Entropy-based Moment Methods for Rarefied Gases and Plasmas · Second, the Hessian matrix of the Newton method used in the dual minimization problem for the Lagrange parameters is

E N T R O P Y- B A S E D M O M E N T C L O S U R E S F O RR A R E F I E D G A S E S A N D P L A S M A S

Von der Fakultät für Mathematik, Informatik und Naturwissenschaften derRWTH Aachen University zur Erlangung des akademischen Grades eines

Doktors der Naturwissenschaften genehmigte Dissertation

vorgelegt von

M.Sc. ETH CSERoman Pascal Schärer

aus Baden, Schweiz

Berichter: Universitätsprofessor Dr. Manuel Torrilhon

Assistenzprofessor James McDonald, PhD

Tag der mündlichen Prüfung: 29.8.2016

Diese Dissertation ist auf den Internetseiten derUniversitätsbibliothek online verfügbar.

To my mother Zita and my father Thomas,To my sister Nadine,

To my love Debra.

Start by doing what’s necessary;then do what’s possible;

and suddenly you are doing the impossible.

— Francis of Assisi

A C K N O W L E D G M E N T S

I would like to share my gratitude towards those, who helped me during mytime as a doctoral student at RWTH Aachen.

My deepest thanks go to my advisor Manuel Torrilhon. For providing theopportunity to pursue research on a fascinating topic, his trust, careful guidanceand inspiring discussions on kinetic theory. Special thanks go to James McDonaldfor providing many helpful tips during his stay in Aachen and supporting thefinal stage of my doctoral studies as referee and co-examiner. Many thanks go toThierry Magin, for his hospitality at VKI during my short visit and stimulatinginputs regarding the description of plasma flows and chemical reactions.

My thanks go to my colleagues Julian Köllermeier, Armin Westerkamp, Zhen-ning Cai, Graham Alldredge, Jonas Kusch and Michael Abdel Malik for numer-ous inspirational discussions on moment equations and life. Special thanks goto Zhenning Cai for computing reference BGK data for the slab geometry case. Ialso thank Pratyuksh Bansal, who co-authored the publication [109], for his greatwork. I would like to thank all of my colleagues at MathCCES for providing anenriching and encouraging environment.

Finally, I want to express my deepest gratitude to my family who has alwaysbeen there to support me throughout my studies.

August, 2016

Roman Schärer

v

A B S T R A C T

The method of moments provides a flexible mathematical framework to derivereduced-order models for the approximation of the kinetic Boltzmann equation.This thesis investigates moment equations based on the principle of entropy max-imization to describe moderately rarefied gas and plasma flows in the transitionregime between the collision dominated continuum and the free molecular flowregime.

The maximum-entropy system has favorable mathematical features: The re-sulting system of partial differential equations is in conservative form, satisfies anH-theorem and is symmetric hyperbolic. However, the robust and efficient numericalsolution of entropy-based moment closures is challenging: First, the maximum-entropy closure can have a singularity in the closing flux around the equilibriumdistribution, rendering initial value problems with data in local thermodynamicequilibrium questionable. Second, the Hessian matrix of the Newton methodused in the dual minimization problem for the Lagrange parameters is arbitrarilyill-conditioned. Third, moments of the maximum-entropy distribution functionare in general not available in closed form and have to be evaluated numerically,which can result in excessive run times.

The first issue can be avoided by bounding the velocity domain. Numerical ex-amples show that enlarging the velocity domain leads to smaller sub-shocks, i.e.,unphysical discontinuities in the continuous shock-structure problem. To studythe effect of the singularity in the closing flux, a closed-form closure for a sim-plified toy model problem is considered. Numerical results demonstrate that thesub-shock in the continuous shock-structure problem can be mitigated by non-linear closures and eventually removed by a singularity in the closing flux.

Several numerical examples are provided for the 35-moment system in slabgeometry, showing promising results for shock-structure and Riemann problems.The use of an adaptive basis method for the dual minimization problem allowsthe robust simulation of strongly nonequilibrium processes.

To reduce the excessive computational run times of the maximum-entropy clo-sure, high-performance implementations of the numerical integration algorithmsare developed for multi-core processors and graphics cards. Additionally, newefficient explicit and semi-implicit time-integration schemes based on a formula-tion in the Lagrange parameters of the dual minimization problem are presented.

vii

Z U S A M M E N FA S S U N G

Die Methode der Momente ist ein flexibles mathematisches Gerüst zur Herlei-tung ordnungsreduzierter Modelle zur Approximation der kinetischen Boltz-manngleichung. Diese Dissertation untersucht auf dem Prinzip der Entropiema-ximierung basierende Momentenabschlüsse zur Beschreibung moderat verdünn-ter Gas- und Plasmaströmungen.

Das resultierende PDE-System hat vorteilhafte mathematische Eigenschaften:Die Gleichungen können in Erhaltungsform geschrieben werden, sind symmetrischhyperbolisch und erfüllen ein H-Theorem. Allerdings stellt die effiziente und robus-te numerische Lösung entropiebasierter Momentenabschlüsse eine Herausforde-rung dar: Erstens kann der Abschluss des Momentensystems eine Singularitätum die Gleichgewichtsverteilung aufweisen, so dass das Anfangswertproblemmit Daten im lokalen Gleichgewicht schlecht gestellt ist. Zweitens kann die Hes-sematrix des Newton-Verfahrens für das duale Minimierungsproblem zur Be-rechnung der Lagrangemultiplikatoren beliebig schlecht konditioniert sein. Drit-tens können Momente der Maximum-Entropie-Geschwindigkeitsverteilung imAllgemeinen nicht in geschlossener Form ausgedrückt werden, so dass auf nu-merische Integrationsmethoden zurückgegriffen werden muss, was den Rechen-aufwand erheblich erhöht.

Das erste Problem kann umgangen werden, indem das Geschwindigkeitsge-biet begrenzt wird. Numerische Beispiele zeigen, dass die Vergrösserung desGeschwindigkeitsgebiets die Sprunghöhe der Sub-shocks verkleinern kann. Umden Effekt der Singularität für unbeschränkte Geschwindigkeitsgebiete genau-er zu untersuchen, wird ein Abschluss in expliziter Form für ein vereinfachtesModellproblem betrachtet. Eine numerische Untersuchung zeigt, dass die Stärkedes Sub-shocks im Grenzfall einer Singularität komplett verschwindet.

Mehrere numerische Beispielrechnungen für das 35-Momentensystem zeigenvielversprechende Ergebnisse für die Approximation eindimensionaler, statio-närer Schockstrukturen und zeitabhängiger Riemannprobleme. Die Verwendungeiner adaptiven Basis für das duale Minimierungsproblem erlaubt die robusteBerechnung starker Nichtgleichgewichtsprozesse.

Um die Laufzeit zur Berechnung der numerischen Quadratur zu reduzieren,werden optimierte Implementierungen für Mehrkernprozessoren und Grafikkar-ten untersucht. Darüber hinaus werden effiziente explizite und semi-impliziteZeitschrittverfahren vorgestellt, welche auf einer Formulierung in den Lagrange-Parametern des dualen Minimierungsproblems basieren.

viii

C O N T E N T S

1 introduction 1

2 kinetic theory of gases 5

2.1 Kinetic Theory of Single-species Gases . . . . . . . . . . . . . . . . 5

2.2 Binary Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Hard-sphere Collisions . . . . . . . . . . . . . . . . . . . . . 9

2.3 Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 H-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.7 BGK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.8 Collisionless Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.9 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.10 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 moment approximations 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Closure Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Properties of Moment Approximations . . . . . . . . . . . . . . . . 20

3.4.1 Conservation of Mass, Momentum, and Energy . . . . . . . 20

3.4.2 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.3 Convex Entropy Extension . . . . . . . . . . . . . . . . . . . 21

3.4.4 Galilean Invariance . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Maximum-entropy Closures . . . . . . . . . . . . . . . . . . . . . . . 24

3.5.1 Dual Formulation . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5.2 Entropy Dissipation . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.3 Realizability and Stability . . . . . . . . . . . . . . . . . . . . 26

3.5.4 Admissible Moment Systems . . . . . . . . . . . . . . . . . . 27

3.5.5 Hierarchy of Moment Systems . . . . . . . . . . . . . . . . . 28

3.6 The Euler System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Slab Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.7.1 Moment Systems in Slab Geometry . . . . . . . . . . . . . . 31

3.8 Grad’s Classical Closure . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.8.1 Connection to the Maximum-entropy Closure . . . . . . . . 33

3.8.2 Function Approximations . . . . . . . . . . . . . . . . . . . . 34

3.8.3 Regularized Grad Closures . . . . . . . . . . . . . . . . . . . 36

ix

x contents

3.8.4 Hyperbolic Grad Closures . . . . . . . . . . . . . . . . . . . 36

3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 numerical methods 39

4.1 The Finite-volume Scheme . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 Conservation Property . . . . . . . . . . . . . . . . . . . . . 40

4.1.2 Numerical Flux Functions . . . . . . . . . . . . . . . . . . . 40

4.2 Semi-discrete Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Compact Form . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.2 A Second-order Scheme . . . . . . . . . . . . . . . . . . . . . 43

4.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Numerical Schemes for the Maximum-entropy Closure . . . . . . . 44

4.3.1 Discretization of the Velocity Domain . . . . . . . . . . . . . 45

4.4 Numerical Implementation of the Minimization Problem . . . . . . 48

4.4.1 Implementation of the BGK-operator . . . . . . . . . . . . . 49

4.4.2 Adaptive Basis Methods . . . . . . . . . . . . . . . . . . . . 50

4.4.3 Realizability Preserving Method . . . . . . . . . . . . . . . . 52

4.5 Efficient Methods for the Maximum-entropy Closure . . . . . . . . 55

4.5.1 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . 55

4.5.2 A Semi-implicit Time-marching Scheme . . . . . . . . . . . 56

4.5.3 Explicit First-order Scheme . . . . . . . . . . . . . . . . . . . 59

4.5.4 Higher-order Schemes . . . . . . . . . . . . . . . . . . . . . . 60

4.6 Efficient Time Stepping Algorithms . . . . . . . . . . . . . . . . . . 62

4.7 Discrete Velocity Model . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.7.1 Conservative BGK-operator by Entropy Minimization . . . 66

4.7.2 Conservative BGK-operator with Least-norm Projection . . 67

4.7.3 Spatial and Temporal Discretizations . . . . . . . . . . . . . 68

4.8 Memory Complexity of the DVM . . . . . . . . . . . . . . . . . . . 69

5 optimized quadrature implementations 71

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Efficient Quadrature Implementations . . . . . . . . . . . . . . . . . 72

5.2.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.2 Serial Implementation . . . . . . . . . . . . . . . . . . . . . . 72

5.2.3 Efficient Single- and Multi-core Implementations . . . . . . 73

5.2.4 Graphics Cards as Accelerators . . . . . . . . . . . . . . . . 73

5.2.5 Performance Measurements . . . . . . . . . . . . . . . . . . 75

5.2.6 Roofline Model . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 numerical examples 81

6.1 Homogeneous relaxation test . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Smooth density perturbation problem . . . . . . . . . . . . . . . . . 85

6.2.1 Time measurements . . . . . . . . . . . . . . . . . . . . . . . 89

6.3 Shock Structure Problems . . . . . . . . . . . . . . . . . . . . . . . . 93

contents xi

6.3.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3.2 Efficient Solvers applied to the Shock Structure Problem . . 98

6.4 Two-beam collision problem . . . . . . . . . . . . . . . . . . . . . . 101

7 regularized singular closures 109

7.1 Moment Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.1.1 Reduced Moments . . . . . . . . . . . . . . . . . . . . . . . . 111

7.1.2 The 5-moment System . . . . . . . . . . . . . . . . . . . . . . 111

7.1.3 Closure Problem . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.1.4 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.1.5 Grad’s Closure Theory . . . . . . . . . . . . . . . . . . . . . 113

7.1.6 The Maximum-entropy Hierarchy . . . . . . . . . . . . . . . 114

7.1.7 Realizability Conditions for the 5-moment System . . . . . 115

7.1.8 Realizability Conditions in 1D . . . . . . . . . . . . . . . . . 116

7.1.9 Realizability Conditions on Bounded Velocity Domains . . 118

7.1.10 Regularized Maximum-entropy Closures . . . . . . . . . . . 121

7.2 A New Closed-form Closure . . . . . . . . . . . . . . . . . . . . . . 122

7.2.1 Characteristic Wave Speeds . . . . . . . . . . . . . . . . . . . 124

7.3 Riemann Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.3.1 Linear Degenerate Waves . . . . . . . . . . . . . . . . . . . . 129

7.3.2 Hugoniot Locus from the Equilibrium State . . . . . . . . . 130

7.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.4.1 Reference BGK Solver . . . . . . . . . . . . . . . . . . . . . . 132

7.5 The Continuous shock structure Problem . . . . . . . . . . . . . . . 133

7.5.1 Vanishing Regularization Limit . . . . . . . . . . . . . . . . 135

7.6 Further Numerical Examples . . . . . . . . . . . . . . . . . . . . . . 135

7.6.1 Symmetric Two-Beam Riemann Problem . . . . . . . . . . . 135

7.6.2 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . 143

7.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8 moment approximations of plasmas 147

8.1 Kinetic Theory of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . 147

8.2 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.2.1 Diffusion Velocity and Mixture Quantities . . . . . . . . . . 151

8.3 Moment Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8.3.1 Ionization and Recombination . . . . . . . . . . . . . . . . . 153

8.4 Kinetic Modeling of Collisions . . . . . . . . . . . . . . . . . . . . . 155

8.5 Production Terms for the One-Dimensional Euler System . . . . . 156

8.5.1 General Momentum and Energy Production Terms . . . . . 158

8.5.2 Hard-sphere Collisions . . . . . . . . . . . . . . . . . . . . . 159

8.5.3 Electrostatic Interaction . . . . . . . . . . . . . . . . . . . . . 159

8.6 Vlasov-Poisson-BGK System . . . . . . . . . . . . . . . . . . . . . . 160

8.6.1 Expansion in Hermite Polynomials . . . . . . . . . . . . . . 161

xii contents

8.6.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . 163

9 conclusions 169

a other closure theories 171

a.1 Multi-Gaussian Closures . . . . . . . . . . . . . . . . . . . . . . . . . 171

a.2 Partial Moment Approximations . . . . . . . . . . . . . . . . . . . . 174

a.2.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . 175

a.2.2 Entropy Dissipation and Hyperbolicity . . . . . . . . . . . . 175

a.2.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . 176

a.3 Kappa Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

b notations and general results 181

b.1 Tensor Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 181

b.1.1 Special Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 181

b.2 Trace-free Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

b.3 Tensor Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

b.4 Moment Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

b.5 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 183

b.6 Gauss-Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

b.7 Jacobian Matrix of the 5-moment System . . . . . . . . . . . . . . . 185

bibliography 187

curriculum vitæ 199

1I N T R O D U C T I O N

On a macroscopic scale, gases are huge collections of interacting particles sepa-rated by large distances compared to their effective range of interaction. In classi-cal gases the trajectory of each particle is determined by Newton’s laws of motionand the state of the gas is fully described by the knowledge of all positions andvelocities of the particles if each particle is assumed to be spherical with no in-ternal degrees of freedoms. Consequently, the state of a gas with N interactingparticles is governed by the evolution equations of Newton’s laws.

By collecting the degrees of freedoms of all particles in a single vector, the stateof the gas is described by the coordinates of a single point in a 6N dimensionalphase-space, which contains all possible states of the gas. The temporal evolutionis then described by the trajectory of this point in the 6N dimensional phase-space.

However, since the state of the gas is usually not know exactly, the point inphase-space is replaced with a probability density, P. In 1838 Liouville publishedan evolution equation in [77] for the probability density, P, which became one ofthe cornerstones of statistical physics.

For gases at standard conditions, one cubic meter contains on the order ofN = 1025 particles [114], rendering the direct solution of Liouville’s equationinfeasible. Furthermore, the detailed description of Liouville’s equation is usuallynot necessary if only macroscopic phenomena of gases are of interest.

In 1872 Boltzmann published in his seminal paper [15] an evolution equationfor the one-particle distribution function, f , by which the phase-space of the gasis reduced dramatically from 6N to only 6 dimensions. Boltzmann’s evolutionequation is based on several assumptions [22, 49, 114], such as:

1. The length scale is much larger than the effective interaction range of theintermolecular forces.

2. The gas is sufficiently dilute, such that only binary collisions have to beconsidered.

1

2 introduction

3. Velocities and positions of particles before a collision are uncorrelated. Thisassumption is called molecular chaos.

The assumption of molecular chaos introduced by Maxwell in [84] is a crucialpart of Boltzmann’s equation. It implies the famous H-theorem [15] and thetime-irreversibility of Boltzmann’s equation. Lanford provides in [72] a rigorousderivation of Boltzmann’s equation, showing the validity for intervals of timemuch shorter than the mean free time, i.e. the average time of a particle betweensuccessive collisions; see [47] for a detailed discussion and further references.

For many gas processes, even the one-particle distribution function is a toodetailed description. As an illustrative example, let us consider an initially per-turbed gas in a container left at rest. As time progresses, the collisions betweenthe particles give rise to a relaxation of the gas to an equilibrium state, for whichthe gas remains at rest at a macroscopic level, even though each individual parti-cle follows a complicated trajectory as it undergoes collisions with other particles.In the equilibrium state, the velocities of the particles are distributed accordingto the Maxwell distribution [82, 83], which only depends on the mass density,macroscopic velocity and thermodynamic temperature of the gas. Thus the ve-locity distribution can be replaced by the macroscopic variables describing massdensity, velocity and temperature.

The collisional effect is quantified by the Knudsen (Kn) number, which is de-fined as the ratio of the mean free path, i.e. the average distance traveled by aparticle between successive collisions and a macroscopic reference length scale.In the limit of small Knudsen numbers Kn → 0, the velocity distribution canbe shown to be Maxwellian and the Boltzmann equation reduces to the Eulersystem, which contains evolution equations for mass, momentum and energydensity.

