-element solver; and

4. a post processor.

The modules that have been important for this work have been the geometry description and

the meshing tool. These two modules will be described in the following sections.

3.1.2 Geometry Module

The geometry module in Gmsh uses a boundary representation to describe the geometries.

Geometric models are created using a bottom up flow. This begins by first defining points. The

points are then connected by defining lines. Next, surfaces are defined as a group of lines, and

finally volumes are described using a group of surfaces. It is then possible to define physical

groups based on these elementary geometric entities. These can all be parametrized using

Gmsh’s scripting language.

As this thesis is reliant on two-dimensional geometries, definition of points, lines, and surfaces is

all that is required. A two-dimensional geometry that has been created using Gmsh is depicted

in Figure 3.1. The figure shows how Gmsh can graphically represent a geometry that has curved

boundaries.


Figure 3.1: Two-dimensional geometry created using Gmsh

3.1.3 Mesh Generation Module

Gmsh has the capabilities of creating both two-dimensional and three dimensional meshes. All

meshes that are created using Gmsh are considered to be unstructured, which implies that

the elementary geometrical elements are defined only by an ordered list of their nodes, but no

predefined ordering is assumed between any two elements.

The meshes are created using various algorithms which use a similar bottom-up flow as the

geometry creation. First all lines and line segments are discretized. The lines are then used

to mesh the surfaces. For a two-dimensional mesh, the procedure would be finished here. For

a three-dimensional mesh, the mesh of the surface are then be used to mesh the volume. The

geometry that was shown in Figure 3.1 can be seen after using Gmsh to create a two-dimensional

mesh in Figure 4.2. The resulting mesh is made up of 398 nodes and 657 elements.

Gmsh has the capability of using several different algorithms when generating a two-dimensional

mesh which will be discussed in Section 3.1.4.


Figure 3.2: Two-dimensional mesh generated using Gmsh

3.1.4 Mesh Generation Procedure

Considering two-dimensional mesh generation, as two-dimensional meshes are of particular

interest to this work, it is worth investigating the various algorithms that Gmsh uses to generate

meshes. Regardless of which algorithm is used, first a Delaunay mesh that contains all the points

of a one-dimensional mesh is constructed using a divide-and-conquer algorithm [54]. When

generating a two-dimensional mesh, the initial one-dimensional mesh represents the boundary

discretization. Missing edges are recovered using edge swaps [51]. After the initial step, three

different algorithms can be applied to generate the final mesh:

1. MeshAdapt;

2. Delaunay; and

3. Frontal.

Each of these three algorithms will be described in the following sections, finishing with a

comparison between the three.


Figure 3.3: Visualization of Mesh Modifications [53]

MeshAdapt

The MeshAdapt algorithm has been based on local mesh modifications. The technique makes

use of local edge swaps, splits, and collapses in parametric space to obtain a better geometrical

configuration. The advantages of this method is that it does not require the computation of

derivatives of the parametrization and is therefore quite robust.

The algorithm requires an initial mesh to be created. This can be done using a divide and

conquer technique [54]. Then, any missing edges can be recovered using edge swaps [51]. The

next steps of the technique makes use of adimensional lengths, which are simply lengths that

have been non-dimensionalized. At this point, local mesh modifications can be applied. These

modifications include splitting every edge that is too long, removing edges that are too short

using an edge collapse, edges for which a better configuration is obtained by swapping are

swapped, and vertices are relocated optimally. These four cell operations are described in more

detail as follows and can be seen represented in Figure 3.3:

1. Edge Splitting: An edge is considered too long when its adimensional length is greater

than le > 1.4. When split, the two new edges will have a minimal size of 0.7. In order to

converge to a stable configuration, an edge of size le = 0.7 should not be considered as a


short edge.

2. Edge Collapsing: An edge is considered to be short when its adimensional length is smaller

than le < 0.7. An edge cannot be collapsed if one of the remaining triangles after the

collapse is inverted in the parametric space.

3. Edge Swapping: An edge is swapped if min (Ye1 , Ye2 ) < min (Ye3 , Ye4 ), unless

(a) it is classified on a model edge;

(b) the two adjacent triangles e1 and e2 form a concave quadrilateral in the parametric

space;

(c) the angle between the triangles normals is greater than a threshold, typically 30

degrees.

4. Vertex Re-positioning: Each vertex is moved optimally inside the cavity made of all its

surrounding triangles. The optimal position is chosen in order to maximize the worst

element quality.

In practice, this algorithm converges in about 6-8 iterations and produces anisotropic meshes

in the parametric space without computing derivatives of the mapping [53].

Delaunay

The Delaunay triangulation method that is implemented in Gmsh is based on the work of

the GAMMA team at INRIA [55]. Delaunay-type methods are used extensively for mesh

generation and engineering purposes. The Delaunay method requires knowledge of the location

of the circumcenter and the circumradius of each cell. The circumcenter of each cell is located

at the intersection of the perpendicular bisectors of the sides of the triangle. The circumradius

of a cell is the radius of the smallest possible circle for which the triangular cell can fit inside.

It is also useful to define an adimensional circumradius, which is simply a non-dimensionalized

circumradius.

The Delaunay method inserts new points sequentially at the circumcenter of the elements that

have the largest adimensional circumradius. The mesh is the reconnected using an anisotropic

Delaunay criterion. The fundamental concept for Delaunay triangulation is that any vertex

must not be contained within the circumcircle of any triangle. The method begins with the

mesh edge. A set of vertices are generated by marching along the existing internal edges at

a given spacing ratio. The spacing ratio is computed based on the edge length and length


Figure 3.4: Hierarchical structure of triangle types [52]

information deduced from the surface mesh vertex distribution. Points are created on all edges

judged too long, then a filtering step is used to remove all points too close to one another.

It is worth noting, that given a set of points, the Delaunay triangulation is unique. Further

information describing the mathematics of Delaunay triangulations can be found in [55].

Frontal

The frontal algorithm is based on the work of Rebay [52]. The approach takes full advantage of

the sequential way in which the Delaunay triangulation is constructed. Additionally, the dis-

tinctive characteristic is that the point positions and connections are computed simultaneously.

In this way, the frontal approach is an efficient unstructured mesh generation method based on

Delaunay triangulation. The method is computationally efficient, and applicable to two- and

three-dimensions.

The frontal algorithm first requires an initial mesh to be generated. At this point, there is no

guarantee that from an arbitrary distribution of points the boundary mesh will be conforming.

It is therefore necessary to recover all missing boundary edges [51]. The next step is to identify

all internal and external triangles. The internal triangles can be further classified as those

that meet the required size, accepted, and those that do not, non-accepted. The triangles are

further categorized as active and waiting. An active triangle is a non-accepted triangle with at

least one accepted neighbor, and a waiting triangle is a non-accepted triangle surrounded by

non-accepted triangles. This hierarchical structure is depicted in Figure 3.4.

A new point is then inserted at the coarsest part of the domain on a segment associated with

an edge of an active triangle. The Bowyer-Watson [56, 57] algorithm is then used to generate

a new triangulation that differs from the previous one locally around the newly inserted point.

This is repeated until all triangles are accepted.


A summary of the frontal algorithm as described by Rebay in [52] is then as follows:

1. Triangulate all the boundary points by means of the Bowyer-Watson algorithm [56, 57].

2. Check if the triangulation is body conforming and recover all missing boundary edges.

3. Divide all triangles into internal and external. Divide internal triangles into accepted and

non-accepted on the basis of their circumcircle radii. Divide non-accepted triangles into

active and waiting.

4. Order the active triangles according to their circumscribed circle radii.

5. Consider the top element of the ordered list of the active triangles and insert the new

point according to the Voronoi segment insertion criterion.

6. Regenerate the triangulation by means of the Bowyer-Watson algorithm.

7. Divide new triangles into accepted and non-accepted on the basis of their circumcircle

radii. Divide new non-accepted triangles into active and waiting.

8. Divide non-accepted triangles adjacent to the new ones into active and waiting.

9. If the active triangle list is empty stop, else go to step 5.

3.1.5 Comparison

Each of the meshing algorithms described above have different relative strengths and weaknesses

which are summarized in Table 3.1, with one representing the best. From this table, it can be

drawn that for very complex curved surfaces the MeshAdapt algorithm is the best choice. When

high element quality is important, the Frontal algorithm will likely yield the best results. For

very large meshes of plane surfaces the Delaunay algorithm is computationally the fastest [53].

Algorithm Robustness Performance ElementQuality

MeshAdapt 1 3 2

Delaunay 2 1 2

Frontal 3 2 1

Table 3.1: Comparison of Meshing Algorithms

For qualitative comparison purposes, Figure 3.5 shows three meshes, each using the same ge-

ometry, but generated using the three different algorithms. It can easily be seen that each

algorithm meshes the entire domain, produces different grids by placing nodes and elements in

different location.


(a) (b)

(c) (d)

Figure 3.5: Various meshing algorithms: (a) Geometry (b) MeshAdapt (c) Delaunay (d) Frontal

3.2 Domain Decomposition and Multi-Block Partitioning

Computing the solution to practical engineering flows becomes extremely computationally ex-

pensive. For example, when computing the solution to fluid flows which involve turbulence, and

chemical reactions, which are typical of combustors, the vastly different spatial and temporal

scales of the flow make the calculation extremely expensive to compute accurately. A domain

decomposition technique can be used to make efficient use of large-scale parallel distributed-

memory computing facilities. In this approach, a computational domain is decomposed into

many smaller domains each residing on a different processor. The solutions on each domain can

then be calculated in parallel, vastly decreasing the elapsed wall-clock time required to achieve

a numerical solution.

The decomposition of the domain is carried out here via a partitioning of the unstructured mesh

into a multi-block mesh containing a specified number of sub-blocks. The mesh partitioning

can be accomplished in a variety of different ways. However, for this work PMETIS, developed

by Karypsis and Kumar [58] has been used, as it is already incorporated within the meshing


software, Gmsh. Algorithms that find a good partitioning of highly unstructured graphs are

critical for developing efficient solutions for a wide range of problems in many application areas

using parallel computers [58]. The algorithms should be able to equally balance the number of

cells per processor, so that each processor has a similar amount of computations, and therefore

will complete the computations in a similar amount of time. Additionally, the distribution

should be done in a way that minimizes the number of cells at the interface, this will minimize

the amount of solution information that must be passed and exchanged from one process or

another.

3.2.1 PMETIS

PMETIS is a software package developed by Karypis and Kumar [58] for partitioning large

irregular graphs, partitioning large meshes, and computing fill-reducing orderings of sparse

matrices. Of particular importance to this work is the ability to effectively partition large

meshes. PMETIS uses a multilevel graph partitioning algorithm, which reduces the size of the

mesh by collapsing vertices and edges and then partitioning the coarser mesh. The mesh can

then be refined. While refining the mesh, PMETIS employs refinement to the partitioning as

well. This is in contrast to traditional methods which employ partitioning algorithms directly

to the original mesh. The multilevel graph partitioning algorithms allow PMETIS to efficiently

and quickly produce high quality partitions of general unstructured mesh [58].

PMETIS provides two different partitioning programs, partnmesh and partdmesh. Partnmesh

first converts the mesh into a nodal graph, where each node of the mesh becomes a vertex

of a graph which is then partitioned. Partdmesh converts the mesh into a dual graph, where

each element of the mesh becomes a vertex of a graph which is partitioned. An in-depth

understanding of the partitioning method is not required here, but the reader is referenced to

further information that can be found in [58, 59].

A mesh that has been partitioned using PMETIS is depicted in Figure 3.6. It can be easily

noticed that the partitions are all of a similar size depicting how PMETIS balances the load

that will be taken by each processor. Additionally, and somewhat more difficult to visualize,

it has been partitioned in a way that minimizes the number of cell interfaces shared between

blocks.


(a) (b)

Figure 3.6: PMETIS Partitioned Mesh showing: (a) Blocks (b) Blocks and Mesh

3.3 Finite-Volume on Unstructured Mesh

The finite-volume method that was discussed in Chapter 2 can be applied to unstructured

meshes by using the semi-discrete form of the finite-volume discretized Euler equations given

in equation 3.1 as follows:

dUi

dt= − 1

Ai

Nf∑l=1

(−→F · −→n∆`)i,l (3.1)

where−→F is the flux dyad which has already been presented, Nf = the number of faces = 3 for

triangular elements, Ai is the area, ∆` is the edge length, and −→n is the unit vector normal to

a given edge. The various time marching schemes described previously can be applied to the

solution of this coupled system of ODEs.

The key consideration when working with unstructured meshes must be knowing the connec-

tivity between neighboring cells. Knowledge of the cell connectivity is essential to compute

fluxes transferred from one cell to the adjacent cell. There are various ways of storing the cell

connectivity and they rely on the data structure of the computational software.


3.3.1 Data Structure

The choice of data structure is a key consideration when dealing with unstructured meshes as

the connectivity between neighboring cells needs to be known and easily accessible to carry out

the required computations. Depending on the type of solution algorithms being applied some

data structure types are more advantageous than others. Typical data structures can range

from finite element data structures [60], edge structures [31], vertex lists [61], and quad-edge

structures [62]. These structures are all described in [33].

The data structure type that has been implemented in this work is an edge type structure. This

type of structure lists the connectivity of vertices and adjacent faces by storing a quadruple of

each edge that consists of the origin and destination of each edge, as well as two faces (cells)

that share that edge. The form of this data structure can be seen in Table 3.2. The table

shows an example with cell and node numbers chosen at random to give the reader a better

understanding how this storage type works.

Celli Cellj Nodei Nodej

5 7 25 4

9 16 37 15

12 26 1 9

14 4 11 7

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

Table 3.2: Example of Edge Type Data Structure

This type of data structure has the advantage that it allows very easy traversal through the

mesh [33]. This is very useful in finite-volume schemes, as the mesh traversal can be done at

each time step, and used to compute the flux at each cell face. Additionally, each face is stored

once, therefore the flux is calculated, added to one cell and subtracted from the adjacent cell.

In this way there is no duplication of the calculation, producing an efficient implementation of

the scheme.

3.3.2 Finite-Volume Example on Unstructured Mesh

A similar example to the case shown in Figure 2.7, has been computed using a Godunov-type

finite-volume method applied on an unstructured mesh. The results are shown in Figure 3.7.

Once again, the example is a shock box calculation, where the initial conditions are shown

on the left, and the solution at 7 ms is shown on the right. The results shown here are on


(a) (b)

Figure 3.7: Finite-Volume Shock-Box Calculation on Unstructured Mesh: (a) Density - Initial

Conditions (b) Density - Results 7 ms

a unstructured multi-block mesh that consists of 8 blocks, 46412 nodes, and 92026 triangular

elements. The spatial reconstruction is second-order, and the computation uses a second-order

Runge-Kutta time marching scheme.

It is evident from comparison between Figure 3.7 with Figure 2.7 that they are very similar

producing the same flow structure. However, there are some slight mesh effects that are visible

in the results of the unstructured case. The discontinuities in the flow are slightly less distinct

and show slight variations that do no occur on a Cartesian mesh that is in line with the

discontinuities. In general, the results are in excellent agreement with the structured case

shown in Figure 2.7 of the previous chapter, and provides first proof that the proposed finite-

volume scheme has been correctly implemented for multi-block unstructured mesh. Further

verification will be presented in Chapter 5.

Chapter 4

High-Order CENO Scheme

The development of high-order methods is currently a very active field of research in CFD.

Such schemes possess the potential to reduce the computational cost associated with achieving a

desired level of accuracy for a numerical simulation. This is accomplished through the improved

numerical efficiency of high-order schemes, and reduction of the number of computational cells

required to achieve the target accuracy. The greater computational efficiency of high-order

methods is particularly useful when computing the solution to complex numerical simulations

such as complex flows involving wide ranges of spatial and temporal scales. These types of flows

are often encountered in such physical phenomenons as combustion, aeroacoustic modelling,

computational electromagnetics, and other physically important situations.

The Central Essentially Non-Oscillatory (CENO) high-order finite-volume scheme has been de-

veloped by Ivan and Groth [12, 17, 18, 19] for the two-dimensional solution of the Euler equations

and the Advection-Diffusion equation on structured meshes. This has been further extended

by Rashad [32] for the solution of the Euler and Navier-Stokes equations in three-dimensions

on Cartesian grids. The focus of this work has been the extension of the CENO high-order

finite-volume method for the solution of two-dimensional compressible flows on unstructured

meshes.

This chapter will begin by outlining some previous high-order work that has been accomplished

relating to similar types of high-order reconstruction, such as ENO [13] and WENO [14] schemes.

This will lead into a description of the k-exact reconstruction method of Barth [31]. The k-

exact reconstruction method is the basis for the CENO method, and therefore an in-depth

explanation of the CENO scheme will follow. Next, an overview of the high-order multi-block

solution method used within this thesis will be described. Finally, a brief example will be

presented. However this will be a qualitative example, and a more complete investigation of

40

Chapter 4. High-Order CENO Scheme 41

the proposed high-order scheme for unstructured mesh is given in Chapter 5.

4.1 Previous High-Order Work

Godunov’s theorem states that among the class of constant coefficient schemes, there are no

schemes that are higher than first-order accurate and monotone [46]. This effectively means

that while more accurate for smooth solutions, higher-order accurate solution techniques will

tend to introduce oscillations near discontinuities. In order to avoid this drawback, and obtain

higher-order accuracy while still maintaining monotonicity it is necessary to introduce non-

linear solution techniques.

Harten introduced the concept of total variation [63], which led to the development of non-linear

solution techniques that could be classified as total variation diminishing (TVD). Additionally,

Essentially Non-Oscillatory (ENO) schemes were introduced by Harten et al. [13] to maintain

monotonicity and produce high-order accuracy. These schemes have less restrictive conditions

for satisfying monotonicity, therefore allowing them to increase the spatial accuracy while still

being able to guarantee convergence of the numerical solution to weak solutions of the solved

PDE. In general, the goal is to have accurate solutions where no new extrema are created.

A general framework for generating second-order, upwind, TVD discretizations was formulated

by Van Leer [64, 65] with the introduction of Monotone Upstream-Centered Schemes for Con-

servation Laws (MUSCL) scheme. In the MUSCL approach, the TVD conditions are ensured

by restricting the amplitude of the gradients appearing in the piecewise linear solution recon-

structions via non-linear slope limiting. Similar results can also be obtained by applying flux

limiting [66]. There are many different slope and flux limiters readily available in the liter-

ature [3, 66, 48]. This research as mentioned previously makes use of the slope limiters of

Barth-Jesperson [31] and Venkatakrishnan [47].