For gas flows close to the equilibrium state, the Navier-Stokes-Fourier (NSF)equations allow for an accurate description [123]. However, if the Knudsen num-ber becomes sufficiently large, the state of the gas can strongly deviate from theequilibrium state into the non-equilibrium regime, rendering the NSF approxi-mation invalid. Strong non-equilibrium states can occur in localized regions evenfor moderately rarefied gases, e.g. in traveling shock-waves or gas-wall bound-ary layers, see [90, 123]. The kinetic non-equilibrium effects occurring at largerKnudsen numbers can be very counterintuitive, such as a growing mass flow ratein a Poiseuille flow for increasing Knudsen numbers or a heat flux from low tohigh temperatures in driven cavity flows [123], which are in sharp contrast to thepredictions of the constitutive laws of NSF.

A promising approach to describe non-equilibrium effects for moderately rar-efied gases are moment equations, which are derived directly from the Boltz-mann equation by a Galerkin type projection in the velocity space, which reducesthe phase-space from six to three dimensions. In contrast to Boltzmann’s equa-

introduction 3

tion, moment systems only describe the evolution of macroscopic quantities. Themost fundamental moment approximation of Boltzmann’s equation are the Eulerequations. While the Euler equations are only valid in the equilibrium state, theinclusion of higher-order moment equations allows to extend the validity intothe non-equilibrium regime, see e.g. [111].

Figure 1.1 visualizes three fundamental levels of gas descriptions. While themicroscopic description takes into account the positions and velocities of all par-ticles, the kinetic theory characterizes the gas by the distribution function f . Afurther model order reduction yields a macroscopic description of the gas, whichonly considers a finite number of fields, which only depend on time and position.

(x, t)

v(x, t)

(x, t)...

f(x, t, c)

Kinetic Theory Macroscopic FieldsMolecular Dynamics

Model order reductions

microscopic mesoscopic macroscopic

Figure 1.1: From microscopic to mesoscopic and macroscopic descriptions of gas flows.

Moment equations require a closure theory, i.e. a constitutive law express-ing the higher-order moments through the known lower-order moments. Thefocus of this thesis is on closure theories based on the principle of entropy-maximization. These closures have promising mathematical properties as theylead to symmetric hyperbolic moment systems with a convex entropy law. Theobject of this thesis is to investigate entropy-based closures for moment approxi-mations of gases and plasmas. The application of entropy-based closures for gasdynamics is challenging, because of several difficulties associated with entropy-based closure theories. In this thesis the following challenges associated withentropy-based closures are addressed:

1. The maximum-entropy closure can become singular in the equilibriumstate, rendering the initial value problem ill-posed.

2. The Hessian matrix in the dual minimization problem for the Lagrangemultipliers can become arbitrarily ill-conditioned.

4 introduction

3. Numerical solutions require long execution times, because of the high com-putational complexity of maximum-entropy closures based on numericalquadrature methods.

This thesis is structured as follows: Chapter 2 presents the kinetic theory forclassical gases. An introduction to moment approximations for gases is providedin Chapter 3. The main properties of entropy-based closures are discussed andcompared with other well-known closure theories. Numerical methods used todiscretize and solve the resulting moment systems and the reference kinetic equa-tions are described in Chapter 4. High-performance implementations and runtime measurements of the maximum-entropy closure are presented in Chapter5. Chapter 6 presents numerical results to several test cases for the 35-momentsystem in slab geometry. In Chapter 7 a closed-form closure, approximating thecorresponding maximum-entropy closure, is investigated for the 5-moment sys-tem in a simplified, one-dimensional setting in order to analyze the effect of thesingularity in the closing flux on the evolution of the gas. Chapter 8 presentsmoment approximations to plasma flows with numerical results to the Vlasov-Poisson-BGK model. The main findings of this work are summarized and dis-cussed in the final Chapter 9. In Appendix A, some related methods, such as theMulti-Gaussian closure and the partial moment method are briefly described. Fi-nally, some general results are presented in Appendix B.

Parts of Chapters 3, 4, 5, and 6 have been published in [109, 112]. The paper [109]was co-authored by Pratyuksh Bansal, who contributed the initial optimized C++

and CUDA implementations, which have been further optimized by the author.Chapter 7 is mainly based on the publications [110, 111].

2K I N E T I C T H E O RY O F G A S E S

Kinetic theory is a statistical description of gases and plasmas composed of manyinteracting particles such as atoms or molecules. This chapter provides a short in-troduction to the kinetic theory of ideal, monatomic gases consisting of identicalparticles. The kinetic theory introduced in this chapter serves as the foundationfor the moment models discussed in Chapter 3. The kinetic theory for single- andmulti-species plasmas is presented in Chapter 8.

2.1 kinetic theory of single-species gases

Let us consider a classical, monatomic and ideal gas composed of identical parti-cles. In kinetic theory this gas is described by the one-particle distribution func-tion

f : Ωx ×Ωt ×Ωc → R≥0, (x, t, c) 7→ f (x, t, c), (2.1)

where x ∈ Ωx ⊆ Rdx is the spatial position, c = (c1, . . . , cdc )T ∈ Ωc ⊆ Rdc

denotes the microscopic velocity and t ∈ Ωt ⊆ R≥0 is the time. The distri-bution function f describes the number density of the gas in the phase-spaceΩ = Ωx × Ωc, such that integration over all microscopic velocities yields thenumber density

n(x, t) = 〈 f 〉(x, t) :=∫

Ωc

f (x, t, c) dΩc (2.2)

in physical space at time t. Similarly, the velocities of particles with position x attime t are distributed according to the probability density function

F(x, t, c) =f (x, t, c)n(x, t)

with 〈F〉(x, t) = 1. (2.3)

In the physically most relevant setting, the spatial and velocity dimensionsdx, dc are both three-dimensional, such that the phase-space Ω is six-dimensional.

5

6 kinetic theory of gases

However, often simplified geometries are used to study specific processes, forwhich either the spatial dimension or both the spatial and velocity dimensionsare reduced.

Macroscopic quantities of the gas are obtained by weighted averaging of thedistribution function f . Let m denote the mass of the particles and kB Boltz-mann’s constant. Analogously to the number density n, the mass density ρ,macroscopic velocity v and temperature θ = kBT/m in energy units are relatedto f by

ρ = m〈 f 〉, v =m〈c f 〉

ρ, θ =

m〈|C|2 f 〉dcρ

, (2.4)

where C = c− v is the central velocity of the gas.The temporal evolution of the distribution function f is described by Boltz-

mann’s equation

∂t f + ci∂xi f + ai∂ci f = C( f ), (2.5)

where the summation over the index i is implied by Einstein’s summation con-vention, see Appendix B.1, which is used throughout this thesis. The vectora = (a1, . . . , adc )

T describes the acceleration of the gas due to long-range forcefields, such as gravitational or electro-magnetic force fields and C denotes thecollision operator, which models the effect of binary collisions on the gas.

2.2 binary collisions

Boltzmann’s equation (2.5) assumes that collisions involving three or more par-ticles are extremely rare and may be neglected, such that only binary collisionshave to be modeled. Let us consider the motion of two identical particles withmass m as they undergo an elastic collision. Let x1, x2 denote their positions andc1, c2 their velocities.

For a classical, elastic collision, the collisional process can be modeled by solv-ing Newton’s equations of motion

mx′′1 = F(x1, x2) = −∇x1 V(|x1 − x2|), (2.6)

mx′′2 = −F(x1, x2) = −∇x2 V(|x1 − x2|), (2.7)

where V = V(r) denotes an interaction potential, which is assumed to onlydepend on the particle distance r = |r|, where r = x1 − x2.

Let g = c1 − c2 and g′ = c′1 − c′2 denote the relative velocities between theparticles before and after the collision process, respectively. First we note thatthe interaction force F and r are co-linear, such that the particle trajectories lie

2.2 binary collisions 7

in a collision plane spanned by g and g′. Figure 2.1 shows a particle trajectoryin the collision plane for the power-law potential V(r) = r−5 in a co-movingframe of reference, where one particle remains at rest at the origin. The so-calledimpact parameter b is the distance of closest approach between the particles ifthe interaction force is neglected, whereas the actual distance of closest approachis denoted by rm. The angle between the vector of closest approach rm and gis denoted by θ. Note that the trajectory is symmetrical about the line co-linearwith rm. The scattering angle χ describes the deviation in angle due to the particleinteraction and is related to θ by χ = π − 2θ.

0

0

ez

erg

g'

χb

θ

m

mrm

Figure 2.1: Collision of two particles in a co-moving frame of reference. The particle at theorigin remains stationary.

Depending on the interaction potential, the trajectories of the particles canbe very complex. However, in the context of kinetic theory the interaction isassumed to occur localized both in space and time. Therefore, the actual trajec-tories of the particles during a collision process are not modeled, but only thepost-collisional state is determined as a function of the pre-collisional data andthe interaction potential.

For elastic collisions between identical particles, the conservation of mass, mo-mentum and energy implies

mc1 + mc2 = mc′1 + mc′2,

m|c1|2 + m|c2|2 = m|c′1|2 + m|c′2|2.(2.8)

These equations allow to express the post-collisional velocity vectors (c′1, c′2) bythe pre-collisional states, see [114]. There exists an insightful geometrical rep-


resentation between the pre- and post-collisional states based on Thales’ circle[67].

For a prescribed interaction potential V and particle mass m, it turns out thatthe scattering angle χ is determined solely by the pre-collisional relative speedg = |g| and impact parameter b. The scattering angle can be expressed as

χ(b, g) = π − 2∫ ∞

rm

br2

[1− b2

r2 −4V(r)mg2

]−1/2

dr, (2.9)

see [12].

2.2.1 Cross Sections

A key quantity in the description of collisional processes is the differential crosssection,

σ(χ, ε, g) =b

sin(χ)

∣∣∣∣dbdχ

∣∣∣∣ , (2.10)

which denotes the flux of scattered particles in direction (χ, ε), where ε denotesthe orientation of the collision plane, see e.g. [12, 114]. The total scattering crosssection

σt(g) =∫ 2π

0

∫ π

0σ(χ, ε, g) sin(χ) dχ dε

= 2π∫ π

0σ(χ, ε, g) sin(χ) dχ

(2.11)

denotes the effective area that results in a scattering. This allows to express themean collision frequency of the particles as

ν(x, t) =

∫Ωc

∫Ωc

σt(g)g f (c) f (c′) dc′dc∫Ωc

f (c) dc, (2.12)

see [114]. If σt is independent of the relative velocity g, then the collision fre-quency simplifies to

ν(x, t) = n(x, t)σt g(x, t), g =

∫Ωc

∫Ωc

g f (c) f (c′) dc′dc(∫

Ωcf (c) dc

)2 , (2.13)

where g denotes the average relative velocity between the particles.

2.3 collision operator 9

2.2.2 Hard-sphere Collisions

As an illustrative example, let us consider the collision of two identical hardspheres with diameter d. The interaction potential can be written as

V(r) =

0, r ≥ d,

∞, r < d,(2.14)

see e.g. [12, 114]. For hard-sphere collisions the scattering angle χ can be deter-mined geometrically to be

χ = π − 2 arcsin(

bd

). (2.15)

The differential and total cross sections for hard spheres evaluate to

σ =d2

4, σt = πd2. (2.16)

Since the cross section is independent of the relative velocity, the average collisionfrequency is given by Eq. (2.13).

2.3 collision operator

The collision operator C models the effect of binary collisions on the distributionfunction f . The classical Boltzmann collision operator can be written as

C( f )(x, t, c) =∫

Ωc

∫ 2π

0

∫ π2

0

[f ′ f ′1 − f f1

]gσ sin(θ)dθdεdc1, (2.17)

see e.g. [114], where

f = f (c), f1 = f (c1), f ′ = f (c′), f ′1 = f (c′1). (2.18)

The pre-collisional and post-collisional velocities are denoted by (c, c1) and (c′, c′1)respectively and the relative pre-collisional speed is given by g = |c1 − c|. Thedifferential cross section σ encodes the physical properties of the interacting par-ticles, such as the interaction potential, as described above.


2.4 h-theorem

Multiplying the Boltzmann Eq. (2.5) with a general function ψ = ψ(x, t, c) andintegration over velocity space yields the evolution equation

〈ψ∂t f 〉+ 〈ψci∂xi f 〉+ 〈ψai∂ci f 〉 = 〈ψC( f )〉, (2.19)

or equivalently

∂t〈ψ f 〉+ ∂xi 〈ciψ f 〉+ ∂ci 〈aiψ f 〉 = 〈(∂tψ) f 〉+ 〈ai(∂ci ψ) f 〉+ 〈ψC( f )〉 (2.20)

for the quantity 〈ψ f 〉. The expression

〈ψC( f )〉 = 14

∫

Ωc

∫

Ωc

[ψ + ψ1 − ψ′ − ψ′1

] (f ′ f ′1 − f f 1

)×

gσ sin(θ) dθ dφ dc1 dc, (2.21)

where Ωc = Ωc × [0, 2π] × [0, π2 ], can be derived using the symmetries of the

collision process as shown in [114].For ψ = ln f we have the evolution equation

∂th + ∂xi ϕi = −Σ, (2.22)

where

h = 〈 f ln f − f 〉, ϕi = 〈ci( f ln f − f )〉 (2.23)

and

Σ = −〈ln( f )C( f )〉

=14

∫

Ωc

∫

Ωln(

f ′ f ′1f f1

)(f ′ f ′1 − f f 1

)gσ sin(θ) dθ dφ dc1 dc.

(2.24)

Since g, σ, and sin(θ) are non-negative over the domain of integration and thelogarithm is strictly monotonic with ln(1) = 0, it follows that Σ ≥ 0 is non-negative. Thus h, ϕi are an entropy-entropy flux pair of Boltzmann’s evolutionequation (2.5) and Σ is the entropy production term.

Equation (2.22) implies that for an isolated gas with volume Ω, the quantity

H(t) =∫

Ωh(x, t) dΩ (2.25)

is non-increasing. This is the famous H-theorem published by Boltzmann in [15].Since H is non-increasing, the system is irreversible unless Σ = 0, in contrast toNewton’s laws of motion.

2.5 conservation laws 11

2.5 conservation laws

The right hand side of Eq. (2.21) vanishes for ψ ∈ span1, c, |C|2. Thus theBoltzmann collision operator satisfies

〈γC( f )〉 = 0, (2.26)

where γ = (1, c, |C|2)T are the collision invariants of C, such that mass, momen-tum and energy are conserved by the Boltzmann Eq. (2.5).

2.6 equilibrium

A gas is in (global) thermodynamic equilibrium if it is in a homogeneous, steadystate, for which the entropy production vanishes, i.e.

Σ = −〈ln(E)C(E)〉 = 0. (2.27)

This equation implies that ln(E) is a collision invariant of Boltzmann’s collisionoperator. Thus the equilibrium can be written in the form

E(c) ≡ Eβ(c) = exp (β · γ) , (2.28)

where β ∈ R2+dc .Consider a perturbed gas with initial condition f0. If the gas is left at rest, the

equilibrium state is reached once the entropy production ceases. Therefore, theequilibrium state maximizes the physical entropy with the constraint that theevolution of f satisfies the conservation of mass, momentum and energy. Thusthe equilibrium state solves the minimization problem

Eβ = arg minf〈 f ln f − f 〉, s.t. 〈γ f 〉 = uγ, (2.29)

where uγ = 〈γ f0〉 and f0 denotes the initial condition averaged over the spatialdomain. The solution to the minimization problem yields the ansatz (2.28) asshown in Chapter 3.

A gas is said to be in local thermodynamic equilibrium at position x if thevelocity distribution at this position is the equilibrium distribution E .

2.7 bgk model

Boltzmann’s equation (2.5) with the collision operator (2.17) is a high-dimensional,non-linear integro-differential equation. In order to reduce the high complexity


of Boltzmann’s equation we consider a simpler collision model, which inheritsimportant properties of the Boltzmann collision operator.

The reduced BGK collision operator [10] is given by

C(BGK)( f ) = − f − E( f )τ

, (2.30)

where τ denotes a constant relaxation time and E is the local equilibrium distri-bution.

Despite the simplicity of the reduced BGK operator it shares some of the prop-erties of Boltzmann’s collision operator:

1. Conservation of mass, momentum and energy: By definition we have

〈γE( f )〉 = 〈γ f 〉, (2.31)

such that the conservation laws follow immediately.

2. Galilean invariance: If the velocity space is unbounded we have E( f ) =M( f ) and TE( f ) = E(T f ), where T denotes an affine transformation op-erator. Thus the BGK operator is Galilean invariant, see [75].

3. Entropy dissipation: Let h( f ) = f log f − f , then we find

∂th( f ) + ∂xi ϕi( f ) = −⟨

h′( f )f − E( f )

τ

⟩

= − 1τ

⟨(h′( f )− h′(E( f ))( f − E( f )))

⟩≤ 0,

(2.32)

where we have used the monotonicity of h′( f ) and h′(E( f )) ∈ span γi.One of the major drawbacks of the BGK collision operator is the incorrect

Prandtl number, which can be shown to be 1, whereas the measured value formonatomic gases is close to Pr = 2/3, see [63, 114]. More complex collision op-erators, such as the ES-BGK [62] or the Shakov model [113], are necessary for aphysically correct Prandtl number.

Note that the mapping E : f 7→ E( f ) is non-linear, because of the requirement(2.31). Thus the resulting BGK equation is a non-linear differential equation.

2.8 collisionless gases

Let us consider the special case C ≡ 0, such that the gas is collisionless. In thiscase the Boltzmann Eq. (2.5) reduces to a linear transport equation

∂t f + ci∂xi f = 0. (2.33)

2.9 boundary conditions 13

This equation can be written in conservative form as

∂t f + divx(c f ) = 0, (2.34)

implying the conservation of f along the characteristics of the flow field. Let(x(c)(t), c(c)(t)) denote a characteristic, then

ddt(x(c), c(c)) = (c(c)(t), 0), (2.35)

such that

x(c)(t) = x(t0) + tc(t0), (2.36)

implying that all particles remain in uniform motion. For the initial data f 0 =f 0(x, c) at time t = t0, the explicit solution is given by

f (x, t, c) = f 0(x− (t− t0)c, c). (2.37)

The collisionless kinetic Eq. (2.33) will be used in subsequent chapters as a simpletest case for moment approximations to the Boltzmann equation.