The ENO concept introduced by Harten et al. [13], provided a means for constructing schemes in

which oscillations on the order of the truncation error were permitted to exist in smooth regions,

thereby allowing the numerical discretizations to provide high-order accuracy while avoiding a

Gibbs-like phenomenon near discontinuities. The ENO reconstruction algorithm is based on

selecting the smoothest reconstruction stencil among several candidate stencils. An extension

to the ENO schemes is the Weighted Essentially Non-Oscillatory (WENO) schemes [14, 15, 16].

These types of schemes assign a weighting to the multiple reconstruction stencils in an attempt

to simplify the stencil selection and to assist the convergence to steady-state. These techniques

unfortunately have proven to be quite computationally expensive, particularly for unstructured


mesh.

A more flexible high-order scheme that is easily extended to multiple dimensions, the k-exact

reconstruction procedure, has been developed by Barth and Fredrickson [31, 40]. Note that

the k-exact reconstruction is used as part of the CENO formulation developed by Ivan and

Groth [12, 17, 18, 19]. The CENO scheme uses a hybrid solution approach consisting of fixed

central stencil, such that the reconstruction must only be computed once. Additionally a

smoothness indicator is calculated to locate under resolved solution content and/or solutions

near shocks. If the solution is under resolved or in the vicinity of a shock, the reconstruction is

reverted to a limited linear piecewise reconstruction. In this way monotonicity can be preserved,

while still achieving high-order solution accuracy in smooth regions.

4.1.1 Overview of the CENO Scheme

The CENO scheme is a variant of the original ENO scheme [67]. However, the CENO scheme

is not based on either selecting or weighting reconstructions from multiple stencils. Instead,

a hybrid solution reconstruction procedure is used that combines the high-order k-exact least-

squares reconstruction technique of Barth [31] on a fixed central stencil, with a monotonicity

preserving limited piecewise linear least-squares reconstruction algorithm. In this way, the

CENO reconstruction method makes use of a single central stencil for each computational cell.

The method is performed in three separate steps:

1. perform k-exact reconstruction in each cell;

2. evaluate the solution smoothness indicator for each cell; and

3. perform piecewise linear reconstruction in those cells that are flagged as having non-

smooth solution content.

The k-exact reconstruction step performs a high-order reconstruction of the solution content

in each computational cell. The details of this step are described further in Section 4.2. Once

the reconstruction of the solution content has been completed, the next step is to calculate the

smoothness indicator. This is calculated based on the solution content of the cells within the

reconstruction stencil.

Once the smoothness indicator is calculated, it is compared to a pass / no-pass cutoff to

determine which cells are flagged as discontinuous or under-resolved. The cells that are flagged

are then treated using a piecewise linear reconstruction which preserves the monotonicity of


the solution. At this stage the solution can be marched forward in time using a variety of time

marching schemes that have already been discussed in Chapter 2.

4.1.2 Motivation

As has previously been discussed, there is a clear motivation for the development of high-

order accurate solution methods to reduce the overall computational cost of obtaining accurate

solutions to complex physical flows. There have been many different proposed approaches to the

construction of high-order accurate schemes. Therefore, it is worth considering the advantages

and disadvantages of the CENO scheme, which make it a promising tool for computing the

solution of complex physical flow problems.

The main advantages to the CENO scheme are as follows:

1. Produces monotone solutions through the hybrid approach of computing high-order solu-

tions where the solution is smooth and reverting to piecewise linear reconstruction when

necessary. In this way, the CENO technique can still capture smooth extrema with high-

order accuracy.

2. Robustly handle under-resolved / non-smooth solution content by reverting to a linear-

piecewise reconstruction technique.

3. The use of a fixed central stencil will provide a more representative reconstruction of the

solution content taking information from all sides of the cell of interest. Additionally,

this eliminates the need to compute the reconstruction using several stencils which can

be computationally expensive.

4. The scheme is well suited for use on unstructured mesh, as it avoids the complexities

of multiple stencils, associated with ENO and WENO schemes, that can produce poorly

conditioned coefficient matrices.

5. The method is straight forward and avoids complications that are associated with different

types of high-order reconstructions such as ENO and WENO’s computation on several

different stencils. Additionally, the procedure of reconstruction, smoothness indicator,

piecewise linear reconstruction is very easy to follow.

As with any method however, there are some disadvantages associated with the CENO recon-

struction technique that are outlined below:


1. Uniform accuracy is formally lost with the CENO reconstruction procedure, this is a

feature of traditional ENO and WENO schemes.

2. There is an increased computational cost associated with the calculation of the smoothness

indicator, and in some cases the requirement of a second reconstruction (piecewise linear)

to be calculated. However, this additional cost is balanced by the requirement of having

to use only one reconstruction stencil.

The scheme clearly has advantages and disadvantages, as would be expected with any numerical

solution procedure. However, it does seem to be a good compromise between the often com-

peting goals of providing accurate solutions in a robust manner while decreasing computational

costs. Additionally, the straight forward concept of the solution procedure does make it a very

attractive option for computing high-order accurate solutions, particularly for unstructured

mesh.

4.2 k-Exact Reconstruction

A high-order accurate finite-volume scheme for solving the Euler equations of gasdynamics, us-

ing a k-exact reconstruction, and high-order flux quadrature formulas was developed previously

by Barth [40]. The method proposed by Barth follows high-order solution techniques developed

previously for structured meshes which had proven to be extremely successful in significantly

improving the quality of the numerical solutions [31]. Therefore, the extension of these tech-

niques to unstructured meshes, that can be created using efficient algorithms, seemed like a

logical progression.

The k-exact higher-order reconstruction formulation begins by assuming that the solution with

each cell can be represented by the form

uki (x, y) =N1∑p1=0

N2∑p2=0

(x− xi)p1(y − yi)p2Dp1p2 (4.1)

where uki represents the reconstructed solution of the primitive solution variable of interest,

(xi,yi) are the coordinates of the cell centroid, and k is the order of the piecewise polynomial

interpolant. The upper bounds of the summations N1 and N2 must satisfy the condition that

(N1+N2) ≤ k. Dp1p2 , are the unknowns in Equation (4.1) which are the polynomial coefficients.

Clearly, depending on the order of accuracy the number of coefficients defining the reconstruc-

tion will vary. In general, the number of coefficients, N, for a particular order k and one solution


variable is determined with the following relation

N =1d!

d∏n=1

(k + n) , (4.2)

where d represents the number of space dimensions. For 2D problems, d=2 and N becomes

N =(k + 1)(k + 2)

2. (4.3)

Therefore, since the order of accuracy exhibited is equal to k+1, third-order accurate solutions

will have six unknowns, and fourth-order accurate solutions will have ten unknowns.

4.2.1 Conditions on Reconstruction

When determining the unknowns it is essential that several conditions are satisfied within the

reconstruction procedure. Specifically, for the k-exact reconstruction procedure, Barth outlines

three essential conditions [40]:

1. conservation of the mean;

2. k-exactness; and

3. compact support for the stencil.

The first condition specifies that the reconstructed polynomial must have the correct cell av-

erage, such that it is equal to the original cell average. This constraint can be defined mathe-

matically as

ui =1Ai

∫∫Ai

uki (x, y) dA (4.4)

where ui is again the primitive variable of interest, in the cell i. Additionally, the second

condition states that the reconstruction must be k-exact such that the reconstructed polynomial

is of degree k or less. Mathematically this condition is equivalent to,

uki (−→X −

−→Xi)− ukexact(

−→X ) = O(∆xk+1). (4.5)


Finally, the reconstructed polynomial in each cell must be independent of the mean solution vari-

able outside a relatively small area (i.e, the stencil for reconstruction should be compact) [40].

It should be noted that for a quadrilateral mesh, the current k-exact reconstruction scheme

uses a fixed central stencil which includes 8 neighbor cells for k=1 and 24 neighbors for k=2,

and k=3 [12]. For unstructured meshes in this work, the convention has been to reconstruct

using a fixed central stencil which includes three neighbor cells for k=1 and approximately 18

neighbor cells for k=2, and k=3.

It is worth noting that the stencil for high-order solutions is described further in Section 4.2.2.

However, the stencil is effectively three layers of neighbor cells around the cell of interest.

Neighbor cells are deemed to be those sharing a face with the cell of interest. This will typically

lead to a stencil of approximately 18 cells, but due to the inconsistent nature of unstructured

meshes can lead to stencils of 16 cells to 20 cells. However, this is still a sufficient stencil size

to solve for the derivatives.

4.2.2 Determination of Reconstruction Stencil

The k-exact reconstruction procedure leads to a system of overdetermined equations. Solu-

tions of this system can be computed using a least-squares technique that is also described by

Barth [68]. The objective of the least squares method is to minimize the sum of the errors taken

over the entire stencil. The errors that are being minimized are between the existing known

average value in the given neighbor and the average value that would result from the extension

of the reconstruction to the given neighbor. The details of setting up and solving this system

in a least squares manner are given in the Section 4.2.3.

The initial challenge that must be overcome in order to apply this method to general unstruc-

tured meshes will be the development of an algorithm to determine the neighbors and next to

nearest neighbors for each cell to be used in the reconstruction procedure. For a structured

mesh, this is a straight-forward task as depicted by Figure 4.1(a). In this figure, cells colored

in red depict the neighbors, while cells colored in blue depict the next to nearest neighbors for

cell i. This leads to a stencil of 24 cells, sufficient to solve the least-squares problem. When

applied to unstructured meshes, this task is not as clear. This is depicted in Figure 4.1(b),

where cells colored in red are obvious choices for neighbor cells. However, cells shown with a

question mark could be possible choices for next to nearest neighbors, or not considered in the

reconstruction of cell i. Nevertheless, once this challenge is overcome the CENO method can be

applied directly within a two-dimensional finite-volume scheme capable of calculating accurate

solutions on arbitrary unstructured meshes.