2.9 boundary conditions

The kinetic Boltzmann equation requires boundary conditions on ∂Ωx for boundedspatial domains. For the kinetic equations described above, the incoming part ofthe distribution has to be specified at the wall, i.e. we require f for n · c > 0,where n is a normal vector at the boundary pointing into the spatial domain Ωx,see e.g. [125].

At gas-wall interfaces, appropriate models are necessary to describe the physi-cal interaction of gas molecules with the wall. One of the most commonly appliedmodel is Maxwell’s accommodation model [85], see e.g. [115, 125]. Let us con-sider a stationary wall with normal n. In Maxwell’s model the incoming part ofthe distribution f is given by

f (c) = χM(c) + (1− χ) f ∗(c), n · c > 0, (2.38)

where

f ∗(c) = f (c− 2(c · n)n) (2.39)

denotes a specularly reflected distribution function andM is a Maxwellian distri-bution. The accommodation parameter χ controls the fraction of particles beingequilibrated by the wall and re-emitted according to the equilibrium Maxwellian


distribution. Note that the model (2.38) implies a discontinuity of the distributionfunction at the wall.

The focus of this thesis is on gas bulk processes far from boundaries, such thatgas-wall interactions can be neglected. Here, open Neumann boundary condi-tions are used, for which the state on the boundary is determined by constantextrapolation from the solution near the wall.

2.10 dimensional analysis

To simplify the analysis of the dimensional equations we non-dimensionalizethe Boltzmann equation. Let l0, t0, v0 denote a characteristic macroscopic length,time and velocity scale, respectively. These quantities are related by the Strouhalnumber St by

St =l0

t0v0. (2.40)

Here we set St = 1, so that the characteristic flow velocity is given by v0 = l0/t0.On the microscopic scale, a characteristic time scale is given by the relaxationtime τ0, which is related to a reference collision frequency by ν0 = 1/τ0. Letθ0 denote a characteristic temperature, then a reference microscopic length scaleis the mean free path measured in a co-moving frame of reference at velocityv0, for which we use λ0 = a

√θ0τ0, where a > 0 is a constant. A characteristic

microscopic random velocity is given by C0 = λ0/τ0 = a√

θ0.Let cs =

√γθ0 denote the characteristic speed of sound, where γ denotes the

adiabatic index, which for a monatomic gas evaluates to γ = 5/3. The Machnumber, defined as the ratio between a macroscopic velocity and the speed ofsound, is given by

Ma =v0cs

=a√γ

. (2.41)

The ratio of the macroscopic velocity to the characteristic random velocity definesa Mach number Ma′, which is related to the standard Mach number Ma by

Ma′ =v0C0

= Ma√

γ

a. (2.42)

Similarly, the ratio of the microscopic mean free path to the macroscopic lengthscale defines the Knudsen number

Kn =λ0l0

. (2.43)

2.10 dimensional analysis 15

Summarizing, the macroscopic parameters are related to the microscopic scalesby

t0 =τ0

Kn Ma′, l0 =

λ0Kn

, v0 = C0 Ma′ . (2.44)

The Reynolds number is given by

Re =ρ0v0l0

µ0, (2.45)

where µ0 defines the dynamic viscosity. The reference dynamic viscosity is givenby µ0 = τ0ρ0θ0, see e.g. [114], so that we find the relation

Re = a√

γMaKn

(2.46)

for the Reynolds number. For flows in the transition regime with Kn ∼ 1, theReynolds number is thus directly proportional to the Mach number.

With the non-dimensional quantities

t t0 = t, xi l0 = xi, ci C0 = ci, f f0 = f ,

n n0 = n, aiC2

0l0

= ai, C( f )f0τ0

= C( f ),(2.47)

where f0 = n0/Cdc0 , the Boltzmann equation can be written in nondimensional

form as

∂t f +ci∂xi fMa′

+ai∂ci fMa′

=C( f )

Kn Ma′, (2.48)

where all bars have been omitted for better readability.The parameter a = C0/

√θ0 defines the ratio between the microscopic velocity

and the square-root of the temperature. Let us consider the following choices:

a1 = 1, a2 =16

5√

2π, a3 =

√8π

, a4 =√

γ. (2.49)

Setting a = a1 leads to the simple relationship C0 =√

θ0. Using ν0 = ρ0θ0/µ0[123] for the characteristic collision frequency and the expression of the meanfree path for hard-spheres given in [11] yields parameter a2, see e.g. [90, 123].The expression a3 results from using the mean random speed CM

0 =√(8/π)θ0

of a particle as a characteristic speed [114]. Finally, setting a = a4 yields C0 = csand thus Ma′ = Ma.


The choices a = a1 and a = a2 are considered in subsequent chapters. Whilethe former choice leads to simple expressions of the scaled variables, the latteris used to compare the numerical results presented in this thesis with publishedresults, e.g. the shock-structure solutions in [90].

3M O M E N T A P P R O X I M AT I O N S

This chapter introduces the method of moments, which are lower-order approx-imations to the kinetic Boltzmann equation. Moment equations require a consti-tutive closure theory, which allows to express unknown higher-order momentsthrough the system variables. The focus of this chapter is on entropy-based clo-sures, i.e. closures based on the principle of entropy maximization and closed-form approximations thereof. After presenting the main properties of the result-ing partial differential systems, the entropy-based closures are compared to otherwell-known closure theories, such as the classical Grad closure [50].

3.1 introduction

As shown in Chapter 2, the main quantity of interest in kinetic theory is thephase-space density f = f (x, t, c). However, for many applications this descrip-tion is too detailed. Let us consider a gas with a very small Knudsen number, i.e.Kn 1, such that the relaxation process to the equilibrium distribution becomesdominant. If the velocity distribution of the gas is assumed to be Maxwellian,then the evolution of the gas is fully described by the knowledge of the massdensity, momentum and energy fields. The evolution of these fields is governedby the famous Euler equations. However, for larger Knudsen numbers, the Eu-ler equations lose their validity and methods modelling non-equilibrium effectshave to be used instead. An often-used correction to the Euler equations are theNavier-Stokes-Fourier equations (NSF), which are based on a Chapman-Enskogexpansion of the Boltzmann equation. The NSF system is a perturbative the-ory as it rests on the assumption of sufficiently small Knudsen numbers. Whilethe NSF model is a de facto standard for the description of fluids, its constitu-tive relations are invalidated by strong non-equilibrium effects as they occur inmoderately rarefied gases, such as gas flows at high altitudes or flows in microdevices.

The widely used direct simulation Monte Carlo (DSMC) method is capable todescribe rarefied gas flows [11]. However, the stochastic nature of DSMC leads

17

18 moment approximations

to noise in the solution. The reduction of noise can require excessive computa-tional resources for flows with small Knudsen or Mach numbers, especially fortransient flow problems, see e.g. [11, 98].

Another approach is the direct discretization of the Boltzmann equation bythe discrete velocity method (DVM) [92, 93]. In contrast to DSMC, the DVMis deterministic and does not suffer from stochastic noise. However, due to thehigh-dimensional phase-space, the method requires excessive memory and com-putational resources for three-dimensional simulations.

Moment approximations offer a systematic way to derive reduced-order mod-els to the Boltzmann equation. These models form a hierarchy of approximations,of which the Euler system is the most fundamental theory. While the Euler equa-tions is valid for gas flows in local equilibrium, the inclusion of higher-ordermoment equations into the Euler system allows to effectively extend the valid-ity of the model to higher Knudsen numbers. These models are derived by aprojection of Boltzmann’s equation on an N-dimensional subspace spanned by apolynomial basis in velocity space. This projection procedure generates a systemof partial differential equations for N macroscopic variables, which only dependon the spatial position x and time t, therefore reducing the dx + dc dimensionalphase-space of Boltzmann’s equation to dx dimensions. Thus moment equationsare lower-dimensional approximations to the Boltzmann equation with reducedcomputational and memory requirements.

Figure 3.1 shows the expected validity in the Knudsen number for severalmodels approximating the Boltzmann equation. Since the Euler system assumesthe gas to be in local equilibrium, its validity is restricted to very small Knud-sen numbers. In contrast, the NSF equations allow the correct description ofsmall deviations from the equilibrium. The R13 equations [116] are based on aChapman-Enskog expansion of the Grad’s classical 13-moment system and havean extended range of validity compared to the NSF model, see [123]. Finally,the ME35 system is based on the maximum-entropy method and is presented indetail in the subsequent sections.

3.2 general formulation

Here we focus on flows with macroscopic velocities on the order of the speedof sound and set Ma′ = 1. Furthermore, macroscopic fields, such as gravity,are neglected. Therefore the acceleration term in the nondimensional Boltzmannequation (2.48) is dropped. Replacing the Boltzmann collision operator with theBGK approximation (2.30) yields the kinetic BGK equation

∂t f + ci∂xi f =C(BGK)( f )

Kn, C(BGK)( f ) = E( f )− f (3.1)

3.2 general formulation 19

Equilibrium Free FlightTransition Regime

Boltzmann Equation

NSF

R13

ME35

0

Euler

0.001 0.01 0.1 1 10 100 Kn

Figure 3.1: Moment-based methods, such as the R13 and the ME35 systems, can extend therange of validity of the NSF equations and allow for the description of gases inthe transition regime.

which will serve as the starting point for the derivation of moment approxima-tions.

Let φ = (φ0, . . . , φN−1)T denote a vector of polynomial basis functions over

the velocity space Ωc, spanning a linear subspaceW of dimension N.A projection of Boltzmann’s equation (3.1) on the basis functions yields the

moment system

∂tu(x, t) + ∂xi fi(u(x, t)) =p(u(x, t))

Kn, (3.2)

where

u(x, t) = 〈φ f 〉, fi(u(x, t)) = 〈ciφ f 〉, p(u(x, t)) = 〈φC(BGK)( f )〉.

Let f = (f1, . . . , fdx ), then the moment system (3.2) can also be written in theform

∂tu(x, t) + div f(u(x, t)) =p(u(x, t)

Kn. (3.3)

This system consists of the transport operator div f, describing the transport ofthe gas in physical space and a production term p, which models the effect ofbinary collisions.

Note that in contrast to the kinetic Boltzmann Eq. (3.1), both the transport andcollision operator in the moment system (3.3) can be highly nonlinear.


3.3 closure theories

The moment system (3.2) is not closed, as both the transport operator and theright hand side require in general higher-order moment projections of f , whichare not contained in the solution vector u. The moment system can be closedeither by a direct mapping from the known moments u to the required velocityprojections of f or by defining a model distribution

f (Model)α : Rdc → R+, c 7→ f (Model)

α (c), (3.4)

where the coefficients α are determined by the consistency condition

〈φ f (Model)α 〉 = u. (3.5)

The required velocity projections are then obtained from the model distribution

f (Model)α , e.g. the flux vector is given by

fi(u) = 〈ciφ f (Model)α(u) 〉. (3.6)

Note that the availability of a model distribution function allows not only thecomputation of higher-order moments, but also enables the application of kinetic-based numerical methods, such as kinetic-based numerical flux schemes or theuse of kinetic boundary conditions.

3.4 properties of moment approximations

The properties of the moment system (3.2) are determined by the closure the-ory and the choice of the subspace W . The following sections present desirableproperties of the resulting moment system.

3.4.1 Conservation of Mass, Momentum, and Energy

The moment system should contain conservation laws for mass, momentum,and energy. This property is achieved by including the collision invariants γ =(1, c, |C|2) in the subspaceW .

3.4.2 Hyperbolicity

The resulting system should be stable. A necessary condition for stability is thehyperbolicity of the moment system.

3.4 properties of moment approximations 21

Definition 1 (Hyperbolicity). The PDE system (3.2) is called hyperbolic in directionn = (n1, . . . , ndx )

T ∈ Sdx−1 at u if and only if the Jacobian

J(u) =∂fi(u)

∂uni (3.7)

is diagonalizable with real eigenvalues. If this condition holds in every direction for allu ∈ D ⊆ RN , then the system (3.2) is hyperbolic over D. The system is called strictlyhyperbolic if additionally all eigenvalues are pair-wise distinct.

Hyperbolicity of the moment system implies that disturbances propagate atfinite speed. As an example, let us consider the conservation laws

∂tu + ∂xf(u) = 0, (3.8)

where u ∈ RN , which can be written as

∂tu + J(u)∂xu = 0, J(u) =∂f(u)

∂u, (3.9)

where J denotes the flux Jacobian. Assuming that this system is hyperbolic, theflux Jacobian can be diagonalized as J = VΛV−1, where the diagonal matrix Λ =diag (λ1, . . . , λN) contains the eigenvalues of J. The linearization u = u0 + εu1about a homogeneous state u0 yields

∂tu1 + J0∂xu1 = 0

⇔ ∂tw1 + Λ0∂xw1 = 0,(3.10)

where

w1 = V−10 u1, J0 =

∂f∂u

(u0). (3.11)

Thus the propagation velocities are the eigenvalues of the Jacobian J0. For non-linear systems, the propagation velocities depend on the state u.

Hyperbolicity can be difficult to show for general nonlinear systems. However,hyperbolicity follows directly for systems in balance-law form with a convexentropy extension, as shown in the next section.

3.4.3 Convex Entropy Extension

Let h = h(u) denote a convex, scalar-valued function, satisfying

h′∂fi(u)

∂u= ϕ′i , h′p(u) ≤ 0 (3.12)


for a scalar-valued function ϕi. Then the moment system (3.2) satisfies an entropylaw

∂th(u) + ∂xi ϕi(u) = −Σ(u) ≤ 0, (3.13)

where (h, ϕ) denotes an entropy-entropy flux pair and Σ is an entropy productionterm.

Proof. Premultiplying Eq. (3.2) with h′ yields

h′∂tu + h′∂xi fi(u) = h′p(u)

⇔ ∂th + ∂xi ϕi(u) = −Σ(u),(3.14)

where we identified h′p(u) = −Σ(u).

Let h = h(u) denote a convex function with its minimum at the equilibriumstate uE . For the BGK operator we find

Σ(u) = h′(u)u− uE

Kn=(h′(u)− h′(uE )

) u− uEKn

≥ 0, (3.15)

due to the convexity of h. Thus the BGK operator dissipates any convex entropy.

It can be shown that there exist potentials h∗ = h∗(α), ϕi∗ = ϕi∗(α), such that

h∗(α) + h(u) = α · u, ϕi∗(α) + ϕi(u) = α · fi, (3.16)

where α ∈ RN , see [105] and the references therein. The potential h∗ is definedas the Legendre dual of the convex function h and is thus convex as well. Due tothe relations

u = h′∗, fi = ϕ′i∗, (3.17)

the moment system can be written in the form

A∂tα + Bi∂xi α = p(u), (3.18)

where A = h′′∗ , Bi = ϕ′′i∗, see [43]. The Hessian A is symmetric positive definite,since the Legendre dual h∗ is strictly convex. Therefore A can be diagonalized asA = QΛQT , where Q denotes a unitary matrix and Λ = diag (λ1, . . . , λN) withλi > 0 for all i ∈ 1, . . . , N, such that A1/2 = QΛ1/2QT exists. The system (3.18)can be written as

A1/2∂tα + A−1/2BiA−1/2A1/2∂xi α = A−1/2p(u)

⇔ ∂tα + Ci∂xi α = A−1/2p(u),(3.19)

3.4 properties of moment approximations 23

where Ci = A−1/2BiA−1/2 and α = A1/2α are the system variables of the sym-metrized system. Due to the symmetry of Bi and thus also Ci, the system hasreal eigenvalues and is therefore hyperbolic.

3.4.3.1 Discussion

For general hyperbolic systems it is not trivial to find an entropy-entropy fluxpair (h, ϕ), such that the condition (3.12) is satisfied. However, in the case ofscalar equations entropy-entropy flux pairs can be easily found. Consider forexample the scalar transport equation

∂tu + a(u)∂xu = 0 (3.20)

on a one-dimensional domain Ωx = [xL, xR], where a = a(u) is a possibly nonlin-ear term controlling the propagation speed of u. Let h denote a convex function,then

h′a(u) = ϕ′, (3.21)

defines ϕ up to a constant. Let us consider the case a(u) = up, such that p = 0and p = 1 yield the linear advection and Burger’s equation respectively.

Clearly, the variables

h(u) = uq, ϕ(u) =q

q + puq+p, for even q > 0, (3.22)

are an entropy-entropy flux pair of (3.20). A natural choice for the entropy isq = 2, such that the entropy law

∂tu2 +2

2 + p∂xu2+p ≤ 0, (3.23)

allows to find an energy bound

∂t

∫

Ωx

u2 dx ≤ − 22 + p

u2+p∣∣∣xR

xL(3.24)

for the solution.

3.4.4 Galilean Invariance

The resulting moment system should be invariant with respect to Galilean trans-formations. This requirement places constraints on the choice of the polynomialsubspaceW , see e.g. [6, 96, 119, 122].


Figure 3.2 shows moment tensors ordered according to their number of con-tractions s and tensorial degree n, see also B.1. A moment tensor of nth degreewith s contractions is given by the tuple (n, s).

A necessary criterion for Galilean invariance is given in [122]: If the tensor(n, 0) is included in the system, then Galilean invariance requires also the inclu-sion of the tensors (n− 2i, i) for i = 1, . . . , n/2.

A simple sufficient condition for Galilean invariance has been considered in[119], where a specific sequence of moment systems is constructed by succes-sively adding moments in a particular pattern to the system starting from (n =0, s = 0), which corresponds to the mass density ρ. See also [122], where severalchoices of Galilean invariant moment systems are discussed.

Rhijikk

Riijj

Qhijki

s

1 2 3 4

0.

1

2

. r . vi

. q . . qi .

. sij . Rhijkli · · ·

...

n0

Figure 3.2: Trace-free tensors sorted by their dimension n and number of contractions s.

3.5 maximum-entropy closures

Closures based on the principle of entropy maximization [34, 75, 95, 96] deter-mine f as the distribution that maximizes a physically relevant entropy underthe constraints that f satisfies a given set of known moments u.

Here we use the mathematical convention of the entropy and define h as thenegative of a physical entropy. The distribution function f is then determined bythe constrained minimization problem

f := arg minf

h( f ), s.t. 〈 f φ〉 = u, (3.25)

where h( f ) = 〈η( f )〉 is a strictly convex functional, φ a vector of basis functionsin velocity space and u ∈ RN a vector of given moments. Let us define theLagrangian

L( f ; α) := h( f )− α · (〈φ f 〉 − u) , (3.26)

3.5 maximum-entropy closures 25

where α ∈ RN denotes a vector of Lagrange multipliers. The first variation of(3.26) yields the extremum

η′( f ) = α ·φ. (3.27)

Assuming that f solves Eq. (3.27), then f is a minimizer of (3.25), due to the strictconvexity of η.

3.5.1 Dual Formulation

The dual Lagrangian of L is given by

g(α) = inffL( f ; α) = αTu− h∗(α), (3.28)

where h∗ denotes the Legendre dual of h. Let f be a feasible solution to theminimization problem (3.25), then g(α) ≤ h( f ). Since the dual Lagrangian isconcave due to the convexity of h∗, the tightest lower bound to h( f ) is given bythe dual problem

supα

g(α) = − infα

h∗(α)− αTu

, (3.29)

see e.g. [16]. Thus the Lagrange parameters can be determined by the finite di-mensional, convex minimization problem

α(u) := arg minα

h∗(α)− αTu. (3.30)

Let us now specifically consider the classical Boltzmann entropy

h( f ) = 〈η( f )〉 = 〈 f log f − f 〉, (3.31)

which is strictly convex for positive f , since η′′(z) = z−1 > 0 for z > 0. For theBoltzmann entropy, Eq. (3.27) immediately yields the maximum-entropy ansatz

f (ME)α (c) = exp(α ·φ(c)). (3.32)

The dual variable h∗(α) = 〈 f (ME)α 〉 follows from Eq. (3.16), such that the mini-

mization problem is given by

α(u) := arg minα

〈 f (ME)α 〉 − αTu. (3.33)


If the minimizer α exists, it satisfies

〈φ f (ME)α 〉 = u, (3.34)

which are the constraint conditions on the velocity distribution.

3.5.2 Entropy Dissipation

The moment system (3.3) with the maximum-entropy closure can be expressedin the potential variables h∗, ϕ∗ as

∂th′∗ + ∂xi ϕ′i∗ =

〈φC〉Kn

. (3.35)

Following [75], a multiplication of (3.35) with αT from the left yields

∂t

(αTh′∗ − h∗

)+ ∂xi

(αT ϕ′i∗ − ϕi∗

)=〈αTφC〉

Kn. (3.36)

Since αTφ = log( f (ME)α ) and

αTh′∗ − h∗ = 〈αTφ f (ME)α − f (ME)

α 〉= 〈 f (ME)

α log( f (ME)α )− f (ME)

α 〉 = h( f (ME)α ),

(3.37)

the moment system (3.35) dissipates the Boltzmann entropy h for any collisionoperator satisfying

〈log( f )C〉 ≤ 0. (3.38)

Assuming (3.38), then Eq. (3.35) is the entropy law

∂th( f (ME)α ) + ∂xi ϕi( f (ME)

α ) ≤ −Σ( f (ME)α ),

Σ( f (ME)α ) = −〈log( f (ME)

α )C〉Kn

≥ 0,(3.39)

where Σ denote a non-negative entropy production term.

3.5.3 Realizability and Stability

Let f denote a non-negative distribution function f ∈ L1. Then the solvability ofthe moment constraints 〈 f φ〉 = u depends on the given moment vector u. Nat-urally, we are interested in the set of all moment vectors, for which the momentconstraints are solvable.

3.5 maximum-entropy closures 27

Definition 2 (Realizability). A moment vector u ∈ RN is called realizable if thereexists a non-negative distribution function f ∈ L1, such that 〈φ f 〉 = u for a givenset of basis functions φ(c) ∈ RN . The set of all realizable moments is the so-calledrealizability domain R ⊆ RN .

Let D denote the set of all moment vectors, for which the dual minimizationproblem (3.33) is solvable. Clearly, the requirement u ∈ R is a necessary condi-tion for (3.33) to have a solution, such that D ⊆ R.

As shown in [60, 64, 66], the dual minimization problem (3.33) is typically notsolvable for all realizable moment vectors if the underlying velocity domain isunbounded. The domain of realizable moments, for which the dual minimiza-tion problem is not solvable, J := R\D, has been named the Junk subspace[90]. Even more problematic, it was proven in [66] that the equilibrium state lieson the boundary of D if the underlying velocity domain is unbounded and thepolynomial of highest degree in φ is even and grows super-quadratically at in-finity, rendering initial value problems with data in local equilibrium ill-posed.Additionally, the closing fluxes can be singular on the Junk subspace, as shownin [64] for the 5-moment system with a one-dimensional velocity space.

In contrast, if the underlying velocity domain is bounded, then the realizabilityof a moment vector is a sufficient condition for the moment problem to have aunique solution, so that D = R, see e.g. [4, 65].

3.5.4 Admissible Moment Systems

The maximum-entropy closure leads to an additional constraint on the choice ofthe moment tensors to be included into the system. Following [75], let us considerthe subset

We = φ ∈ W|〈exp(φ)〉 < ∞ ⊆ W , (3.40)

which contains all exponentially integrable polynomials. A natural constraint onthe moment system is the requirement that the interior of We is non-empty. Thecondition 〈exp(φ)〉 < ∞ is fulfilled only for polynomials satisfying

lim|c|→∞

exp(φ(c)) = 0. (3.41)

This requirement excludes all moment theories, for which the highest polyno-mial degree is odd.

A moment system is called admissible in the sense of Levermore [75] if the fol-lowing conditions are fulfilled:


1. The moment equations contain conservation laws for mass, momentumand energy density.

2. The system is Galilean invariant.

3. The interior ofWe is non-empty.

An example of admissible moment systems is given by the choiceW = Πn forany even n ≥ 2, where Πn denotes the subspace spanned by all polynomials upto degree n. These moment systems are full tensor systems, since they contain allelements of each tensor up to degree n, see [122].

3.5.5 Hierarchy of Moment Systems

The conservation laws of mass, momentum, and energy density yield the Eulersystem, which is the most fundamental admissible moment theory in the hierar-chy of moment equations.

The additional inclusion of equations for the stress-tensor σij yields the 10-moment system, which has been investigated by many authors, see e.g. [9, 17,53, 86–88, 117]. The 10-moment system with the maximum-entropy closure yieldsa Gaussian velocity distribution, allowing the moment variables to be expressedexplicitly in terms of the Lagrange multipliers. However, since the 10-momentsystem does not include the heat flux vector qi, it is not of great interest for manystandard non-equilibrium gas flow problems, such as shock-structure problemsor gas-wall boundary layers, see e.g. [123].

By adding the heat flux vector qi to the 10-moment system we obtain the classi-cal 13-moment system, which has been investigated by Grad [50, 51]. Clearly, thissystem is not admissible, since the heat flux is generated by a weight functionof third degree. Adding the fully contracted fourth order tensor (n = 0, s = 2)to the 13-moment system yields the 14-moment system, which has been investi-gated in [73, 90] for the description of shock-structure problems. Note that the14-moment system is the smallest admissible system with equations for both thestress tensor and the heat-flux vector. It is also the smallest admissible systemwith the formally correct Navier-Stokes approximation by asymptotic expansion[73].

In this thesis we consider the admissible choiceW = Π4, which yields the 35-moment system in the three-dimensional setting. This system is expected to yieldmore accurate results than the 14-moment system, since the 14-moment systemis fully contained in the 35-moment system.

Figure 3.3 visualizes the above mentioned admissible moment systems, whereeach moment theory is represented by a collection of tensors in the ns-plane, seealso Figure 3.2.

3.6 the euler system 29

n0.

1 2 3 4

0.

12

n0.

1 2 3 4

12

n1 2 3 4

12

N = 14 N = 35

0.

0. 0

.n1 2 3 4

12

0. 0

.

N = 5 . N = 10 .

s

ss

s

Conservation laws

Stress tensor andheat flux vector

Higher-order tensors

Figure 3.3: Examples of admissible moment systems for the maximum-entropy closure.

3.6 the euler system

Let φ = (1, ci, |C|2)T , then the resulting moment system is given in non-dimensionalform by

∂tρ + ∂xj

(ρvj

)= 0,

∂t (ρvi) + ∂xj

(ρvivj + ρθδij + ρθ〈ij〉

)= 0,

∂t(ρe) + ∂xj

(ρ(e + θ)vj + ρviθ〈ij〉 + qj

)= 0,

(3.42)

where θ〈ij〉 = θij − θδij is the deviatoric part of the temperature tensor θij and theenergy density ρe is given by

ρe =〈c2

i f 〉2

= ρε +ρv2

2, ρε =

〈C2i f 〉2

, (3.43)


where ε denotes the specific internal energy, such that ρε is the internal energydensity. Analogously, the system can be written in the Lagrangian frame as

Dρ

Dt+ ρ

∂vj

∂xj= 0,

ρDviDt

+∂p∂xj

+∂σij

∂xj= 0,

ρDε

Dt+ p

∂vj

∂xj+ σij

∂vi∂xj

+∂qj

∂xj= 0,

(3.44)

where σij = ρθ〈ij〉 denotes the deviatoric stress tensor.The maximum-entropy closure yields the velocity distribution

f (ME)α (c) = exp(α(0) + αici + α(1)c2

i ), (3.45)

which corresponds to a Maxwellian distribution M if the underlying velocitydomain is unbounded. The Maxwellian distribution satisfies σij = 0, qi = 0,such that the moment system 3.42 reduces to the Euler equations

∂tρ + ∂xj

(ρvj

)= 0,

∂t (ρvi) + ∂xj

(ρvivj + ρθδij

)= 0,

∂t(ρe) + ∂xj

(ρ(e + θ)vj

)= 0.

(3.46)

The entropy of the Euler system is given explicitly by

h = 〈M lnM−M〉 = −ρ

(h0 + ln

(p

ργ

)), (3.47)

where h0 is a constant and γ = 5/3 for a three-dimensional velocity space. Theentropy h is related to the nondimensional, specific physical entropy s by

ρs = −h, (3.48)

where s denotes the specific entropy of the Euler equations, see e.g. [39].The significance of the Euler equations lies in the fact, that this system is ob-

tained from Boltzmann’s equation in the limit of small Knudsen numbers. For-mally, this can be seen by considering the asymptotic expansion f = f0 + ε f1 +O(ε2) in the Boltzmann Eq. (2.48). Since the kernel of the Boltzmann collisionoperator is spanned by the collision invariants γ = (1, ci, |C|2), the limit of smallKnudsen numbers yields formally f = E( f ), which agrees with the maximum-

3.7 slab geometry 31

entropy ansatz (3.45). Thus the maximum-entropy method yields the physicallycorrect model in the limit of small Knudsen numbers.

3.7 slab geometry

For simplicity, we use the notation (x, y, z) = (x1, x2, x3) in the following. Letus consider the special case of a one-dimensional gas process along the ex =(1, 0, 0)T direction, such that ∂y f = ∂z f = 0. The initial velocity distribution isassumed to be rotationally invariant under rotations about the cx-axis. Since theevolution of the BGK equation preserves the rotational invariance of the velocitydistribution, f only depends on the four independent variables x, t, c with

c = (cx, cr)T ∈ Ωcx ×Ωcr , where cr =

√c2

y + c2z ∈ Ωcr (3.49)

is the velocity component in the radial direction. Here we consider the boundedrectangular velocity domain given by

Ωc = Ωcx ×Ωcr = [−cMx , cM

x ]× [0, cMr ], (3.50)

where cMx , cM

r > 0 are the velocity bounds. Moments of the distribution functionare then given by

〈ψ f 〉 := 2π∫

Ωc

ψ f cr dΩc, where ψ = cixc2j

r for i, j ∈N0. (3.51)

In the geometrical setting considered here, the BGK equation simplifies to

∂t f + cx∂x f = −f − Eβ(uγ)

Kn, (3.52)

where

uγ = 〈γ f 〉, γ = (1, cx, C2x + C2

r )T (3.53)

and Cx = cx − vx, Cr = cr denote the central velocities.

3.7.1 Moment Systems in Slab Geometry

The one-dimensional moment system is given by

∂tu + ∂xf(u) = p(u), (3.54)


where

u = 〈φ f 〉, f(u) = 〈cxφ f 〉, p(u) =〈φC(BGK)( f )〉

Kn. (3.55)

Let Πn denote the space spanned by all polynomials up to degree n in the velocityspace Ωc. In slab geometry, the number of basis functions spanning Πn is givenby

N := N(n) = dim Πn =

(2 + n)2/4 for even n,

(1 + n)(3 + n)/4 for odd n.(3.56)

The choice n = 4 yields the 35-moment system in the fully three-dimensionalsetting, which reduces to N = 9 equations in slab geometry, which are generatedby the basis functions

φ = (1, cx, c2x, c3

x, c4x, c2

r , c4r , cxc2

r , c2xc2

r )T . (3.57)

In addition to the collision invariants γ = (1, cx, C2x +C2

r )T , the resulting moment

equations contain non-equilibrium moments, such as the directional tempera-tures and a heat flux given by

θxx = 〈C2x f 〉, θrr =

〈C2r f 〉2

, qx =〈Cx(C2

x + C2r ) f 〉

2. (3.58)

3.8 grad’s classical closure

Grad proposed in [50, 51] the ansatz distribution

f (Grad)ρ,v,θ,α (c) =Mρ,v,θ(c)

N−1

∑k=0

φk(c)αk =Mρ,v,θ(c)φ · α, (3.59)

where Mρ,v,θ denotes the local equilibrium distribution, (ρ, v, θ) are the localmass density, velocity and temperature of the gas, φ = (φ0, . . . , φN−1)

T is apolynomial basis, and α = (α0, . . . , αN−1)

T denote expansion coefficients. Theexpansion is usually performed in orthogonal polynomials with respect to theweight functionMρ,v,θ .

Clearly, the ansatz distribution (3.59) is not non-negative, which might leadto unphysical effects. In contrast to the maximum-entropy closure, the ansatzis only nonlinear in the variables v, θ and linear in the mass density ρ and theexpansion coefficients α. The linearity leads to significant simplifications in thecomputations and higher-order moments can be written explicitly in terms of thelower-order moments, see e.g. [123].

3.8 grad’s classical closure 33

It is well known that Grad’s closure for moment equations results in locallyhyperbolic systems, which become unstable for large deviations from the equi-librium distribution, see e.g. [90]. Therefore, Grad’s closure cannot be used forthe simulation of strongly non-equilibrium gas flows, such as shock-structureproblems with high Mach numbers. Furthermore, Grad’s closure suffers fromthe generation of strong unphysical sub-shocks in the continuous shock-structureproblem. While sub-shocks are an inherent property of hyperbolic moment sys-tems, the strength of the sub-shock can be decreased by considering stronglynonlinear closure theories, see Chapter 7.

Kauf investigated in [67] the applicability of spline functions as basis functionsin the expansion (3.59). While it was found that spline functions can lead to betterfunction approximations, the issue of local hyperbolicity remains for these ansatzfunctions.

3.8.1 Connection to the Maximum-entropy Closure

A formal linearization of the maximum-entropy distribution about a state

f 0(c) = exp(α0 ·φ(c)) (3.60)

with 〈 f 0〉 < ∞, where α0 ∈ RN and φ = (φ0, . . . , φN−1)T is a polynomial basis,

yields

f (LME)α (c) = f 0(c) (1 + ∆α ·φ(c)) , (3.61)

where α = α0 + ∆α. The moment constraints

〈φ f (LME)α 〉 = u (3.62)

with u = u0 + ∆u and 〈φ f 0〉 = u0 yield a linear system

〈φφT f 0〉∆α = ∆u, (3.63)

for the expansion coefficients ∆α. This system has a unique solution, since thematrix 〈φφT f 0〉 is symmetric positive definite. Let φ be an orthonormal basiswith respect to the weight function f 0, so that the linear system simplifies to∆α = ∆u.

Let us consider a bounded velocity domain Ωc, for which the equilibriumdistribution E is in the interior of the realizability domainR. Then a linearizationabout the equilibrium state f 0 = E is well defined and given by

f (LME)α (c) = E(c)

(1 +

N−1

∑i=0

∆uiφi(c)

), (3.64)


where φ is an orthonormal basis with respect to E .Assuming that E is integrable on the unbounded velocity domain, the distribu-

tion E converges toM, as the size of the velocity domain is enlarged. Therefore,the linearized maximum-entropy closure approximates the corresponding Gradclosure

f (Grad)α (c) =M(c)

(1 +

N−1

∑i=0

∆uiφi(c)

)(3.65)

on an unbounded velocity domain, so that Grad’s closure can be considered as alinearization of the maximum-entropy closure around the equilibrium state.

3.8.2 Function Approximations

Let us consider a one-dimensional velocity space Ωc = R, for which Grad’sansatz is given by

f (Grad)α (c) =Mρ,v,θ(c)

N−1

∑i=0

αiHei(c), (3.66)

where Hei denotes the ith Hermite basis function, satisfying

〈HeiHejMρ,v,θ〉 = ρδiji! (3.67)

for 0 ≤ i, j ≤ N − 1. The corresponding maximum-entropy distribution reads

f (ME)α (c) = exp

(N−1

∑i=0

αiφi(c)

). (3.68)

In the following we consider the bi-Gaussian distribution

f (BM)(c) =2

∑i=1

wi√2πθi

exp(− (c− vi)

2

2θi

)(3.69)

with parameters

w1 =34

, v1 =−12

, θ1 =16

, w2 =14

, v2 =32

, θ2 =12

(3.70)

in order to compare the approximation properties of both the classical Grad andthe maximum entropy closures. Note that the correct description of bi-modaldistributions is important in various rarefied gas and plasma applications, seee.g. [8, 12].