(a) (b)

Figure 4.1: (a) Nearest and next to nearest neighbors on structured grid (b) Nearest and next

to nearest neighbors difficult to determine on unstructured grid

(a) (b) (c)

Figure 4.2: (a) Nearest neighbors (b) Nearest and next to nearest neighbors (c) All neighbors

The algorithm that has been used to determine all required neighbors for the reconstruction

process makes use of the grid data structure that has been described in Section 3.3.1. It is

easy to determine cells that share a common face as the data structure used stores adjacent

faces. Therefore, the nearest neighbors can be quickly determined as any cell sharing a face

with the cell of interest. This is shown in Figure 4.2(a). The next to nearest neighbors are then

determined as any cell that is sharing a face with the neighbors as depicted in Figure 4.2(b).

This should typically lead to a system with nine neighbors and one central cell, giving a total

stencil of ten cells. However, this does not lead to an overdetermined system of equations when

considering fourth-order accurate solutions, with piecewise cubic reconstructions which have

ten unknowns. It is therefore necessary to extend the stencil once more in a similar manner to

determine an additional ring of cells. This results in the stencil 4.2(c).


The three-layer stencil of Figure 4.2(c) will lead to an overdetermined system of equations when

fourth-order accuracy is sought. It should be noted that while a smaller stencil could be used

when seeking third-order accuracy and piecewise quadratic reconstruction, the same stencil has

been implemented to alleviate additional programming situations, and remain consistent with

the structured solver; which uses the same stencil for third- and fourth-order accurate schemes.

4.2.3 Evaluation of the Derivatives for the Reconstruction

The next few sections will be focused on presenting a mathematical description of the overde-

termined system of equations that are required to determine the derivatives of the high-order

spatial reconstruction. This is done by dividing the development into three components:

� forming the overdetermined system;

� calculating the geometric coefficients and moments; and

� solving the system using a least-squares technique.

It is worth first considering a matrix of the form Ax6=b. Here A is a coefficient matrix, x is

the vector of solution quantities, and b is a vector of the right-hand sides of the equations.

It should be pointed out that for overdetermined systems of this type there is not a unique

solution that will satisfy the equations. Therefore, the goal of solving this system is such that

the least squares problem is minimized, i.e. that ‖Ax−b‖2 is minimized.

For convenience, we now introduce the vector c, which allows us to write our over-determined

system of equations in the form of an equality: Ax−b=c. The vector c is not actually formed

prior to solving the least-squares problem, but is representative of the resulting error incurred

by solving the least-squares problem. This system can then be solved. In this work we make use

of the pseudo-inverse matrix via Singular Value Decomposition (SVD) which will be described

later. In previous implementations of the CENO high-order schemes on structured meshes, a

Householder QR factorization method has also been used, and the reader is encouraged to review

the textbook by Lawson [69] for further information. However, the pseudo-inverse technique

has been used as it was shown in previous work to speed up the computation quite considerably.

The Overdetermined System

The evaluation of the coefficients, Dkp1p2, requires the least-squares solution of an overdetermined

system of linear equations Ax−b=c, where the coefficient matrix, A, of the linear system depends


only on the mesh geometry and can be partially calculated in a pre-processing step. The average

solution data at each reconstruction step is contained in the matrix, b, and the mean value error

in each control volume is in matrix, c, which has the norm minimized in the least-squares sense.

The preservation of the solution variable average value ui within the reconstructed cell is ex-

plicitly enforced by expressing the coefficient, Dk00, as a function of the other unknowns as

follows:

D00 = ui −k∑

p1=0

k∑p2=0

k∑p3=0

(p1+p2+p3 6=0)

Dp1p2p3

(xp1yp2

)i. (4.6)

where the term(xp1yp2

)i

is defined in the next section. Thus, the linear system that needs to

be solved for all other unknowns, except Dk00, has the form

(x0y1

)i1

. . .(xp1yp2

)i1

. . .(xky0

)i1

......

......

......(

x0y1)ij

. . .(xp1yp2

)ij

. . .(xky0

)ij

......

......

......(

x0y1)iM

. . .(xp1yp2

)iM

. . .(xky0

)iM

D01

...

Dp1p2...

Dk0

i

−

(u1 − ui)......

(uj − ui)......

(uM − ui)

=

c1......

cj......

cM

i

,

M ×N N × 1 M × 1 M × 1

where M represents the total number of neighboring cells in the reconstruction stencil, and N

represents the total number of unknown derivatives, minus one, since D000 has been eliminated

by using Equation (4.6).

At this point we can apply geometric weighting to the system of equations. The geometric

weights, wij , are applied to each corresponding row in the matrix A and vector b. The geometric

weight is specific to each control volume and serves the purpose of improving the locality of

the reconstruction. Geometric weights of the form presented below are assigned to each control

volume in order to have a more localized reconstruction:

wij =1

|−→r j −−→r i|, (4.7)


or

wij =1

|−→r j −−→r i|2, (4.8)

where −→r is the position vector from the indicated cell. In general, the inverse squared distance

weighting has been found in previous work to be more accurate. However, the inverse distance

was still implemented. In the next section we will explore a computationally efficient technique

by which the elements of the matrix A are calculated, followed by a technique for solving the

resulting overdetermined system.

Calculating the Geometric Coefficients

It is noted that the matrix A depends entirely on the geometry of the computational cells in

the stencil. Therefore, we seek to compute its entries in a computationally efficient manner,

taking into account both computational cost and memory usage. A computationally efficient

approach involves pre-computing and storing only the geometric coefficients of each cell about

its own cell center. That is, the quantity given by

(xp1yp2

)j≡ 1Aj

∫∫Aj

(x− xj)p1 (y − yj)p2 dA . (4.9)

We can make use of these stored coefficients to calculate the elements found in matrix, A, by

using the appropriate values for the powers (p1 and p2) of(xp1yp2

)j. Without going through all

the details, as they are well described within [70], it is then possible to express a final equation

for calculating the elements of matrix A as

(xp1yp2

)ij

=p1∑ξ=0

p2∑`=0

[Cξp1C

`p2 · ∆xξij∆y

`ij ·

(x (p1−ξ) y (p2−`)

)j−(xp1yp2

)i

]. (4.10)

Upon examination of Equation (4.10), it is noted that the RHS only contains geometric coeffi-

cients of neighboring cells taken about their own cell centers. To clarify this point, notice that

the term denoted(x (p1−ξ) y (p2−`)

)j, is simply a shift in the powers of the stored

(xp1yp2

)j

term. Hence, when computing the elements of matrix, A, it is a straightforward matter of car-

rying out the summations on the RHS of Equation (4.10), by selecting the appropriate values

for the powers of the pre-computed(xp1yp2

)j

belonging to the given neighboring cell j.

Finally, the coefficients of the binomial expansion can be computed, with initial coefficients

C0p1 = C0

p2 = 1, as follows:


Cξp1 =p1 − ξ + 1

ξCξ−1p1 , C`p2 =

p2 − `+ 1`

C`−1p2 , (4.11)

It is therefore essential to compute(xp1yp2

)j

in an accurate and efficient manner. For general

triangular elements, of primary concern for the two-dimensional unstructured meshes considered

within this work, it is possible to map the triangles to a barycentric coordinate system. At which

point it is possible to apply a high-order numerical quadrature formula [71] to compute the

integration. In practise however, it is also possible, and slightly simpler to map the barycentric

quadrature points onto the triangle of interest and calculate the quadrature. This can be

accomplished through the use of the following relations:

x = λ1x1 + λ2x2 + λ3x3, (4.12)

y = λ1y1 + λ2y2 + λ3y3, (4.13)

where the λi’s are the barycentric quadrature locations, xi, yi are the vertex locations of the

triangle, and x, y are the mapped quadrature location on the triangle of interest. It is then

possible to apply a high-order quadrature equation to these positions to integrate the desired

function for(xp1yp2

)j

over the domain of the triangle. The quadrature equation implemented

is a 13 point, integration of Hammer [72] with seventh order accuracy.

Least-Squares solution of the Overdetermined System

While the system discussed can be solved in several different manners, we will limit the dis-

cussion here to solution via the pseudo-inverse matrix, which is found by the orthogonal SVD

of matrix A. The pseudo-inverse matrix, A+, has dimensions N ×M , can be used to solve the

lease-squares problem through the following equation:

x = A+b , (4.14)

where the resulting vector x, which is of length N , is such that we have minimized ||Ax− b||2.

The SVD method used to obtain the pseudo-inverse matrix, consists of a decomposition of the

the matrix A, such that

A = QΣVT , (4.15)


where Q is an M×M orthogonal matrix, Σ is an N×N diagonal matrix with non-negative real

values along the diagonal, and V is an N ×N orthogonal matrix. The values on the diagonal

of the matrix Σ are referred to as the singular values. The reader is encouraged to consult

the textbook by Nocedal [73] for details on finding the matrices Q and V, however now the

pseudo-inverse may be calculated through the above decomposition. This is obtained simply

by using:

A+ = VΣ−QT . (4.16)

The pseudo-inverse method is very appealing as it is a technique which is very robust and

numerically stable. Additionally, the pseudo-inverse depends solely on the geometry of the

mesh. It is therefore possible to compute the pseudo inverse once, and store it throughout the

computation. This has the effect of reducing computational time, as the matrix, A, does not

need to be re-computed every time a reconstruction is performed.

The description of the solution to the least-squares problem has been kept very brief here,

but intended to give the reader a general understanding of the solution technique. For addi-

tional information regarding the pseudo-inverse matrix the reader is encouraged to refer to the

literature [74, 69, 73].

4.3 CENO Reconstruction

A high-order central essentially non-oscillatory finite-volume scheme in combination with a

block-based adaptive mesh refinement algorithm has recently been developed by Ivan and Groth

for the solution of the one- and two-dimensional Euler equations on structured meshes [12, 17].

The scheme has additionally been applied to the Advection-Diffusion equation [18, 19]. Unlike

other essentially non-oscillatory (ENO) schemes [13, 25], or weighted essentially non-oscillatory

schemes (WENO) [14, 16], the CENO scheme is not based on information from multiple stencils.

Instead the CENO scheme is based on a central stencil which makes it well suited for extension

to three dimensions and unstructured meshes. The extension to three dimensions has been

completed by Rashad [32], and the extension to unstructured meshes is the focus of this work.

4.3.1 Reconstruction

The high-order CENO method uses a hybrid solution reconstruction process. This process is

based on the high-order k-exact least-squares reconstruction technique of Barth [31], which


has been described in the previous section, and on a fixed central stencil with a monotonicity

preserving limited piecewise linear least-squares reconstruction algorithm [68]. After the high-

order reconstruction, the hybrid CENO procedure calculates a smoothness indicator described

in Section 4.3.2. This smoothness indicator determines if the solution is resolved on the current

computational mesh. The limited piecewise linear (k=1) reconstruction procedure is applied to

computational cells with under-resolved solution content. Where the k-exact procedure is used

for cells where the solution is fully resolved [12, 17]. The limited piecewise linear reconstruc-

tion uses the limiters of Barth-Jesperson and Venkatakrishnan to preserve monotonicity near

discontinuities. In this way, the CENO reconstruction avoids the issues of capturing smooth

extrema [31, 47]. The slope limiters implemented are discussed in further detail in Section 4.3.3.

4.3.2 Smoothness Indicator

The smoothness indicator, S, that is used is a function of a smoothness parameter, α, the degree

off freedom, DOF , and the size of the stencil, SOS, used for the particular reconstruction in

question. The smoothness indicator is calculated as below,

S =α

max((1− α), ε)(SOS −DOF )DOF − 1

(4.17)

where ε is taken as a tolerance such that a divide by zero is avoided. The numerical value taken

for ε is 10−8. The smoothness parameter, α, is then calculated as follows

α = 1−∑

p(ukp(xp, yp)− uki (xp, yp))2∑p(ukp(xp, yp)− ui)2

(4.18)

where the ranges of the indexes p are the cells in the stencil, and i is the cell of interest. It can

be seen that α can have a value between negative infinity and one.

Considering those values for α in equation 4.19, it can easily be seen that as α approaches

unity the smoothness indicator, S will grow very rapidly. This is depicted in Figure 4.3, as

the term α1−α grows very quickly with increasing alpha. Once the smoothness indicator, S, has

been calculated it can be compared to a pass / no-pass cutoff limit. Experience has shown that

typical cutoff limits are in the range of 1,000-5,000 [12, 17]. Additionally, it should be noted

that typical values for S for smooth solutions tend to be orders of magnitude greater than the

cutoff limit [12, 17].


Figure 4.3: Graph of f(α) = α1−α

4.3.3 Monotonicity Enforcement

It has previously been discussed that an important consideration in high-order finite-volume

schemes is an affective approach to maintain monotonicity of the solution. The CENO scheme

uses a hybrid solution reconstruction procedure. This procedure combines the unlimited high-

order k-exact least-squares reconstruction with a monotonicity preserving limited piecewise

linear least-squares reconstruction algorithm. The limited reconstruction procedure is applied to

any computational cells that are deemed non-smooth or under-resolved based on the calculation

of the smoothness indicator.

Without the additional step of applying a limited reconstruction procedure, the k-exact recon-

struction would introduce spurious numerical oscillations that would become large in under-

resolved regions generating a Gibbs-like phenomenon. Therefore, a reliable smoothness indi-

cator must be able to identify correctly the areas which are under-resolved and revert to the


monotonicity preserving limited piecewise linear least-squares reconstruction.

For systems of conservation laws, the reconstruction procedure can be applied to different sets

of interrelated solution variables, such as conserved, primitive or characteristic variables. This

work is concerned primarily with the Euler equations, and the reconstruction procedure has

been calculated on the primitive solution variables. For situations where the piecewise linear

least-squares reconstruction procedure has bee applied, slope limiters must be used. Both the

slope limiters of Barth-Jesperson [31] and Venkatakrishnan [47] have been used.

4.3.4 Flux Evaluation

The numerical fluxes are determined for each cell by solutions of a Riemann problem. Given

the left and right solution states at the quadrature points, the numerical flux is given by

−→F · −→n = F(Ul,Ur,

−→n ), (4.19)

where the numerical flux, F , is evaluated by solving the Riemann problem in a direction that

is defined by the outward normal to the cell face [12]. The left and right solutions states

are calculated at the cell quadrature points by performing a piecewise k-order CENO poly-

nomial solution reconstruction as has been previously described. The spatial accuracy of the

finite-volume scheme is dependent on the order of the solution reconstruction. A k-order re-

construction provides a k+1 order accurate spatial reconstruction.

It is essential to compute the flux at the correct number of locations to preserve the order of

the scheme. Current practise has been to use one quadrature point for second-order schemes

(piecewise linear k=1 reconstruction), and two quadrature points for third- and fourth-order

schemes (piecewise quadratic, k=2, and cubic, k=3 reconstruction). This ensures that the order

of accuracy of the scheme is preserved. This is required at each cell face. A depiction of the

quadrature points on a particular cell is shown in Figure 4.4 where cell i is shown to have six

quadrature points, two on each cell face.

4.4 Multi-block Implementation

The proposed CENO high-order scheme also makes use of a multi-block implementation using

domain decomposition. This implementation takes advantage of the mesh partitioning de-

scribed in Section 3.2. The computational mesh is broken into smaller sub-meshes on which


Figure 4.4: Quadrature points on a typical high-order unstructured cell

solutions can be obtained in a simultaneous fashion. This has the potential to greatly reduce

the time to obtain a solution as calculations on various sections of the mesh can be carried out

simultaneously.

The important consideration for this multi-block implementation is proper treatment of solution

content at block boundaries. This issue is overcome with ghost cells. Ghost cells in one block

overlap physical cells in the adjacent block. The ghost cells must be updated at every time step

throughout the computation with the average values of the cells they are overlapping. Since

the cell average values are being passed to the ghost cells, before the flux can be computed

the ghost cell solution must be reconstructed. Therefore, it is essential to ensure that enough

cells are passed to compute the solution reconstruction in the cell immediately adjacent to the

block. This is depicted with the use of Figure 4.5. The figure shows two blocks separated by

the highlighted line. If cell i is in the block of consideration, cell j is a ghost cell that is needed

to compute the flux between blocks. Therefore, all cells colored in blue must also be passed to


Figure 4.5: Ghost cells required for message passing

the block of consideration to reconstruct the value of cell j.

It is worth noting that particular attention must be paid to the ghost cells required when

there are many blocks, and various cells required for the reconstruction are coming from several

different blocks. This situation has been treated within the current computational framework

of the code.

4.5 High-Order Treatment for Curved Boundaries

Correct high-order treatment of boundary conditions (BCs) is a crucial element for developing

accurate numerical schemes. It is especially important for high-order methods, where errors

due to geometrical approximation may dominate the discretization error, mitigating the full

capabilities of a high-order scheme [30]. In order to obtain high-order accuracy at boundaries,


the geometric data (i.e., cell area, centroid, geometric moments, normals, edge lengths, locations

of the Gauss quadrature integration points) are computed to the same order of accuracy as that

of the interior scheme. Implementing BCs with additional layers of ghost cells can work well

for relatively straight boundaries, but may give rise to large errors when the geometry is highly

curved.

Curved boundaries present several immediate challenges, such as the problem of accurately

obtaining the geometric data. For instance, in order to compute the geometric moments, a

highly-accurate integration must be calculated over the area of the cell, which now includes

a curved surface. This is not a straightforward integration procedure. Similarly, the other

types of geometric data is not as straightforward to determine with the presence of a curved

boundaries.

Additionally, if we consider the example of an inviscid solid wall boundary, providing a suitable

normal for the mirroring process which results in accurate high-order reconstructions for the

velocity components is problematic for a curved boundary. Therefore, a better alternative is to

enforce the boundary conditions using only information from the interior domain and from the

required boundary values at the Gauss integration points along the boundary. By constraining

the least-squares reconstruction in control volumes adjacent to the boundary, complex boundary

conditions can be enforced such as those of Olivier-Gooch and Van Altena [75].

The treatment of curved boundaries has not been considered in this thesis. However, this issue

must be dealt with in future follow on studies.