3.8 grad’s classical closure 35

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

c

f

Grad Approximations

BM Maxwellian Grad5

Grad11 Grad17

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

c

f

ME Approximations

BM Maxwellian ME5

ME11 ME17

Figure 3.4: Approximations to the bi-Gaussian distribution (3.69) with the Maxwellian,Grad and maximum-entropy (ME) distributions.

The first three moments of (3.69) with the coefficients (3.70) are given by

ρ = 1, v = 0, θ = 1, (3.71)

and the third and fourth non-equilibrium central moments evaluate to

Q = 〈C3 f (BM)〉 = 98

, R = 〈C4 f (BM)〉 = 5516

. (3.72)

Thus the bi-Gaussian distribution with coefficients (3.70) has a positive heat flux.In order to compare the approximation qualities of both closure theories, the

model distribution functions are shown in Figure 3.4 for various numbers ofmoments, N.

Since both Grad’s closure and the maximum-entropy closure contain the equi-librium distribution as a subsystem, both closures reduce to the Maxwellian ap-proximation M1,0,1 on the unbounded velocity domain for N = 3. For higher-order moment approximations, the maximum-entropy distribution yields betterapproximations to the bi-Gaussian distribution. Furthermore, the Grad distribu-tions have unphysical negative densities around the velocity c = −2.

Figure 3.5 shows a direct comparison between the Grad and the maximum-entropy distributions. Clearly, for both N = 5 and N = 15, the maximum-entropydistribution yields a much better approximation to the bi-Gaussian distribution.

A convergence analysis in the number of moments N is shown in Figure 3.6.For both ansatz distributions, spectral convergence can be observed in the num-ber of moments.


-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

c

f

Moment Approximations (N=5)

BM Grad5 ME5

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

c

f

Moment Approximations (N=15)

BM Grad15 ME15

Figure 3.5: Approximations to the bi-Gaussian distribution (3.69) with the Grad and MEansatz functions using N = 5 (left) and N = 15 (right) moments.

3.8.3 Regularized Grad Closures

In [116] a regularization of Grad’s 13-moment system was proposed. The result-ing system, known as the R13 moment system, was shown to allow for accuratedescriptions of non-equilibrium effects in moderately rarefied gases, see e.g. [114,123]. Furthermore, the parabolic nature of the system allows for smooth shock-structure solutions. A disadvantage of the R13 system is that the parabolic termsin the equations lead to a smaller time step restriction for explicit time-steppingschemes and a greater sensitivity with regard to mesh irregularities, see [86, 90].

3.8.4 Hyperbolic Grad Closures

Another approach to improve Grad’s closure is based on a modification of theresulting moment equations, which renders the resulting system globally hyper-bolic [18–20]. A specific case are quadrature-based projections methods, whichhave been proposed in [68], where the system is modified by replacing the ex-act moment projections in velocity space with Gauss quadratures. The inexactprojection was shown to generate globally hyperbolic systems.

An advantage of these hyperbolic moment equations is the availability of ex-plicit expressions of the eigenvalues and eigenvectors of the system, which canbe exploited in numerical schemes. Unfortunately, the resulting moment systemscannot be written in conservative form and special numerical methods for non-conservative systems have to be used.

3.9 discussion 37

3 5 7 9 11 13 15 17 19

1

0.3

0.1

0.03

0.01

N

Rel.Error

Convergence Analysis

Euler Grad ME

Figure 3.6: Relative L1 errors for the Grad and ME approximations of the bi-Gaussian dis-tribution (3.69) as a function of the number of moments N, together with theEuler approximation.

3.9 discussion

The maximum-entropy closure has attractive mathematical properties as it yieldssymmetric hyperbolic moment equations with an entropy law. Since the closuretheory is non-perturbative, it allows for an accurate description of some stronglynon-equilibrium states, as shown for the bi-modal distribution above. Comparedto the globally hyperbolic moment closures based on the Grad closure [18, 68],the maximum-entropy closure leads to conservative systems, for which standarddiscretization methods can be applied as shown in the next chapter.

4N U M E R I C A L M E T H O D S

This chapter presents numerical discretization methods for the moment systemspresented in Chapter 3. Section 4.1 introduces the finite volume scheme in thecontext of non-linear hyperbolic partial differential systems. Semi-discrete meth-ods based on the method of lines are considered in Section 4.2. In contrast toclosures with closed-form expressions for the closing moments, the maximum-entropy closure requires numerical methods for the evaluation of higher-ordermoments. Several numerical methods for the maximum-entropy closure and theresulting moment system are presented in Sections 4.3 to 4.6. Finally, Section 4.7discusses numerical methods for the discrete velocity model, which are used tocompute reference solutions to the moment approximations.

4.1 the finite-volume scheme

The method of moments generates hyperbolic systems of partial differentialequations in balance-law form

∂tu + ∂xf(u) = p(u), (4.1)

where, in general, ∂xf(u) is a nonlinear transport operator and p(u) a nonlinearsource term.

The spatial domain Ωx = [xL, xR] is uniformly discretized into Nx cells

Ij = [xj−1/2, xj+1/2] for j = 1, . . . , Nx (4.2)

of length ∆x = xj+1/2 − xj−1/2, where x1/2 = xL, xNx+1/2 = xR. A finite-volumediscretization of this system is given by

∂tuj +F j+1/2 −F j−1/2

∆x= pj, for j = 1, . . . , Nx, (4.3)

39

40 numerical methods

where

uj(t) =1

∆x

∫

Ij

u(x, t) dx, pj(t) =1

∆x

∫

Ij

p(x, t) dx (4.4)

denote cell-averaged moment vectors over cell Ij and

F j−1/2 = F (u−j−1/2, u+j−1/2), F j+1/2 = F (u−j+1/2, u+

j+1/2) (4.5)

are the numerical fluxes at the interfaces xj−1/2 and xj+1/2 respectively, where

u±j+1/2 = limε→ 0+

u(xj+1/2 ± ε) (4.6)

are the right and left limits of u at the cell interface at position xj+1/2. The nu-merical flux function F j+1/2 is considered to be an approximation to the exactflux f(u(xj+1/2, t)).

4.1.1 Conservation Property

If p ≡ 0, then (4.1) is a system of conservation laws and integration over a timeinterval ∆t = [tn, tn+1] yields the conservation form

un+1j = un

j −∆t∆x

(Fn

j+1/2 − Fnj−1/2

), (4.7)

where unj denotes the cell averaged solution at time tn and

Fnj+1/2 =

1∆t

∫ tn+1

tnF j+1/2 dt (4.8)

is a time averaged numerical flux. This form implies conservation of u, i.e. ujonly varies in time through fluxes at the cell boundaries. This feature is of greatimportance in the context of moment approximations to the Boltzmann equation,since it guarantees numerical conservation of mass, momentum and energy.

4.1.2 Numerical Flux Functions

Following Godunov’s approach [46], the solution at each cell interface u(xj+1/2, t)is determined by the solution to a Riemann problem, where the left and rightstates are given by u−j+1/2 and u+

j+1/2, respectively. While for simple systems theexact solution to the Riemann problem can be determined analytically, for gen-eral non-linear hyperbolic systems the exact solution is both complex and com-

4.1 the finite-volume scheme 41

putationally expensive to compute. Therefore, inexpensive numerical approxi-mations to the exact Riemann solution are often used in practice.

The following condition is necessary for the numerical flux to be consistentwith the original continuous partial differential equation:

Definition 3 (Consistency). The numerical flux function F is consistent if F (u, u) =f(u) and F is Lipschitz continuous in both arguments [74].

Let us consider numerical flux functions of the form

F (uL, uR) =12(f(uL) + f(uR)) +

D(uL, uR)

2(uL − uR) , (4.9)

where D ∈ RN×N denotes a dissipation matrix. Note that the numerical flux(4.9) is consistent for any choice of D. The dissipation matrix

D(uL, uR) =∆x∆t

I (4.10)

yields the classical Lax-Friedrichs (LF) method. While this flux is very robust andparticularly simple to compute, since it does not depend on the eigenstructureof the system, it suffers from a large diffusivity, which leads to low resolutionsof discontinuities.

An alternative choice is the dissipation matrix

D(uL, uR) = maxλ(uL), λ(uR)I, (4.11)

where λ(u) denotes the spectral radius of the flux Jacobian

J(u) =∂f(u)

∂u. (4.12)

This yields the local Lax-Friedrichs (LLF) scheme, which is also called Rusanovscheme. The LLF scheme is less diffusive than the LF flux, but requires the com-putation of the spectral radii of the Jacobian matrices J for all spatial cells.

Harten, Lax, and van Leer proposed in [59] a simplified Riemann solver re-solving only two waves. The numerical flux is given by

F (HLL)(uL, uR) =

fL 0 ≤ λL,λRfL − λLfR + λLλR(uR − uL)

λR − λLλL ≤ 0 ≤ λR,

fR λR ≤ 0,


where λL, λR denote the fastest left- and right-going characteristic velocities inthe solution to the Riemann problem. The numerical flux can also be written inthe equivalent compact form

F (HLL)(uL, uR) =12(f(uL) + f(uR))

+k02(f(uL)− f(uR))−

k12(uL − uR) , (4.13)

where

k0 =|λL| − |λR|

λL − λR, k1 =

|λL|λR − |λR|λLλL − λR

, (4.14)

see [121]. A simple estimate for the fastest left and right going characteristicvelocities is

λL = minλmin(uL), λmin(uR),λR = maxλmax(uL), λmax(uR),

(4.15)

where

λmin(u) = min σ(J(u)), λmax(u) = max σ(J(u)). (4.16)

See [118] for a discussion of several estimates for the fastest characteristic veloci-ties. Note that the HLL flux reduces to the LLF flux if we set λL = −λ, λR = λ.

4.2 semi-discrete methods

In order to evaluate the limits u−j+1/2, u+j+1/2 a local reconstruction of the nu-

merical solution is required. In the simplest case, the solution is assumed to beconstant over each cell. Applying a first-order forward finite-difference approx-imation to the temporal derivative of the semi-discrete system (4.3) yields theEuler scheme

un+1j = un

j −∆t∆x

(Fn

j+1/2 −Fnj−1/2

)+ ∆tpn

j , (4.17)

where the numerical flux Fnj+1/2 = F (un

j , unj+1) is considered to be an approxi-

mation to f(u(xj+1/2, tn)). This scheme is first-order accurate in space and time.

4.2 semi-discrete methods 43

4.2.1 Compact Form

The semi-discrete system (4.3) can be written compactly as

∂tu = g(u), (4.18)

where

u =(

uT1 , . . . , uT

Nx

)T∈ RN·Nx , g =

(gT

1 , . . . , gTNx

)T∈ RN·Nx , (4.19)

with

gj = −F j+1/2 −F j−1/2

∆x+ pj. (4.20)

This notation allows to write the Euler discretization (4.17) as

un+1 = un + ∆tg(un). (4.21)

4.2.2 A Second-order Scheme

The spatial accuracy can be improved with higher-order spatial reconstructionsof u. Let uj = (uj,1, . . . , uj,N)T denote the components of the numerical solutionover cell Ij. Let us consider the linear reconstruction

uj,k(x, tn) = unj,k + (x− xj)σ(un

j−1,k, unj,k, un

j+1,k), x ∈ Ij, (4.22)

where σ : R3 → R is a slope given by

σ(uj−1, uj, uj+1) =uj+1 − uj

∆xφ(SL)(θj), θj =

uj − uj−1

uj+1 − uj. (4.23)

The function φ(SL) : R → R is a so-called slope-limiter function, which allowsto reduce the accuracy of the method to first order around discontinuities toprevent unphysical oscillations. Note that for φ(SL) ≡ 1 the reconstruction doesnot involve any limiting and is simply a linear interpolation of the neighboringcell averages, which, however, leads to spurious oscillations in the vicinity ofshock waves of hyperbolic equations. In this work, we consider the total variationdiminishing minmod limiter

φ(SL)MM (θ) = max (0, min(1, θ)) . (4.24)

In order to obtain a second-order method in space and time, the linear recon-struction in space has to be coupled with a second-order temporal discretization


method. Let us consider Heun’s method, which is an explicit, two-stage Runge-Kutta method given by

u(1) = un + ∆t g(un),

u(2) = u(1) + ∆t g(u(1)),

un+1 =un + u(2)

2.

(4.25)

Both the explicit Euler and Heun’s method are strong-stability preserving (SSP)methods. The SSP property guarantees that the total-variation of the numericalsolution does not increase over time.

4.2.3 Boundary Conditions

We consider open, periodic and Dirichlet boundary conditions. Here, the bound-ary conditions are implemented with the ghost cell method. For open boundaryconditions, the ghost cells are set to

un−1 = un

0 := un1 , un

Nx+2 = unNx+1 := un

Nx, (4.26)

while for periodic boundary conditions, the ghost cell values are given by

un−1 = un

0 := unNx

, unNx+2 = un

Nx+1 := un1 . (4.27)

In contrast, for the Dirichlet conditions, the boundary values are set according toa prescribed boundary functions bL = bL(t), bR = bR(t), i.e.

un−1 = un

0 := bnL, un

Nx+2 = unNx+1 := bn

R, (4.28)

where bnL = bL(tn), bn

R = bR(tn).

4.3 numerical schemes for the maximum-entropy closure

For moment systems with polynomial weights of order higher than two, integrals

of the maximum-entropy distribution f (ME)α are in general not available in closed-

form. Therefore, we consider numerical approximations of the moments of f (ME)α

with a quadrature rule Q.

4.3 numerical schemes for the maximum-entropy closure 45

Let us consider the quadrature projection of the kinetic BGK equation givenby

⟨φ

(∂t f + cx∂x f − C

(BGK)( f )Kn

)⟩

Q= 0, (4.29)

where 〈·〉Q denotes a quadrature approximation to the exact integral 〈·〉.Choosing a quadrature rule Q with fixed abscissas and positive weights allows

us to write Eq. (4.29) as

∂t

⟨φ f (ME)

α

⟩Q+ ∂x

⟨cxφ f (ME)

α

⟩Q= −

⟨φ(

f (ME)α − Eβ(uγ)

)⟩Q

Kn, (4.30)

where the maximum-entropy ansatz f (ME)α has been inserted. The equilibrium

distribution satisfies the constraints

〈γEβ(uγ)〉Q = 〈γ f (ME)α 〉Q, where uγ = 〈γ f (ME)

α 〉Q. (4.31)

The quadrature-based moment system (4.30) inherits important properties of themoment system (3.2): The system can be cast into symmetric hyperbolic formand dissipates the discrete Boltzmann entropy

hQ( f (ME)α ) = 〈η( f (ME)

α )〉Q, (4.32)

see [112]. In addition, the minimization problem (3.33) allows for a solution onthe set of quadrature-realizable moments vectors, RQ, which is a subset of R,see [4]. However, in contrast to the moment system (3.2), the quadrature-basedmoment system is not Galilean invariant, since the quadrature nodes are fixed.

4.3.1 Discretization of the Velocity Domain

For the slab geometry case, we consider the tensorized quadrature rule

〈φ f 〉Q = 2πNcx−1

∑i=0

Ncr−1

∑j=0

wxi wrj crj φ(cxi , crj ) f (cxi , crj ), (4.33)

where (wxi , wrj ) are the quadrature weights and (cxi , crj ) the quadrature nodes.The total number of quadrature nodes is denoted by Nc = Ncx Ncr . As before, thevelocity domain is given by Ωc = [−cM

x , cMx ]× [0, cM

r ].Since the velocity domain is bounded, a natural choice for the numerical inte-

gration is the Gauss-Legendre quadrature.


-10 -5 0 5 10

1

2

4

8

16

cx

Nbx

Distribution of Abscissas

Figure 4.1: Abscissas of the block-wise Gauss-Legendre quadrature with Ncx = 64 on thevelocity domain Ωc = [−10, 10] for different numbers of blocks Nb

cx .

As the nodes of the Gauss-Legendre quadrature tend to cluster near the bound-ary of the integration domain, the rectangular velocity domain Ωc is divided intoequally sized blocks

Ωi,jc = [−cM

x + (i− 1)dcx ,−cMx + idcx ]× [(j− 1)dcr , jdcr ],

dcx =2cM

x

Nbcx

, dcr =cM

r

Nbcr

(4.34)

for i = 1, . . . , Nbcx

and j = 1, . . . , Nbcr

, such that ∪i,jΩi,jc = Ωc. On each block a ten-

sorized Gauss-Legendre quadrature with Ngcx · N

gcr quadrature nodes is applied.

The total number of quadrature nodes in the velocity variables cx and cr are thengiven by

Ncx = Nbcx

Ngcx and Ncr = Nb

crNg

cr , (4.35)

respectively.

4.3.1.1 Convergence Study of the Integration Rule

Figure 4.1 shows the distribution of the abscissas using a total of Ncx = 64quadrature points and different numbers of blocks. Clearly, using a higher num-ber of blocks at the expense of quadrature points per block leads to a more evenlyspaced distribution of the abscissas.

4.3 numerical schemes for the maximum-entropy closure 47

◼

◼

◼

◼

◼

8 32 64 96 128

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

Ncx

Rel.Error

ρ = 1, v = 0, θ = 1

◼

◼

◼

◼

◼

8 32 64 96 128

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

Ncx

Rel.Error

ρ = 1, v = 1, θ = 1

◼◼

◼ ◼◼

◼

◼◼

◼

◼

◼

8 32 64 96 128

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

Ncx

Rel.Error

ρ = 1, v = 0, θ = 0.1

◼ ◼◼

◼◼

◼◼

◼◼

◼◼

◼

◼

8 32 64 96 128

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

Ncx

Rel.Error

ρ = 1, v = 1, θ = 0.1

Nbx=1 Nbx=2 Nbx=4 Nbx=8

Figure 4.2: Relative error of the block-wise Gauss-Legendre quadrature approximation ofthe mass density of several Maxwellian distributions as a function of the totalnumber of quadrature points Ncx for different numbers of blocks.