4.6 Example

Continuing with the example that was investigated in the previous chapter, a two-dimensional

shock box, Figure 4.6 shows a second-order solution compared to a fourth-order result. Both

calculations were computed by the proposed Godunov-type finite-volume schemes on the same

computational mesh consisting of four blocks, 2840 nodes and 5482 elements. The flux in both

situations was computed using Roe’s approximate Riemann Solver [41]. Both simulations were

calculated until 7 ms.

It can be seen that both calculations compute a flow structure in good agreement with what

has previously been investigated. However, the high-order solution provides more detail, at

least qualitatively. In general, the discontinuities appear much sharper. It is also worth noting

that the area circled in red in block solutions shows much more symmetry in the high-order


(a) (b)

Figure 4.6: Finite-Volume Shock-Box Calculation on Unstructured Mesh: (a) 2nd-order (b)

4th-order

solution than in the low-order case. In general, the high-order solution much closer resembles

the earlier results given in previous chapters. It is also worth noting that the mesh used for

these simulations is much coarser than what was previously investigated, and the high-order

solution does a fairly good job of providing a solution of equivalent quality.

This outcome is not a surprise as the goal of generating high-order solution schemes is to obtain

a greater level of accuracy on the same computational mesh. Alternatively, the goal is to obtain

the same level of accuracy but using a coarser mesh, but in this case it is more beneficial for

qualitative comparison on the same mesh. This situation will be investigated further, along with

other cases in the following chapter, where we will focus, not only on qualitative appearance of

the solution, but quantitative results.

Chapter 5

Numerical Results

An essential step in the development of algorithms in CFD is the verification and validation

process. The concepts of verification and validation have been defined in the AIAA guide-

lines [76]. Verification is the “process of determining that a model implementation accurately

represents the developer’s conceptual description of the model and the solution to the model”

[76], whereas validation is the “process of determining the degree to which a model is an accurate

representation of the real world from the perspective of the intended uses of the model” [76].

In this chapter, we present the verification of the high-order CENO finite-volume scheme as de-

veloped for multi-block unstructured mesh. Several key features of the scheme are highlighted,

and the numerical results from selected test cases are presented. The validation process will

be an important future consideration as the scheme is applied to engineering applications in

future follow-on studies. It is through the process of verification and validation that we build

confidence that the solutions generated by the algorithms we have implemented are producing

accurate results that represent the correct solutions to the flow problems of interest.

The CENO method has previously been through an extensive verification and validation process

by Ivan and Groth [12, 17, 18, 19], and additionally by Rashad [32] on structured meshes.

However, with the new implementation on multi-block unstructured meshes, it is once again

necessary to verify that the theoretical accuracy of the schemes is achieved on the unstructured

meshes to again gain confidence in the numerical results.

This chapter will begin by presenting the theory necessary to evaluate the scheme, followed

by several examples of function reconstructions in two-dimensions in order to first verify the

CENO reconstruction procedure for unstructured mesh. These reconstructions will consider

both smooth and non-smooth functions. Finally, there will be a demonstration of the application

of the proposed high-order CENO finite-volume scheme to the solution of two-dimensional Euler

60

Chapter 5. Numerical Results 61

equations, for several example flow problems.

5.1 Theory for Evaluating High-Order Algorithms

There are various examples in the literature of the verification of high-order methods such as

those described by Roach [77] and Zingg [78]. This research has followed these methods by

comparing high- and low-order solutions on the same grid resolutions, as well as systematic

grid convergence studies. The methods that are necessary for the convergence studies are

described in detail below focusing first on how to conduct a convergence study, followed by the

methodology for error calculations which are required.

5.1.1 Determining the Order of Accuracy

One critical step of determining the order of actual numerical accuracy that is achieved by a

scheme is determined via a systematic convergence study. The convergence study involves a

systematic increase in grid resolution while monitoring the L1, L2 and L∞ error norms. By

plotting this increase in grid resolution versus the error norms on a log-log plot, the best fit

through the results in log space yields the order of accuracy, as achieved by the numerical

scheme.

The convergence study therefore relies on the ability to calculate two things. First the exact

solution in each cell must be know, or at least accurately estimated. If the exact solution is

known, an average of the exact solution for each computational cell can then be computed as

Ui =1Ai

∫∫Ai

Uexact(x, y)dA , (5.1)

which is useful for beginning the computation. The second is the error between the recon-

structed solution and the exact solution. This calculation is described in the following section.

It is important to note that, for the unstructured mesh considered herein, the increase in

grid resolution is done through decreasing a characteristic length input into Gmsh [53]. This

characteristic length is connected to the size of the cells, and decreasing this length has the

effect of increasing the grid resolution. Therefore, for various orders of accuracy, the same

characteristic lengths can be used, which will ensure the same grids are used throughout the

various convergence studies of the different order methods.


5.1.2 Accuracy Assessment

The accuracy of the solution must be computed on each grid for the convergence study as

described above. This assessment includes calculating the L1, L2 and L∞ error norms, which

can be computed as

L1 = |E|1 =1AT

∑i

∫∫Ai

[Uk

i (x,y)−Uexact(x,y)]dA , (5.2)

L2 = |E|2 ={ 1AT

∑i

∫∫Ai

[Uk

i (x,y)−Uexact(x,y)]2dA}1/2

, (5.3)

L∞ = |E|∞ = max{ 1Ai

[ ∫∫Ai

[Uk

i (x,y)−Uexact(x,y)]dA}

over all cells i, (5.4)

where AT is the total area, Uki (x,y) is the reconstructed solution as a function spatial coordi-

nates, and Uexact(x,y) is the exact solution.

The integrations are carried out through high-order numerical quadrature rules for triangles.

As described in Chapter 4, the quadrature rules previously presented [71] have been used. This

numerical quadrature rule consists of 13 points and is accurate to seventh-order, which is higher

than the order of the spatial reconstructions considered herein.

5.2 Reconstruction of Two-Dimensional Functions

A useful first step in the verification of a high-order finite-volume scheme is to perform a spatial

reconstruction of various specified functions. We want to ensure that the reconstruction order

maintains the desired order of accuracy when a convergence studies is completed. We can

evaluate the error by first initializing the mesh with correct cell averages, then performing a

high-order reconstruction, and finally evaluating the error that has been incurred. Through this

reconstruction of functions. we can be sure that the error is a result of the spatial reconstruction,

and not due to the time marching schemes.

Several two-dimensional functions have been reconstructed and are presented below along with

the order of accuracy that was computed for each case. Initially, smooth functions are considered

to ensure that the desired order of accuracy is maintained. Then a non-smooth function will be

considered to verify the effectiveness of the smoothness indicator at maintaining monotonicity

when presented with solution content containing many discontinuities. In all cases throughout

this section it should be assumed that a smoothness indicator cutoff value of 1000 was used in


the CENO reconstruction, unless otherwise noted.

5.2.1 Spherical Cosine Function

The first case considered is the reconstruction of a smooth spherical cosine function. The

function is smooth in both the x and y directions and the exact solution is described by

uexact = 1 +13

cos(r) , (5.5)

where r is the radial position given as r =√x2 + y2. The solution is computed on a domain

that is 20×20 centered on x = y = 0, with various grid resolutions considered to complete the

convergence study.

The results for this reconstruct on are shown in Figure 5.1. More specifically, the exact solution

is given as Figure 5.1(a), a first-order (k=0) reconstruction is presented in Figure 5.1(b), and

a fourth-order (k=3) reconstruction is shown in Figure 5.1(c). The reconstructions were all

computed on the same mesh with 92040 triangular elements and 46419 nodes. This was the

most refined mesh for which the reconstruction of this function was computed. It should

be seen that the fourth-order reconstruction more closely matches the results of the exact

solution. The first-order reconstruction suffers from the mesh effects of prescribing the cell

average throughout the entire cell. Therefore, near areas of function variation the contours

are not a smooth transition as they are in the fourth-order reconstruction. As a result, the

first-order variation in the function appears fuzzy in the figure.

The L1, L2 and L∞ error norms associated with the reconstruction of the smooth trigonometric

function are depicted in Figure 5.2. This is shown on a log-log plot ranging from a course mesh to

a finer mesh. It can be seen that the error is decreased as we move from fist-order reconstruction

to fourth-order reconstruction as would be expected. Additionally, the slopes of the plot are

increased as the order of accuracy increases.

Finally, Table 5.1 presents the computed orders of accuracy for the L1, L2 and L∞ error norms.

In this case, each error norm was calculated by using the final three points that are shown in

the Figures above.

The reconstructions are reproducing the order of accuracy that are expected in the case of

the L1 error. In the other cases the reconstruction order of accuracy is fairly consistent with

what is expected, although slightly below the ideal value. However, the slopes of the error

are continuing to increase and it is possible that the asymptotic convergence has not yet been


(a) Exact solution.

(b) First-order CENO (k = 0) reconstruction. (c) Fourth-order CENO (k = 3) reconstruction.

Figure 5.1: Reconstruction of spherical cosine function.

achieved on the finest unstructured mesh considered.

5.2.2 Exponential Function

The next case considered is the reconstruction of a smooth exponential function. The form of

the exact function considered is


(a) |E|1 convergence study. (b) |E|2 convergence study.

(c) |E|∞ convergence study.

Figure 5.2: Convergence studies for spherical cosine function reconstruction.