Let us consider the numerical approximation of 〈Mρ,v,θ〉 with the block-wisequadrature rule introduced above on the one-dimensional velocity space Ωc =[−10, 10]. Figure 4.2 shows the relative error

e(rel)(ρ, v, θ) =|〈Mρ,v,θ〉Q − 〈Mρ,v,θ〉|

〈Mρ,v,θ〉(4.36)

of the block-wise Gauss-Legendre quadrature approximation 〈Mρ,v,θ〉Q of themass density 〈Mρ,v,θ〉.

Since Mρ,v,θ is analytic, the error converges exponentially in Ngcx for a fixed

number of blocks Nbcx

. Clearly, the error can be reduced significantly in compari-son to the standard Gauss-Legendre quadrature by choosing an optimal numberof blocks.


4.4 numerical implementation of the minimization problem

Let us consider the iterative Newton method for the solution of the convex min-imization problem. For a given moment vector u, the objective function to beminimized is

f (o)(α) = 〈 f (ME)α 〉Q − αTu. (4.37)

The gradient and Hessian of f (o) evaluate to

g(α) = 〈φ f (ME)α 〉Q − u, H(α) = 〈φφT f (ME)

α 〉Q. (4.38)

If u is realizable, then there exists a unique solution α∗ to the non-linear systemg(α) = 0, due to the strict convexity of the objective function f (o).

Let α0 denote an initial guess for α∗. Newton’s method determines a sequenceof approximations α1, α2, . . . to the minimizer α∗. Given the approximationαk, Newton’s method determines the next iterate αk+1 by the update αk+1 =αk + ∆αk, where ∆αk solves the linear system

H(αk)∆αk = −g(αk). (4.39)

A simple stopping criterion for the Newton iteration is given by ||g(αk)|| ≤ τ,where τ > 0 is an error tolerance.

Since the Hessian is symmetric positive definite, the linear system always hasa unique solution. However, the Hessian matrix can be ill-conditioned, such thatthe search direction ∆αk contains large numerical errors. This problem will beaddressed in Section 4.4.2.

The robustness of Newton’s method can be improved with a line search in thesearch direction ∆αk. Here we consider a simple line search based on backtrack-ing, for which the update formula is given by αk+1 = αk + βk∆αk, where βk de-notes a damping factor. Let p, σ ∈ (0, 1) denote two user-defined parameters. Thedamping factor βk is determined as the largest number in the set 1, p, p2, . . .,for which the Armijo condition

f (o)(αk + βk∆αk) ≤ f (o)(αk) + σβk∆αk · g(αk) (4.40)

is fulfilled, see e.g. [58].Newton’s method can be shown to convergence (at least) quadratically in the

vicinity of the root if the conditions of Thm. 19.1 in [58] are fulfilled. The super-linear convergence rate in the vicinity of the solution renders Newton’s methodvery attractive, since it significantly reduces the required number of iterationscompared to simpler methods with a linear convergence rate, such as the steep-est descent method.

4.4 numerical implementation of the minimization problem 49

The use of a good initial guess for the minimization problem is critical to de-crease the number of required Newton iterations. Since the minimization prob-lem has to be solved at every time step, a simple and effective choice is to usethe solutions to the minimization problems of the last time step as initial valuesfor the Newton procedure.

Let αni denote a vector of Lagrange parameters with

〈φ f (ME)αn

i〉Q = un

i . (4.41)

A simple initial guess for the minimizer αn+1i is given by αn

i . If the macroscopicmoments un+1

i are available, then a slightly improved heuristic, such as

αn+1i = αn

kn+1(i), kn+1(i) = arg minj∈i−1,i,i+1

||un+1i − un

j || (4.42)

can be used.

4.4.1 Implementation of the BGK-operator

In order to evaluate the BGK-operator, the Lagrange parameters β ∈ R3 of theequilibrium distribution Eβ = exp(β · γ) have to be computed numerically. Anal-ogously to the Lagrange parameters of the maximum-entropy distribution func-

tion f (ME)α , the parameters are determined by a dual minimization problem given

by

β(u) := arg minβ∈R3

〈Eβ〉Q − βTuγ, (4.43)

where uγ = 〈γ f (ME)α 〉Q with 〈φ f (ME)

α 〉 = u.For the two-dimensional velocity domain considered here the numerical quadra-

ture has a computational complexity of O(Ncx Ncr ) for general functions. How-ever, for factorizable functions, such as the equilibrium distribution, the compu-tational complexity can be reduced to O(Ncx + Ncr ) as shown in the following.

Let us consider the additive decomposition γ = γx + γr, where

γx = (1, cx, c2x)

T , γr = (0, 0, c2r )

T . (4.44)

Then the equilibrium distribution E ≡ Eβ can be written as E = ExEr with

Ex := exp(γx · β), Er := exp(γr · β). (4.45)


Now the moments of E can be evaluated by one-dimensional quadratures

〈γE〉Q = 〈γxEx〉Qx 〈Er〉Qr + 〈γrEr〉Qr 〈Ex〉Qx , (4.46)

where Qx, Qr denote numerical quadrature approximations to the integrals

〈·〉x =∫

Ωx

· dcx, 〈·〉r = 2π∫

Ωr

· rdcr. (4.47)

Similarly, the Hessian matrix H can be expressed as the sum of products of one-dimensional integrals given by

H = 〈γxγTx Ex〉Qx 〈Er〉Qr + 〈γrγT

r Er〉Qr 〈Ex〉Qx

+ 〈γxEx〉x〈γTr Er〉Qr + 〈γrEr〉Qr 〈γT

x Ex〉Qx . (4.48)

The minimization problem (4.43) can be solved with a Newton iteration, anal-ogously to the Newton method for the Lagrange parameters α. As an initialguess for the Newton iteration, a natural choice are the Lagrange parameterscorresponding to the Maxwellian distribution Mρ,vx ,θ on the unbounded veloc-ity domain, which can be expressed in closed form.

4.4.2 Adaptive Basis Methods

The Hessian matrix H can become ill-conditioned for the minimization problem(3.33) if a fixed basis is used, causing the search direction to be very inaccurate [4].Since the robustness of the Newton algorithm depends on the condition numberof H, the use of basis functions other than monomials can be necessary to findsolutions to the minimization problem.

Let us consider a one-dimensional velocity space Ωc ⊆ R for simplicity. In thefollowing, all terms are in non-dimensional form with a = 1 in Eq. (2.49), suchthat

√θ0 = v0 = c0. Let h = (h0, . . . , hN−1)

T denote a vector of orthonormalHermite basis functions, so that

∫

RhihjM1,0,1 dc = δij (4.49)

for all 0 ≤ i, j < N − 1, where

M1,0,1(c) =1√2π

exp(− c2

2

)(4.50)

is the standard normal distribution.


◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼

0 0.5 1 1.5 2 2.5 31

102

104

106

108

1010

v

κ

◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼

◼

◼

-0.04 -0.03 -0.02 -0.01 0 0.011

102

104

106

108

1010

α4

κ

Monomial Basis ◼ Hermite Basis

Figure 4.3: The plot on the left shows the condition number, κ, of the Hessian matrix asa function of the velocity, v, for the equilibrium distribution E1,v,1. The rightplot shows the condition number of the Hessian matrix for the non-equilibriumdistribution exp(α4c4)E1,0,1 as a function of the coefficient α4.

A natural choice for the basis functions are the scaled and shifted Hermitepolynomials p = (p0, . . . , pN−1)

T given by

pi(c) = hi(ξ) with ξ =c− v√

θ, (4.51)

since these diagonalize H for f (ME)α (c) =Mρ,v,θ(c) and Ωc = R, as we have

H =∫

R(ppT)(c)Mρ,v,θ(c) dc = ρ

∫

R(hhT)(ξ)M1,0,1(ξ) dξ = ρI. (4.52)

Figure 4.3 shows the condition number of H in the monomial and the adaptiveHermite basis for an equilibrium distribution Eρ,v,θ with ρ = θ = 1 as a func-tion of the macroscopic velocity v and for a perturbed equilibrium distributionE1,0,1eα4c4

as a function of α4 on a bounded velocity domain Ωc = [−10, 10].The results show that while the Hermite basis diagonalizes the Hessian for aMaxwellian distribution, deviations from the Maxwellian distribution can stillcause large condition numbers also in the case of the adaptive Hermite basis.

Alldredge et al. presented in [4] an adaptive change of basis algorithm thatdetermines at each iteration of the Newton algorithm a new set of basis func-

tions p that diagonalize the Hessian H = 〈ppT f (ME)α 〉. This fully adaptive basis

method is more robust than the adaptive Hermite basis, but computationallymore expensive.


4.4.3 Realizability Preserving Method

For a time-stepping method in the moment vectors u, it is crucial that the nu-merical solution un+1 at the next time step is realizable, so that the dual mini-mization problem for the Lagrange parameters αn+1 has a solution. Numericalschemes that guarantee the realizability of un+1

j for all cells j = 1, . . . , Nx arecalled realizability preserving methods. In [3] a realizability preserving schemeis presented for moment systems in the context of radiation transport. Follow-ing [3], the subsequent sections introduce a realizability preserving method formoment systems in the context of rarefied gases.

4.4.3.1 Kinetic Flux Scheme

Let us consider the second-order kinetic Riemann solver presented in [3], forwhich the numerical flux is given by

F j+1/2 = 〈cxφ f (ME)j+1/2〉, for j = 0, . . . , Nx, (4.53)

where

f (ME)j+1/2(c, t) =

f (ME)j (c, t) + ∆x

2 sj(c, t), cx ≥ 0

f (ME)j+1 (c, t)− ∆x

2 sj+1(c, t), cx < 0(4.54)

and f (ME)j := f (ME)

α(uj). The slopes are given by

sj = φ(SL)GMM

Θ

f (ME)j − f (ME)

j−1

∆x,

f (ME)j+1 − f (ME)

j−1

2∆x, Θ

f (ME)j+1 − f (ME)

j

∆x

(4.55)

with parameter Θ ∈ [1, 2], see e.g. [4, 71] and the references therein. The function

φ(SL)GMM denotes the generalized minmod limiter defined as

φ(SL)GMM(x1, x2, . . .) =

mini xi, xi > 0 for i = 1, 2, . . . ,

maxi xi, xi < 0 for i = 1, 2, . . . ,

0, otherwise.

(4.56)

The boundary conditions are implemented with the ghost cell method, see alsoSection 4.2.3.


4.4.3.2 Realizability Preserving Conditions

Alldredge et al. presented in the publication [4] realizability preserving condi-tions for the maximum-entropy closure in the context of the linear Boltzmannequation. As shown in the following, these conditions can be easily generalizedto the gas dynamics case.

Let us consider the Heun time-stepping scheme (4.25). Since each stage of thisscheme is equivalent to an Euler time step it suffices to show that the Euler stepis realizability preserving. The final step in (4.25) is then a convex combinationof two realizable moment vectors, which for bounded velocity domains is againrealizable due to the convexity of the realizability domain RQ.

In order to simplify the notation, in the following the maximum-entropy dis-tribution is denoted by G ≡ f (ME). Let m denote the stage of the Runge-Kutta

method and G(m)i the reconstructed maximum-entropy distribution in cell i at

stage m. Since the dual minimization problem can in general not be solved ex-actly, there is a discrepancy between the numerical solution, denoted by G, andthe exact solution G. Following [4], we define

γ(m)i :=

G(m)i

G(m)i

, γmax := maxm∈0,1j=1,...,Nx

c∈Ωc

γ(m)i (c), (4.57)

where G(m)i denotes a numerical approximation to the maximum-entropy distri-

bution G(m)i . Then the realizability of the moment vectors uj for j = 1, . . . , Nx is

ensured at each (intermediate) time step for the bounded velocity domain (3.50)by fulfilling the realizability maintaining time-step restriction

∆t(

cMx γmax

∆x(2 + Θ)

2+

1Kn

)< 1. (4.58)

The following proof of the time-step restriction (4.58) is an extension of the proofof Thm. 2.5 in [3] for the bounded velocity domain (3.50).

Proof. Let E and G(m−1)i+1/2 denote the numerical approximations to the equilibrium

distribution E and kinetic flux G(m−1)i+1/2 , respectively. The numerical solution at

stage m can be expressed as

u(m)i = 〈φχ

(m)i 〉, (4.59)


where the distribution χ(m)i is given by

χ(m)i = G(m−1)

i − cx∆t∆x

(G(m−1)

i+1/2 − G(m−1)i−1/2

)

+∆tKn

(E (m−1)

i − G(m−1)i

). (4.60)

In the following we assume that u(m−1)i is realizable for all cells. Then we have

G(m−1)i ≥ 0 and G(m−1)

j+1/2 ≥ 0 due to the use of the generalized minmod limiter.Note that the use of open boundary conditions directly implies the realizabilityof the ghost-cells at stage (m− 1).

Let us restrict to the case cx ≥ 0 (the case cx < 0 can be shown analogously).

Clearly, the term χ(m)i is bounded below by

χ(m)i ≥ G(m−1)

i − cx∆t∆x

G(m−1)i+1/2 −

∆tKn

G(m−1)i . (4.61)

If sj > 0, then all arguments to the generalized minmod limiter are positive, suchthat

s(m−1)j ≤ Θ

G(m−1)i − G(m−1)

i−1∆x

(4.62)

and

G(m−1)i+1/2 = G(m−1)

i +∆x2

si

≤ G(m−1)i

Θ + 22− Θ

2G(m−1)

i−1 ≤ G(m−1)i

Θ + 22

. (4.63)

The term χ(m)i is then bounded below by

χ(m)i ≥ G(m−1)

i − cx∆t∆x

γ(m−1)i G(m−1)

i2 + Θ

2− ∆t

KnG(m−1)

i , (4.64)

such that χ(m)i ≥ 0 for all cx ∈ [−cM

x , cMx ] if the time-step restriction (4.58) is

fulfilled.

On the other hand if s(m−1)i ≤ 0, then

G(m−1)i+1/2 = G(m−1)

i +∆x2

s(m−1)i ≤ G(m−1)

i . (4.65)

4.5 efficient methods for the maximum-entropy closure 55

A lower bound is then given by

χ(m)i ≥ G(m−1)

i − cx∆t∆x

γ(m−1)i G(m−1)

i − ∆tKn

G(m−1)i , (4.66)

which is non-negative for

∆t(

cMx γmax

∆x+

1Kn

)< 1. (4.67)

As shown in [4], the factor γmax can be bounded by 1+ εγ with high confidenceif the stopping criteria

||g(α)|| ≤ τ and exp(5cnmax||∆α||1) ≤ 1 + εγ (4.68)

are used in the Newton iteration, where n denotes the highest polynomial powerin φ and cmax = maxcM

x , cMr for the slab geometry considered here. The factor

5 in (4.68) is a safety factor to ensure γmax ≤ 1 + εγ with high confidence [4].

4.5 efficient methods for the maximum-entropy closure

The realizability preserving scheme with the adaptive change of basis methodhas been found to be numerically very robust. Unfortunately, the numerical costsare very high, due to the numerical complexity of the quadrature, the kinetic-based flux computation and the adaptive change of basis algorithm.

While the adaptive change of basis method has been found necessary for nu-merical problems with very strong non-equilibrium states in the solution, someimportant problems for rarefied gas flows can be solved using a fixed monomialbasis, such as the classical shock-structure problem if a suitably smoothed initialcondition is used, see also Chapter 6.

In the following sections, several numerical schemes with reduced computa-tional costs are presented. These schemes use a monomial basis and a simplenon-kinetic numerical flux.

4.5.1 Spatial Discretization

The moment system (4.30) can be expressed in terms of the Lagrange parametersas

∂tu(α) + ∂xf(α) = p(α), (4.69)


where

u(α) =⟨

φ f (ME)α

⟩Q

, f(α) =⟨

cxφ f (ME)α

⟩Q

,

p(α) = −

⟨φ(

f (ME)α − Eβ(uγ(α))

)⟩Q

Kn.

(4.70)

Analogously to the super vector u ∈ RN·Nx defined in Eq. (4.19), the correspond-ing Lagrange multipliers can be collected in a single vector as

α = (αT1 , . . . , αT

Nx)T ∈ RN·Nx , (4.71)

where αi satisfies ui = 〈φ f (ME)αi 〉Q. The semi-discrete scheme (4.18) can now be

written in terms of the Lagrange parameters as

∂tu(α) = g(α), (4.72)

where g = (gT1 , . . . , gT

Nx)T ∈ RN·Nx and u = (uT

1 , . . . , uTNx)T ∈ RN·Nx with

gi(α) = ∂tui(α) = −F i+1/2(α)−F i−1/2(α)

∆x+ pi(α) ∈ RN (4.73)

for i = 1, . . . , Nx.

4.5.2 A Semi-implicit Time-marching Scheme

Let us consider a semi-implicit Euler discretization of Eq. (4.73), yielding thenonlinear system

u(αn+1) = u(αn) + ∆tg(αn, αn+1), (4.74)

where

gi(αn, αn+1) = −F i+1/2(α

n, αn+1)−F i−1/2(αn, αn+1)

∆x+ pi(α

n+1). (4.75)

The numerical flux is of the form

F i+1/2(αn, αn+1) =

12

(f(αn+1

i ) + f(αn+1i+1 )

)+

Di+1/2(αn)

2

(u(αn+1

i )− u(αn+1i+1 )

), (4.76)

where Di+1/2(αn) = D(αn

i , αni+1), such that all terms, except the dissipation ma-

trix, are treated implicitly.


The system (4.74) is solved with Newton’s method. Let αn+1,0 denote an initialguess and αn+1,k the vector of Lagrange multipliers at iteration k. Then the kthNewton iteration reads

JI(

αn, αn+1,k)

∆αn+1,k = −r(

αn, αn+1,k)

,

αn+1,k+1 = αn+1,k + ∆αn+1,k,(4.77)

where

r(

αn, αn+1,k)= u

(αn+1,k

)− u (αn)− ∆tg

(αn, αn+1,k

)(4.78)

and JI ∈ RNNx×NNx is the Jacobian

JI(

αn, αn+1,k)= Du

(αn+1,k

)

− ∆t(

JF(

αn, αn+1,k)+ JP

(αn+1,k

)), (4.79)

where the D denotes the derivative operator with respect to the Lagrange multi-pliers.

The Jacobian Du(α) is given by the block-diagonal matrix

Du(α) = diag (Du(α1), . . . , Du(αNx )) ∈ RNNx×NNx , (4.80)

where

Du(αi) = 〈φφT f (ME)αi 〉Q ∈ RN×N . (4.81)

The block matrices Du(αi) are symmetric positive definite and thus also Du(α).

4.5.2.1 Transport Operator

The numerical flux function (4.76) yields the block-tridiagonal flux Jacobian

JF =

JF1,1 JF

1,2

JF2,1 JF

2,2 JF2,3

. . .. . .

. . .

JFNx−1,Nx−2 JF

Nx−1,Nx−1 JFNx−1,Nx

JFNx−1,Nx

JFNx ,Nx

, (4.82)


where the blocks JFi,j = JF

i,j(αn, αn+1) ∈ RN×N are given by

JFi±1,i =

∓Df(

αn+1i

)−Di±1/2 (α

n) Du(

αn+1i

)

2∆x(4.83)

and

JFi,i =

− (Di−1/2(αn) + Di+1/2(α

n))

2∆xDu

(αn+1

i

), i = 2, . . . , Nx − 1. (4.84)

The block matrices JF1,1, JF

Nx ,Nxdepend on the boundary conditions. For open

boundary conditions with α0 := α1 and αNx+1 := αNx we find

JF1,1 =

Df(αn+11 )−D3/2(α

n)Du(αn+11 )

2∆x,

JFNx ,Nx

=−Df(αn+1

Nx)−DNx−1/2(α

n)Du(αn+1Nx

)

2∆x,

(4.85)

while for Dirichlet boundary conditions we have

JF1,1 =

− (D1/2(αn) + D3/2(α

n))

2∆xDu(αn+1

1 ),

JFNx ,Nx

=−(DNx−1/2(α

n) + DNx+1/2(αn))

2∆xDu(αn+1

Nx).

(4.86)

4.5.2.2 Collision Operator

The implicit treatment of the BGK collision operator requires the computation ofthe Jacobian JP = Dp(α), which is given by the block-diagonal matrix

JP = diag(JP1 , . . . , JP

n ) (4.87)

with

JPi = − 1

Kn

(⟨φφT f (ME)

αn+1i

⟩

Q−

⟨φγTEβ(uγ(α

n+1i ))

⟩Q

⟨γγTEβ(uγ(α

n+1i ))

⟩−1

Q

⟨γφT f (ME)

αn+1i

⟩

Q

)(4.88)

for 1 ≤ i ≤ Nx. The matrix 〈γγTEβ〉Q ∈ R3×3 is symmetric positive definite, suchthat its inverse exists for all β ∈ R3. An alternative to (4.88) is to approximate the


equilibrium distribution Eβ by the MaxwellianMβ = exp(β · γ), which satisfies

〈Mβγ〉Rdc = 〈 f (ME)α γ〉Q. The resulting Jacobian matrix is given by

JP = diag(

JP1 , . . . , JP

Nx

), (4.89)

where

JPi = − 1

Kn

(⟨φφT f (ME)

αn+1i

⟩

Q−⟨

φDMβ(uγ(αn+1i ))

⟩Rdc

). (4.90)

While the evaluation of JP is computationally less expensive than the exact Ja-cobian JP, the number of required Newton iterations can increase considerably,since JP is only an approximation to the exact Jacobian JP. For this reason onlythe exact Jacobian JP is used in the numerical examples.

4.5.2.3 Discussion

The Newton iteration in the Lagrange parameters has a distinct advantage overan iteration in the moment variables: the Lagrange parameters are readily avail-able after each Newton update to evaluate the Jacobian matrices, while in thecase of a Newton iteration in the moment variables the Lagrange parametershave to be computed by solving the dual optimization problem (3.33) in eachiteration. Thus the Newton iteration in the Lagrange multipliers can reduce thecomputational costs.

A disadvantage of the semi-implicit scheme is that the collision invariants areconserved only up to the error tolerance of the Newton iteration. However, sincethe Newton algorithm converges quadratically in the vicinity of the root, only afew iterations are expected to be necessary for convergence, even for very smallerror tolerances.

4.5.3 Explicit First-order Scheme

In order to evaluate the efficiency of the semi-implicit scheme, let us also considerthe fully explicit Euler scheme

u(αn+1) = u(αn) + ∆tg(αn). (4.91)

Even though this scheme is fully explicit in the moment variables, it requires thesolution to a nonlinear system for the Lagrange multipliers at the next time step.


As before, this system is solved with a Newton iteration. Let αn+1,0 denote theinitial guess, then the kth Newton iteration is given by

JE(αn+1,k)∆αn+1,k = −r(αn, αn+1,k), (4.92)

αn+1,k+1 = αn+1,k + ∆αn+1,k, (4.93)

where JE(α) = Du(α) and

r(αn, αn+1,k) = u(αn+1,k)− u(αn)− ∆tg(αn). (4.94)

In contrast to the semi-implicit scheme, the nonlinear system decouples into Nxnonlinear systems since JE is block-diagonal.

4.5.3.1 Explicit Euler in Moment Variables

The explicit Euler scheme (4.91) can also be formulated in the moment variables,yielding the update

un+1 = un + ∆tg(un), (4.95)

The Lagrange parameters αn+1 are then computed using the Newton methoddescribed in Section 4.4, which yields the iteration

Du(αn+1,ki )∆αn+1,k

i = −g(αn+1,ki ) (4.96)

αn+1,k+1 = αn+1,k + ∆αn+1,k, (4.97)

where g is the gradient of the objective function (4.37).The main advantage of the formulation in the moment variables is the exact

conservation of the conserved quantities.

4.5.4 Higher-order Schemes

We also consider a second-order scheme based on a Taylor series expansion. Thefully explicit time-marching scheme of Lax-Wendroff type

u(αn+1i ) = u(αn

i )−∆t∆x

(fLW

i+1/2(αn)− fLW

i−1/2(αn))+ ∆tpLW

i (αn) (4.98)

with numerical flux fLW and source term pLW given by

fLWi+1/2(α

n) = fi+1/2 (αn)−

∆t2

Df(Du)−1

i+1/2

(αn)(δ∆xfi+1/2(α

n)− pi+1/2 (αn))

(4.99)


and

pLWi (αn) = p(αn

i ) +∆t2

(Dp (Du)−1 (αn

i ) (−δ2∆xfi(αn) + p(αn

i )))

, (4.100)

where

·i+1/2(αn) :=

(·)(αni ) + (·)(αn

i+1)

2,

δk∆x(·)i(αn) :=

(·)(αni+k/2)− (·)(αn

i−k/2)

k∆x

(4.101)

has a consistency order of 2.

Proof. Let u satisfy the continuous moment equation

∂tu(α) = −∂xf(α) + p(α). (4.102)

A second-order Taylor series expansion of u(αn+1i ) around u(αn

i ) yields

u(αn+1i ) = u(αn

i ) + ∆t∂tu(αni ) +

(∆t)2

2∂ttu(αn

i ) + O((∆t)3) (4.103)

with

∂ttu(αi) = ∂x

(Df(Du)−1(αi)(∂xf(αi)− p(αi))

)+

Dp(Du)−1(αi) (−∂xf(αi) + p(αi)) . (4.104)

Replacing the spatial derivative ∂xf with the finite difference approximation δ∆xfyields

u(αn+1i ) = u(αn

i ) + ∆t(−δ2∆xfi(α

n) + p(αni ) + O

((∆x)2

))

+(∆t)2

2∆x

(Df(Du)−1(αn

i+1/2)(

δ∆xfi+1/2(αn) + O

((∆x)2

)− p(αn

i+1/2))

− Df(Du)−1(αni−1/2)

(δ∆xfi−1/2(α

n) + O((∆x)2

)− p(αn

i−1/2))

+ O((∆x)2

))

+(∆t)2

2

(Dp (Du)−1 (αn

i )(−δ2∆xfi(α

n) + O((∆x)2

)+ p(αn

i )))

+ O((∆t)3

),


where αni+1/2 = α(xi+1/2, tn), αn

i−1/2 = α(xi−1/2, tn). Approximating the termsDf(Du)−1(αn

i±1/2), p(αni±1/2) with their averages Df(Du)−1n

±1/2, pn±1/2

yields the scheme (4.98). Since the average

hi+1/2 =h(xi) + h(xi+1)

∆x(4.105)

satisfies hi+1/2 = h(xi+1/2)+O((∆x)2) for any smooth function h, the method

has a local error given by O(∆t(∆x)2 + (∆t)2(∆x) + (∆t)3). The time-step ∆t =

c∆x for some constant c > 0 yields the global error O((∆x)2).

4.5.4.1 Discussion

Analogously to the first-order schemes, the Lax-Wendroff type scheme presentedabove can easily be solved semi-implicitly in time by treating all terms exceptDf(Du)−1 and Dp(Du)−1 in Eqs. (4.98)-(4.100) implicitly. Thanks to the com-pactness of the Lax-Wendroff scheme in space, the resulting sparse Jacobian ma-trix has the same block-tridiagonal form as the semi-implicit first-order schemes.

4.6 efficient time stepping algorithms

This section presents efficient implementations of the semi-implicit and explicittime stepping schemes that have been introduced in the previous sections. Al-gorithm 1 shows a general time step implementation, which makes use of thefunctions NewtonIteration<Scheme> and GetNewTimeStep<Scheme>, whereScheme refers to the time stepping method.

In case of the explicit first and second-order methods, the time step in thefunction GetNewTimeStep <Scheme> is chosen according to the CFL condition

∆t =cs∆xλmax , λmax = max

iλ(αi), (4.106)

where λ(α) denotes the largest characteristic speed of the moment system instate α and cs ∈ (0, 1) is a safety factor.

For the semi-implicit scheme, the time step is chosen in an adaptive way, sothat the number of Newton iterations is expected to be in a specified interval[minOptNewtIter, maxOptNewtIter]. Let ∆tk denote the time step length of thekth time step. If the actual number of Newton iterations in time step k is largerthan maxOptNewtIter, then the next time step is set according to ∆tk+1 ← ∆tkβd,where βd ∈ (0, 1) is a user-defined constant. Conversely, if the number of Newtoniterations is below minOptNewtIter, then ∆tk+1 ← ∆tkβi with βi ∈ (1, ∞).

4.6 efficient time stepping algorithms 63

Should the Newton iteration fail to converge in either the explicit or the semi-implicit schemes, then the current time step is rejected and repeated with a re-duced time step length ∆tk ← ∆tkβ f with β f ∈ (0, 1).

Alg. 2 shows the Newton iteration for the semi-implicit Euler time steppingscheme. The Function EvalSystem<SemiImplicitEuler> evaluates the Jacobianmatrix JI and vector r. The resulting linear system is solved with an LU factor-ization, for which we use the MKL Pardiso library by Intel. In each Newton iter-ation, a simple test is used to detect divergence of the solution α. This is achievedwith the function LIS, which computes the longest sequence of consecutively in-creasing residuals (e0, . . . , ek). If this number exceeds the user-defined parametermaxIncIter, then the Newton iteration aborts and the time step is rejected.

In Alg. 3, an implementation of the Newton scheme for the explicit Eulermethod is shown. The function EvalExplicitSystem evaluates both the Hessianmatrix JE and the right-hand side vector g0. The vector u0 can be extracted fromthe Hessian matrix. In contrast to the implicit scheme, the vector g only has to beevaluated once and can be used in the Newton iteration to compute r. Since thematrices JE

i ∈ RN×N are symmetric positive definite, a Cholesky factorization isused to solve the linear systems.

The implementation of Alg. 3 follows the idea of Garrett et al. in [44] to com-pute the Hessian matrices in batches. Here, this is achieved by storing the indicesof all spatial cells for which the Newton iteration has not yet converged in a listP, so that Pi is the index of the ith sub-vector αi in α. Initially, P contains all min-imization problems. The vector αP ∈ RN|P| stores the corresponding Lagrangeparameters. The function UpdateProblemSet computes the errors eki

= ||ri||2for each minimization problem, copies the converged solutions to α and updatesthe matrix JE and vectors r, αP, so that they only contain the data of the remainingminimization problems, which have not yet converged.

In both the explicit and semi-implicit schemes, the solution to the last timestep αn is used as the initial guess α0 for the Newton iteration in the next timestep.


Algorithm 1 General time step update implementation.

1: function TimeStep<Scheme>(α0)2: status← notConverged3: while status 6= converged do4: (α1, iter, status)← NewtonIteration <Scheme>(α0, t, ∆t)5: if status = converged then6: t← t + ∆t7: ∆t← GetNewTimeStep <Scheme>(∆t, iter)8: else9: ∆t← ∆t · β f

10: end if11: end while12: return α1

13: end function

Algorithm 2 Newton type optimization for the semi-implicit Euler time steppingscheme

1: function NewtonIteration<SemiImplicitEuler>(α0, t, ∆t)2: α← α0

3: (JI , r)← EvalSystem <SemiImplicitEuler>(α, t, ∆t)4: k← 05: ek ← ||r||26: while ek > newtonTol do7: k← k + 18: Solve JI∆α = −r9: α← α + ∆α

10: (JI , r)← EvalSystem <SemiImplicitEuler>(α, t, ∆t)11: ek ← ||r||212: if k > maxNewtonIter or LIS(e0, . . . , ek) > maxIncIter then13: return (α, k, notConverged)14: end if15: end while16: return (α, k, converged)17: end function

4.6 efficient time stepping algorithms 65

Algorithm 3 Newton minimization for the explicit Euler time stepping scheme

1: function NewtonIteration<ExplicitEuler>(α0, t, ∆t)2: P← (1, 2, . . . , Nx)3: (JE, u0, g0)← EvalSystem <ExplicitEuler>(α0, t, ∆t)4: r← −∆tg0

5: αP ← α0

6: k← 07: (P, JE, r, αP, α, ek)← UpdateProblemSet(P, JE, r, αP, α) . ek = ||r||28: while |P| > 0 do9: k← k + 1

10: Solve JEi ∆αP

i = −ri for i = 0, 1, . . . P− 111: αP ← αP + ∆αP . ∆αP = ((∆αP

0 )T , . . . , (∆αP

P−1)T)T

12: JE ← EvalHessians(αP, t, ∆t) . JE = (JE0 , . . . , JE

|P|−1) ∈ RN×N|P|

13: ui ← JEi e1 for i = 0, 1, . . . P− 1 . e1 = (1, 0, . . . , 0)T ∈ RN

14: ri ← ui − u0Pi− ∆tg0

Pifor i = 0, 1, . . . P− 1

15: (P, JE, r, αP, α, ek)← UpdateProblemSet(P, JE, r, αP, α)16: if k > maxNewtonIter or LIS(e0, . . . , ek) > maxIncIter then17: return (α, k, notConverged)18: end if19: end while20: return (α, k, converged)21: end function


4.7 discrete velocity model

Discrete velocity models (DVM) are based on a direct discretization of the ve-locity on a lattice, see e.g. [92] and the references therein. Let us consider aone-dimensional gas-process on a spatial domain Ωx = [xL, xR] with a one-dimensional velocity domain Ωc ⊆ R.

The truncated velocity domain Ωc = [cL, cR] is uniformly discretized into aset of Nc non-overlapping cells Ici = [ci−1/2, ci+1/2] of size ∆c = ci+1/2 − ci−1/2for i = 1, . . . , Nc. A finite volume discretization in the velocity space yields thesemi-discrete system

∂t fi + ci∂x fi =C(BGK)

i (f)Kn

for i = 1, . . . , Nc, (4.107)

where f = ( f1, . . . , fNc )T and

fi(x, t) =1

∆c

∫

Ici

f (x, t, c) dc, C(BGK)i (f) =

1∆c

∫

Ici

C(BGK)(f) dc. (4.108)

We define the quadrature approximation

Q(φ, f) := ∆cNc

∑i=1

φ(ci) fi ≈ 〈φ f 〉, (4.109)

so that ui = Q(ci, f) defines the ith convective moment of the discrete velocitydistribution.

4.7.1 Conservative BGK-operator by Entropy Minimization

For the discrete system (4.107) to be conservative, the BGK operator has to satisfy

Q(γi,C(f)) = 0 for i = 1, 2, 3, (4.110)

where C = (C1, . . . , CNc )T and γ = (1, c, c2)T are the collision invariants. In [92]

the discrete equilibrium distribution E = (E1, . . . , ENc )T is determined by mini-

mizing the discrete Boltzmann entropy

h(f) := ∆cNc

∑i=1

( fi log fi − fi). (4.111)

The discrete equilibrium distribution function has then the form

Ei = exp(β · γ(ci)) (4.112)

4.7 discrete velocity model 67

with β ∈ R3 if and only if a strict realizability condition is fulfilled [91, 92].The resulting scheme is both conservative and dissipates the discrete Boltz-

mann entropy, see [92, 93]. However, a major drawback of this scheme is thecomputational cost of the unconstrained minimization problems, which have tobe solved for all cells and time steps.

In the next section, a conservative scheme with reduced computational costs ispresented.

4.7.2 Conservative BGK-operator with Least-norm Projection

Let us consider a modified BGK operator of the form

C(BGK)i (f) = −( fi −Mi(f)) + ∆ fi, (4.113)

where instead of the discrete equilibrium Ei, a discrete Maxwellian distributionMi is used. The additional term ∆f = (∆ f1, . . . , ∆ fNc )

T is required to fulfill theconstraints

Q(γi, C(BGK)

(f)) = 0, for i = 1, 2, 3. (4.114)

Here, the correction ∆f is determined by the constrained minimization problem

min ||∆f||2 s.t. AijC(BGK)j (f) = 0, (4.115)

where Q(γ, f) = Af with A = (Aij) ∈ R3×Nc and Aij = γi(cj). Therefore, ∆f isthe smallest correction in the L2 norm that renders the modified BGK-operator(4.113) conservative. The solution to the minimization problem is given by theleast-norm solution

∆f = AT(AAT)−1b, (4.116)

where b = (b1, . . . , bNc )T with bi = Aij( f j −Mj). By performing a QR decompo-

sition of AT = QR, the least-norm solution can be determined numerically in arobust way by ∆f = QR−1b.