Slopes

k |E|1 |E|2 |E|∞

0 -0.998 -1.009 -0.984

1 -1.984 -1.512 -0.844

2 -2.998 -2.681 -2.502

3 -4.443 -3.979 -3.508

Table 5.1: Orders of accuracy for the Spherical Cosine Function.


uexact = exp(xy) , (5.6)

and reconstruction results for a domain of 2×2 centered on x = y = 0 were assessed. Again,

many different grid resolutions have been considered in the convergence study.

The solutions to this function are depicted in Figure 5.3. Once again, the sub-figures show the

exact solution, a first order reconstruction, and a fourth-order reconstruction. This reconstruc-

tion was computed on an unstructured grid consisting of 22552 triangular elements and 11475

nodes. This mesh was selected as it was not as refined as what was shown in the previous

case, therefore more easily depicting the mesh effects caused by the first order reconstruction.

Once again, the fourth-order reconstructed solution is much more accurate at reconstructing

the exact solution.

The results of the convergence study for the reconstruction of the exponential function listed

above are shown in Figure 5.4. Once again, the L1, L2 and L∞ are shown on a log-log plot

verses the increased number of nodes. We can see that as the order of accuracy is increased

the error is decreased, and the slope of the error becomes greater achieving a higher-order of

accuracy in the reconstruction.

The slopes have been computed using the final three points on the plot, and are shown in

Table 5.2. We see that the L1 error norm again achieves the desired level of accuracy. The

plots detailing the convergence of the L2 and L∞ error norms are slightly below the expected

value, but are showing an increase in slope. Further grid resolutions were not computed due

to computational cost. However, it can still be seen that a higher-order of accuracy is being

achieved by the reconstruction, and certainly in the L1 error norms the expected order of

accuracy is well represented.

Slopes

k |E|1 |E|2 |E|∞

0 -1.015 -1.021 -0.972

1 -2.017 -1.540 -0.886

2 -3.051 -2.555 -2.154

3 -4.310 -3.801 -3.034

Table 5.2: Orders of accuracy for the Exponential Function.


(a) Exact solution.

(b) First-order CENO (k = 0) reconstruction. (c) Fourth-order CENO (k = 3) reconstruction.

Figure 5.3: Reconstruction of an Exponential Function.

5.2.3 Abgrall’s Function

The Abgrall function [25] is a function that presents a significant challenge for any high-order

scheme wishing to maintain monotonicity. The problems arise because the function has a

number of solution discontinuities. In order to capture these discontinuities accurately, mono-

tonicity must be maintained in the reconstruction. Therefore, this function is used to test

that the CENO schemes smoothness indicator can maintain monotonicity when presented with

many discontinuities.


(a) |E|1 convergence study. (b) |E|2 convergence study.

(c) |E|∞ convergence study.

Figure 5.4: Convergence studies for Exponential Function Reconstruction.


The Abgrall function is defined in as follows [25]:

Uexact(x, y) =

{f(x− cot

√π2 y) x ≤ cos πy

2 ,

f(x+ cot√

π2 y) + cos(2πy) x > cos πy

2 ,(5.7)

with

f(r) =

−r sin

(3π2 r

2)

r ≤ −13 ,

| sin(2πr)| |r| < 13 ,

2r − 1 + 16 sin(3πr) r ≥ 1

3 ,

(5.8)

where

r =√x2 + y2 . (5.9)

The reconstructed solutions to this problem have been depicted in Figure 5.5. These solutions

were computed of a mesh consisting of 91890 unstructured triangular elements and 46344 nodes.

The smoothness indicator cutoff value for this problem was selected as 2500. It is worth pointing

out that the fourth-order solution much more closely represents the exact solution then does

the first-order result. As has been seen in previous cases, the first-order reconstruction exhibits

mesh effects that cause transitions to appear less smooth. Additionally, the solution has been

compared with the exact solution computed on a structured mesh shown in Figure 5.5(b). The

unstructured high-order reconstruction does a very nice job producing a solution that closely

matches the exact solution, even when computed on a structured mesh of similar resolution.

The L1 error norms for this case are shown in Figure 5.6(a). In this case we see that the first-

and forth-order reconstructed solutions have similar orders of accuracy as shown from the slopes

in the plot. However, the error norm line is shifted downwards in the case of the fourth-order

reconstruction. This shows that while the order of accuracy may revert to first-order, due to

the many discontinuities present in this function, the solution is still more accurate. In fact, the

first-order reconstruction requires 91890 elements to produce an error less than 0.045, while the

fourth order reconstruction only requires 9916 elements to produce a similar error. Effectively,

this mean that the same error can be achieved using roughly 10 percent of the elements. This

is one of the key advantages to high-order methods, it is possible to use much fewer elements,

and still compute a solution to the same desired accuracy.

At this point it is worth inspecting the role of the smoothness indicator for this function. The

role of the smoothness indicator is to detect non-smooth areas in the solution reconstruction,

so that a limited linear piecewise reconstruction procedure can be applied in these areas in

order to maintain monotonicity. Inspecting the locations where the smoothness indicator has

deemed non-smooth in the reconstruction of Abgrall’s function can be done with reference


(a) Exact solution on unstructured mesh. (b) Exact solution on structured mesh.

(c) First-order CENO (k = 0) reconstruction. (d) Fourth-order CENO (k = 3) reconstruction.

Figure 5.5: Reconstruction of Abgrall’s function.

to Figure 5.6(b). Locations shown in red have been flagged as non-smooth solution content

and revert to a limited linear piecewise reconstruction. In this case, the smoothness indicator

does a very nice job at locating the discontinuities that require a lower-order reconstruction

to maintain monotonicity. The discontinuities can be seen with reference to Figure 5.5 and

compared with Figure 5.6(b) to ensure the correct locations are flagged as non-smooth. It

is because of the hybrid switching method and limited linear reconstruction that the scheme

reduces to first-order accurate for the reconstruction of this function.

Finally, the results of the slopes of the L1 error are shown in Table 5.3. In the two cases,


(a) L1 Error. (b) Smoothness Indicator.

Figure 5.6: Results for Abgrall’s function.

first-order (k = 0) and fourth-order (k = 3) reconstruction the order of accuracy of the scheme

is roughly first-order. This is an expected result, due to the discontinuities in Abgrall’s function

the high-order scheme will revert to first-order accurate near solution discontinuities lowering

the overall order of the scheme. However, it is worth once again mentioning that while the

overall order of the schemes are the same, the fourth-order reconstruction does produce more

accurate results, and a lower absolute error.

Slopes

k |E|1

0 -0.995

3 -1.143

Table 5.3: Orders of accuracy for the Abgrall’s Function.

5.3 Solution of Two-Dimensional Euler Equations

While the spatial reconstruction procedure has been proven through the above cases to produce

high-order accurate reconstruction, and the smoothness indicator has been shown to reduce the

order of accuracy as necessary, it is also essential to consider the high-order reconstructed

solutions to equation of fluid flow. We are considering the solution to inviscid compressible flow

governed by the Euler Equations.

This section will show that the unstructured solver yield similar results to those computed on a


structured mesh. Additionally, the smoothness indicator will once again be shown to behave as

expected, and that high-order accuracy is achieved. Moreover, the qualitative appearance of the

solutions will be shown to be better than those on computed using a second-order reconstruction

procedure.

Three cases are being investigated in the following section. First will be the solution to a

two-dimensional shock box. This will be followed by a spherical expansion case, and finally

a boundary value problem of a flow over a bump. In all cases it should be assumed, unless

otherwise noted, that a smoothness indicator cutoff value of 1000 was used with a CFL number

of 0.7. It can also be assumed that Roe’s approximate Riemann solver [41] and the slope-limiter

of Venkatakrishnan [47] have been used.

5.3.1 Unsteady Shock-Box Problem

The first case considered for high-order reconstruction of the Euler Equations is the Shock-

Box problem that has been discussed throughout this thesis. For completeness, the problem

is defined as a situation where two states are initially prescribed separated by a discontinuity.

The states considered are described below, where 1/4 of the domain is initiated as the left-state,

defined as

ρl = 1.408 kg/m3 ,

ul = vl = 0 m/s ,

pl = 101.1 kPa ,

(5.10)

and the remainder of the 10×10 m domain is initialized to right-state, defined as

ρr = 4.9 kg/m3 ,

ur = vr = 0 m/s ,

pr = 404.4 kPa .

(5.11)

The results are shown in Figure 5.7. The results are computed on an unstructured mesh

consisting of 46548 triangular elements with 23559 nodes. The figure shows a comparison of the

first-order solutions with the fourth-order solutions for density, pressure, and mach number. We

can see that the fourth-order solution content is much better resolved. The increased resolution

can be attributed to the reconstruction taking information from nearby cells to compute a

polynomial in each cell, as opposed to the piecewise constant representation of solution content

in the low-order case.


Figure 5.7 also shows the capability of the high-order solver to handle multi-block meshes. The

fourth-order solutions are shown as computed on four block meshes, which matches the single

block computations shown throughout earlier Chapters in this thesis.

Additionally, Figure 5.8 shows the impact of the smoothness indicator. Areas which are shown

in red have been flagged by the code as non-smooth and are then reconstructed using a limited

piecewise linear reconstruction. It can be seen that the non-smooth areas correspond with areas

of discontinuity in the density plot, and therefore it can be determined that the CENO hybrid

scheme is acting appropriately.

Finally, Figure 5.9 shows the density variation computed on a structured mesh, which should

be compared with Figure 5.8(a) computed on an unstructured mesh. The structured solver

uses second-order limited linear piecewise reconstruction with a second-order time-marching

scheme. Flux calculations are computed with Roe’s approximate Riemann solver [41] and

monotonicity is preserved using the slope limiter of Venkatakrishnan [47]. The structured grid

is composed of 40,000 quadrilateral elements. As expected, the results computed on the on

the structured mesh are more symmetric, and somewhat better resolved compared with the

solution on the unstructured mesh. The improvements that the structured case show compared

to the unstructured case can be attributed to the regularity of the structured grid.

5.3.2 Unsteady Cylindrical Expansion

The next case considered is a two-dimensional cylindrically symmetric expansion, taken from

Van Leer [79]. The expansion occurs as a homentropic process, and therefore entropy is uniform

and constant throughout the calculation. The initial values represent a purely radial expansion

extending from r=12 to r=3

2 . The radial velocity is given by

ur =

0 r ≤ 1

2 ,

1γ (1 + tanh r−1

0.25−(r−1)2) 1

2 < r < 32 ,

2γ r ≥ 3

2 ,

(5.12)

with other flow quantities obtained using the following,

ur +2

γ − 1a =

2γ − 1

, (5.13)

ρ = γa2

γ−1 , (5.14)


p =ρa2

γ, (5.15)

where

r =√x2 + y2 , (5.16)

and

θ = arctany

x. (5.17)

The comparison between the density for the first-order reconstruction and the fourth-order

reconstruction are shown in Figure 5.10. These solutions are shown at a time of T = 0.5 s.

Both have been computed on unstructured meshes with 12546 triangular elements and 6422

nodes. It is not difficult to see that the solutions with the high-order reconstruction procedure

and time marching scheme have a more symmetric appearance, and much smoother transitions

as the wave is expanding outwards over the solution domain.

In each case, several different meshes were used to provide the convergence study that is shown

for the L1 error norms of Figure 5.11. The third- and fourth-order reconstruction provides

much smaller errors, and also have a steeper slope indicating that the order of accuracy which

is being achieved is greater than in the case of the first-order computation.

The order of accuracy which are being computed using the last three points on each slope are

given in Table 5.4. As expected both the first- and third- orders of accuracy are being achieved

throughout this convergence study. The fourth-order case is not quite achieving fourth-order

accuracy, however, it is obtaining a greater order of accuracy, and less error than the third-order

case. The regime for the asymptotic convergence has possibly not been yet achieved on the

finest mesh considered here.

Slopes

k |E|1

0 -1.061

2 -3.091

3 -3.446

Table 5.4: Orders of accuracy for the Cylindrical Expansion.


5.3.3 Steady Supersonic Flow Over a Bump

The final case considered within this work has been a boundary driven problem of a compressible

inviscid flow over a bump. The west side of the domain is given by inlet boundary conditions

given by

ρ = 1.225 kg/m3 ,

u = 1361.18 m/s ,

v = 0 m/s ,

p = 101.325 kPa .

(5.18)

For these inlet conditions, the free stream Mach number is four. The east side of the domain

is prescribed as a constant extrapolation outlet condition, while the north and south boundary

conditions are reflecting walls.

The results are shown n Figure 5.12. The figure shows a comparison of the first-order solutions

with the fourth-order solutions for density, pressure, and mach number. The meshes used for

both cases are identical consisting of 10986 triangular elements and 5652 nodes.

It can easily be seen that the results in the high-order case show much smoother transitions

between the various states in the problems. This can be attributed to less effects of the mesh,

as the solution in each cell is represented by piecewise smooth functions, as opposed to a

piecewise constant representation. Therefore, the high-order solution does not exhibit the same

degradation in solution content due to mesh effects.

An error analysis has not been conducted for this case as the current high-order solver does not

include the functionality to correctly treat the presence of curved boundaries, and therefore the

desired orders of accuracy will not be achieved. However, the solution content of the high-order

reconstruction still shows a qualitative improvement compared to the first-order reconstruction

and the capability to handle curved boundaries remains an area of future research.


(a) First-order density. (b) Fourth-order density.

(c) First-order pressure. (d) Fourth-order pressure.

(e) First-order mach number. (f) Fourth-order mach number.

Figure 5.7: Unsteady Shock Box


(a) Fourth-order density. (b) Smoothness Indicator for density.

Figure 5.8: Demonstration of the Smoothness Indicator.

Figure 5.9: Density variation on a structured mesh.


(a) Fourth-order density in 3D.

(b) First-order calculation of density. (c) Fourth-order calculation of density.

Figure 5.10: Unsteady Cylindrical Expansion


Figure 5.11: L1 Error for the Cylindrical Expansion


(a) First-order calculation of density. (b) Fourth-order calculation of density.

(c) First-order calculation of pressure. (d) Fourth-order calculation of pressure.

(e) First-order calculation of mach number. (f) Fourth-order calculation of mach number.

Figure 5.12: Flow over a bump.

Chapter 6

Conclusions and Future Research

High-fidelity advanced numerical solution methods are today used extensively in engineering

design in a wide range of fields, including aerospace, automotive, energy, medical, and others.

Using such CFD tools, computer simulations can be conducted prior to building and testing

prototypes, reducing cost and development time. However, high-order methods have not yet be-

come widespread use by designers due to issues including solution monotonicity, computational

cost, robustness and simplicity of available high-order solution techniques. This research has

attempted to address these concerns by further developing a high-order finite-volume scheme

to compute solutions on two-dimensional unstructured meshes, in a manner that does not com-

promise solution monotonicity, while avoiding the complications of other high-order methods.

This development is a step in the direction of a solution method that could compute high-order

accurate solutions to physically complex phenomenon such as combustion, through the geome-

tries typically found in engineering applications. This future research could lead to improved

combustor designs capable of meeting power generation requirements while responding to in-

creasing environmental concerns and regulations. The overall outcome will assist in achieving

sustained improvements in society, quality of life, and the environment.

6.1 Conclusions

A high-order CENO finite-volume algorithm has been extended for the solution of the Eu-

ler equations governing compressible, inviscid, gaseous flows on two-dimensional unstructured

meshes. The scheme is based on a Godunov type finite-volume method and the CENO recon-

struction method of Ivan and Groth [12, 17, 18, 19]. The development began by first imple-

menting a two-dimensional solver using meshes created using Gmsh [53] as has been described

81

Chapter 6. Conclusions and Future Research 82

in Chapter 3. The solver was initially first-order accurate in both spatial and temporal dis-

cretizations. This was then extended to second-order accuracy by implementing a least-squares

limited piecewise linear reconstruction and a second-order time marching scheme. The spatial

reconstruction was then improved to third- and fourth-order accurate using the hybrid recon-

struction CENO method, and a fourth-order Runge-Kutta time marching scheme. The solver

is also capable of calculating solutions on multi-block meshes which have been partitioned using

PMETIS [58].

The CENO reconstruction procedure and the proposed high-order finite-volume method have

been thoroughly validated. Initial validation was provided through the high-order reconstruc-

tion of several functions. Third- and fourth-order accuracy was achieved and demonstrated

through the error analysis of the solutions. Additionally, the smoothness indicator was shown

to perform as anticipated, lowering the order of accuracy of the scheme near areas of disconti-

nuities and/or under-resolved solution content in order to maintain monotonicity.

Further validation was shown with solutions to the Euler equations. Three cases were inves-

tigated including an unsteady shock-box problem, an unsteady cylindrical expansion, and a

steady supersonic flow over a bump. The shock box case showed more accurate results with

the high-order solutions and the smoothness indicator worked as expected. The cylindrical

expansion cases proved that high-order solutions were achieved when computing the solution

to smooth inviscid compressible fluids. Finally, the solution to the bump case was shown to

produce improved results when using the CENO method.

6.2 Future Research

This thesis research represents the first steps in the development and application of the CENO

high-order finite-volume method to predictions of flow problems on multi-block unstructured

meshes. High-order accuracy has been achieved to the solution of test cases proving the effi-

ciency of the solver. However, there is still much work that can be done in order to apply this

method to real-world engineering problems.

First, the multi-block solver should be improved to be applied with a parallel implementation

on multiple processors to speed up computations. Additionally, the high-order reconstruction

should be extended to include the integrations, and flux computations over curved boundaries

as are often typically of engineering designs.

Following this, the unstructured high-order method as has been described in this thesis should

Chapter 6. Conclusions and Future Research 83

be extended to the solution of the Navier-Stokes equations, including turbulent effects as are

often encountered in most engineering applications of interest. Additionally, the application to

reacting flows should be included. These flows have a wide variation in spatial and temporal

scales, and show promise for improved solution content through the use of high-order solvers.

Finally, research should be carried out to combine the structured and unstructured CENO

schemes for application to hybrid meshes. These types of meshes may be extremely useful for

combining the improved solution content of structured meshes in the vicinity of solid bound-

aries with the ease of mesh generation in unstructured meshes elsewhere. The unstructured

solver should also be extended to three dimensions for handling complex geometries typical in

engineering applications, such as combustors.

While there is still much to be done, this study has been an effective proof of concept that the

CENO reconstruction method is applicable to high-order finite-volume schemes on unstructured

meshes. The use of unstructured meshes removes much work from the very user intensive

process of structured mesh generation, as unstructured meshes can be computed with efficient

algorithms as described in this thesis. Unstructured mesh solution methods have already been

shown to be extremely beneficial to the overall engineering design process. They have the

potential to reduce both development time and costs, while achieving improved designs.

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