Note that for a fixed velocity grid, the QR decomposition has to be computedonly once. For this method to yield reliable results the correction term ∆f shouldbe small. This requires a fine velocity mesh with large velocity bounds, so thatMaxwellian distributions are well represented on the discrete grid.


4.7.3 Spatial and Temporal Discretizations

Let us consider the discrete velocity evolution equation

∂t fi + ci∂x fi = Ci(f), i = 1, . . . , Nc. (4.117)

The spatial domain is discretized with a uniform grid with cells

Jj = [xj−1/2, xj+1/2] (4.118)

and cell length ∆x = xj+1/2 − xj−1/2. Applying the finite volume method to Eq.(4.117) yields

∂t fi,j + ciFi,j+1/2 −Fi,j−1/2

∆x= Ci,j, (4.119)

where Fi,j−1/2, Fi,j+1/2 denote the numerical fluxes at the cell interfaces and

fi,j(t) =1

∆x

∫ xj+1/2

xj−1/2

fi(x, t) dx, Ci,j =1

∆x

∫ xj+1/2

xj−1/2

Ci(f) dx (4.120)

are the spatially averaged distribution fi,j over cell Jj and the discretized sourceterm Ci,j. The upwind flux is given by

F (Up)i,j+1/2 = F (Up)( f−i,j+1/2, f+i,j+1/2) =

f−i,j+1/2, ci ≥ 0,

f+i,j+1/2, ci < 0,(4.121)

where f±i,j+1/2 = limε→0 fi(xj+1/2 ± ε). Here, fi,j is assumed to be a constant

function over each cell, so that f+i,j+1/2 = fi,j+1, f−i,j+1/2 = fi,j.Applying the explicit Euler discretization to the semi-discrete system (4.119)

with the least-norm projection method for the BGK operator yields the fullydiscrete update formula

f n+1i,j = f n

i,j +ci∆t∆x

(Fn

i,j−1/2 −Fni,j+1/2

)+ ∆t

(Mni,j − f n

i,j + ∆ f ni,j

Kn

).

A necessary condition for stability is the CFL condition

∆t =cs∆x

maxi |ci|, (4.122)

4.8 memory complexity of the dvm 69

where cs ∈ (0, 1) denotes a user-specified safety factor. Empirically, the CFLcondition with cs ≈ 0.9 was found to be sufficient to maintain stability for flowswith Knudsen numbers in the transition regime, e.g. Kn ∼ 1.

4.8 memory complexity of the dvm

Both the maximum-entropy closure and the DVM rely on an underlying velocitygrid. However, in contrast to the maximum-entropy closure, each quadraturepoint in the DVM adds one degree of freedom to the system and has to bestored in memory. Let us consider the general case of a three-dimensional gasflow with a three-dimensional underlying velocity domain. Assuming that boththe maximum-entropy closure and the DVM use the same velocity grid with Ncquadrature points and Nx grid cells, then the memory complexity of the DVM isO(Nc Nx), while the memory complexity of the 35-moment system is only O(Nx).For the DVM to have about the same memory requirements as the 35-momentsystem it should not exceed Nc = 35 degrees of freedoms in velocity space.However, typically the number of quadrature points used for the DVM are muchhigher, see e.g. [93]. The low memory consumption of the moment equationsrenders this approach very attractive for multi-dimensional flows, for which therequired memory resources of the DVM can be excessive.

5O P T I M I Z E D Q U A D R AT U R E I M P L E M E N TAT I O N S

This chapter presents optimized software implementations of the computation-ally expensive numerical quadrature algorithm used to compute moments of themaximum-entropy distribution function.

5.1 introduction

Moore predicted in 1965 that the number of transistors would double about ev-ery two years [94]. And indeed, his prediction, which became known as Moore’slaw, held true for over half a century. The growth of transistors was accompaniedby an increase in the clock frequency of processors, which increased the per-formance of serial algorithms. However, since around the year 2005, the growthin the clock frequencies came to a standstill, see [27, 30]. Instead, manufactur-ers began to include more and more cores in processors, increased the size ofSIMD units and developed more complex memory hierarchies [107]. Unfortu-nately, these architectural changes have made it much more difficult to achieveclose to optimal performance of software implementations [107]. To gain advan-tage of modern CPUs and or GPUs, efficient implementations are required toutilize all levels of parallelism, such as

• instruction level parallelism,

• data level parallelism,

• and thread level parallelism.

The following sections show how the exploitation of the inherent parallelism ofthe quadrature algorithm improves the performance of the quadrature computa-tion, when compared to a naive, single threaded implementation in C++.

71

72 optimized quadrature implementations

5.2 efficient quadrature implementations

In each Newton iteration of the time-marching algorithms presented in the pre-vious chapter, moments of the maximum-entropy distribution have to be com-puted, e.g. to set up the Hessian matrices and gradient vectors. Fortunately, dueto the inherent parallelism in the quadrature rule, the computation can be spedup by exploiting multiple levels of parallelism offered by modern CPUs andGPUs.

In the following, we use the Hessian computation

Hα = 〈φφT f (ME)α 〉Q

= 2πNcx−1

∑i=0

Ncr−1

∑j=0

wxi wrj crj f (ME)α (cxi , crj )φφT(cxi , crj )

(5.1)

to compare the computational efficiency of several implementations.For full tensor theories with φi ∈ Πn for i ∈ 0, . . . , N − 1 we have φiφj ∈ Π2n

for all 0 ≤ i, j < N. Plugging the highest possible tensorial degree 2n into Eq.(3.56) yields the number of unique terms in the Hessian matrix, which for the 35-moment system evaluates to N? := 25 moments in the slab geometry consideredhere. Thus, the Hessian matrix Hα can be evaluated by populating the matrix

with the unique moments in the vector u?α = 〈φ? f (ME)

α 〉Q, where φ? is a vectorof monomials spanning Π2n.

5.2.1 Hardware

The implementations presented in the following sections are run on a systemwith 16 Intel Xeon E5-2670 cores and one NVIDIA K20C graphics card. Thesoftware is written in the C++ programming language and all floating point op-erations are performed in double-precision.

5.2.2 Serial Implementation

Algorithm 4 shows a serial implementation of the Hessian computation, whichserves as a basis of comparison for the optimized implementations. The functionEvalBasis evaluates the vector of monomials φ? at the quadrature point (cxi , crj )and stores some of the power evaluations of crj in memory to eliminate unnec-essary reevaluations. In the final step, the function PopulateHessian assembles

the matrix Hα from the vector u? = 〈φ? f (ME)α 〉Q, which contains all unique terms

of the Hessian.

5.2 efficient quadrature implementations 73

Algorithm 4 Single-core Hessian computation.1: function Hessian(α)2: u? ← 0 ∈ RN?

3: for j ∈ 0, . . . , Ncr − 1 do4: for i ∈ 0, . . . , Ncx − 1 do5: φ? ← EvalBasis(cxi , crj )6: u? ← u? + φ? exp(αTφ?)wxi wrj crj

7: end for8: end for9: H← PopulateHessian(2πu?)

10: end function

5.2.3 Efficient Single- and Multi-core Implementations

Note that the Hessian computation given in Eq. (5.1) requires the same oper-ations to be applied at each quadrature point. This data-level parallelism canbe exploited by single-instruction-multiple-data (SIMD) instructions, which per-form a simultaneous execution of the same operation on all elements of a packeddata vector, increasing the performance considerably. The Xeon E5-2670 coresallow the execution of AVX instructions, so that 256-bit SIMD instructions canbe carried out, which operate on four double-precision floating point numberssimultaneously. Algorithm 4 can be vectorized by computing multiple terms inEq. (5.1) with SIMD instructions in parallel.

The Hessian matrix has to be computed for all optimization problems in thetime-marching schemes. In [44] Garrett et al. proposed a batch-wise computa-tion of the independent Hessian for an explicit time-marching scheme. Here wefollow the idea of batch-wise Hessian evaluations both for the explicit and semi-implicit time-marching schemes introduced in Chapter 4. For the semi-implicitscheme, the batch size, denoted by NB, is always set to the total number of cells,Nx. Alternatively, a maximum batch size could be specified if the available mem-ory resources are not sufficient. Here, the independent Hessian computations areparallelized with Intel’s TBB library. Additionally, each thread computes a blockof B(CPU) Hessians at once, which allows to further improve the performance,thanks to a more efficient use of the cache. We found the best performance witha block size of B(CPU) = 4.

5.2.4 Graphics Cards as Accelerators

In contrast to multi-core CPUs, graphics cards are many-core architectures thatprovide high peak performances compared to CPUs. However, for graphics cards


to perform well, the algorithm under consideration has to exploit the fine-grainedparallelism offered by GPUs. Furthermore, if graphics cards are used as acceler-ators, the data transfer between host and GPU has to be taken into account, seee.g. [102].

Algorithm 5 is a general implementation for the computation of a batch ofNB Hessians on some compute device, such as a graphics card. The nodes andweights of the quadrature grid are assumed to reside in the memory of thedevice, such that only the Lagrange parameters α ∈ RNNB have to be trans-ferred to the device. The function PopulateHessians is run on the CPU in orderto reduce the communication cost with the device. The total number of float-ing point numbers transferred from host to device and back is therefore only(N + N?)NB = 34NB.

The HessianKernel is written in the CUDA programming language and runsthe quadrature computation on the graphics card. Figure 5.1 presents a graphicaldepiction of the HessianKernel using the CUDA programming model, see e.g.[76, 97]. The Hessian computations are parallelized using NB CUDA blocks, suchthat the kth block computes the moment vector u?

k , where 0 ≤ k < NB. Addition-ally, each block runs 32 threads in parallel, which corresponds to the warp sizeon the K20C graphics card.

In Stage 1 of the algorithm, the thread with index (tx, ty), where

0 ≤ tx < BlockSizex, 0 ≤ ty < BlockSizey, (5.2)

computes a partial moment vector by performing the quadrature over a rectan-gular sub-grid with Ncx Ncy /(BlockSizexBlockSizey) quadrature points. In Stage2, each block performs a binary reduction to compute the moment vector u?

k ,where k is the block index. While the binary reductions can be performed usingthe shared memory, we found the use of shuffle (SHFL) intrinsics [28] to yield asuperior performance.

Note the function EvalBasis contains dependent operations, since e.g. c3xi

re-quires c2

xiin order to be evaluated in a single operation. By reordering the in-

structions for the quadrature computation in such a way that the same operations

Algorithm 5 Evaluation of NB Hessian matrices on a compute device, such as agraphics card.

1: function HessianWithAccelerator(α) . α = (αT1 , αT

2 , . . . , αTNB

)T ∈ RNNB

2: Transfer α from host to compute device3: Run HessianKernel(α) on compute device4: Transfer u? ∈ RN?NB from device to host5: (Hα1 , . . . , HαNB

)← PopulateHessians(u?)6: end function


Implementation Description

CPU Single-core implementation of Alg. 4 with no optimizations.

CPU (16x) Multi-core implementation of Alg. 4 using 16 Intel Xeon E5-2670 cores but using neither AVX instructions nor blocking.

SIMD Single-core implementation with AVX instructions and block-ing.

SIMD (16x) Multi-core implementation using 16 Intel Xeon E5-2670 coreswith AVX instructions and blocking.

GPU Alg. 5 with one TESLA K20C used as an accelerator and 16Intel Xeon E5-2670 cores with AVX instructions and blocking.

Table 5.1: Summary of the quadrature implementations. The blocking sizes for the op-timized CPU and GPU implementations are set to B(CPU) = min4, NB andB(GPU) = min2, NB.

in the basis evaluation are applied consecutively for B(GPU) quadrature points,the instruction level parallelism can be increased. We found the blocking sizeB(GPU) = 2 to yield the best performance.

5.2.5 Performance Measurements

Figure 5.2 shows the total wall clock time of the implementations CPU, SIMD,SIMD (16x) and GPU, listed in Table 5.1, as a function of the batch size, NB, andthe number of quadrature nodes, Nc. For large problem sizes, such as Nc = 642

with NB ≥ 27 or NB = 210 with Nc > 322, the GPU implementation has a muchsmaller execution time than the optimized multi-core CPU implementation. Onthe other hand, for small values of NB or Nc, the parallel efficiency of the GPUimplementation decreases dramatically. This is due to multiple reasons: the ef-ficiency drops as the number of active threads on the GPU decreases for smallproblem sizes, resulting in a small occupancy. Furthermore, the overhead causedby CUDA, the data transfer between host and GPU, and the execution time ofPopulateHessians become dominant, causing the run time of a single Hessiancomputation on the GPU to be slower by almost an order of magnitude, as com-pared to the optimized single-core implementation.

The execution time of the GPU implementation can be broken down into threemain parts: the GPU kernel, the data transfer and the CPU run time, whichincludes the PopulateHessians function. Figure 5.3 displays the relative execu-tion times of these components. While for large numbers of quadrature nodes,the overall run time is dominated by the GPU kernel, for decreasing values of Nc


Peak Performance Theoretical [GFlop/s] Measured [GFlop/s]

16 Intel Xeon E5-2670 cores 332.8 321.8

NVIDIA Tesla K20C 1170 1043

Table 5.2: Theoretical [106] and measured peak performances of the 16 Intel Xeon coresand the NVIDIA Tesla K20C graphics card. The measured peak performanceshave been determined by running highly optimized DGEMM implementationsof Matlab’s parallel computing toolbox.

the relative run time of the CPU increases and dominates the total run time forNc ≤ 322. We found the main reasons to be the overhead of CUDA and the runtime of the PopulateHessians function. Note that the relative execution time ofthe data transfer between host and device is insignificant in this case, as even forNc = 162 the data transfer only accounts for about 6% of the total execution time.

Figure 5.4 shows strong speed-up measurements of the optimized quadratureimplementations, as compared to the unoptimized, single-core CPU implemen-tation for NB = 210 and Nc = 322, 1282. The results highlight the importanceof in-core optimizations, as the SIMD (16x) allows to improve the performanceof the CPU (16x) implementation by almost an order of magnitude. The GPUimplementation has a smaller execution time compared to the SIMD (16x) imple-mentation if the number of quadrature points is large enough, e.g. Nc > 322 forNB = 210. For the case Nc = 1282, the GPU implementation is roughly 3.3x fasterin comparison to the optimized SIMD (16x) implementation.

In Figure 5.5 we show the efficiency in terms of the measured peak perfor-mances given in Table 5.2 for both the optimized multi-core and GPU kernels forthe quadrature computation of the unique moments in u? ∈ RNNB . Here theefficiency has been computed by using the estimate

Op(NB, Nc) = NB(67Nc − 25) (5.3)

for the required number of operations. The function Op is a lower bound on theactual number of operations, since it assumes all computed terms, such as themonomial powers, to be stored in memory and retrieved if necessary, whereasthe implementations considered here recompute some terms for each Hessianevaluation in order to reduce memory transfers at the expense of a higher num-ber of operations. Let ∆T denote the measured runtime and P(Peak) the measuredpeak performance, then we can define the efficiency as

Eff =Op(NB, Nc)/∆T

P(Peak). (5.4)


Clearly, for both the GPU and the SIMD (16x) implementations the efficiencyincreases monotonically as the number of quadrature points is increased. Forvery large numbers of quadrature points, the efficiencies of both methods levelsoff at about 34% and 42% for the SIMD (16x) and GPU implementations, respec-tively. The expensive evaluation of the exponential function was found to be oneof the main reasons why the kernels do not attain even higher efficiencies forlarge Nc.

5.2.6 Roofline Model

The roofline model [128] allows to estimate the maximum achievable perfor-mance of an algorithm on a specific hardware architecture. The model takes intoaccount the memory bandwidth and peak performance of the system, as wellas the arithmetic intensity of the algorithm under consideration. The arithmeticintensity of an algorithm is defined by the number of floating point operationsper Byte of off-chip memory traffic.

Figure 5.6 shows the rooflines for the 16 Intel Xeon E5-2670 cores, the NVIDIAK20C graphics card and the combined system of 16 Intel Xeon E5-2670 cores withthe NVIDIA K20C used as an accelerator. For the combined system, the data isassumed to reside in the main memory of the CPU, such that the memory band-width is limited by the communication between the host device and the graphicscard. Additionally, the performance measurements of the Hessian implementa-tions have been included for a batch size of NB = 210 and different values ofquadrature nodes, Nc.

Here, the quadrature nodes satisfy Ncx = Ncr , so that the arithmetic intensityincreases monotonically in the number of quadrature points Ncx , since the datatraffic grows linearly in Ncx but the number of operations scales as O(N2

cx). Note

that the arithmetic intensity of the combined system is slightly higher than themulti-core implementation, since the velocity grid already resides in the mainmemory of the GPU when the graphics card is used as an accelerator.


cr

| z

cMr

0

|z

0

2

1

3

0 1 2 3

cxcM

x

tx

ty

cMx

BlockSizex

Blo

ckSiz

e y

↵0 ↵1 · · · ↵k · · ·· · · · · ·B0 B1 Bk BNB1

↵NB1

cr

| z

cMr

0

|z

0

2

1

3

0 1 2 3

cxcM

x

tx

ty

cMx

BlockSizex

Blo

ckSiz

e y

State 1: Partial Moment Computation Stage 2: Binary Reduction

Figure 5.1: Graphical depiction of the HessianKernel implementation. In this example4× 4 threads compute partial moments over 2× 2 quadrature nodes in the firststage. In the second stage of the algorithm the moments are computed from thepartial moments of the first stage by performing a binary reduction, of whichthe first step is shown in the visualization.

1 24 27 29 21021121210-4

10-3

10-2

10-1

1

Nx

Wallclocktime[s]

Nc=642

NB

162 322 642 1282 2562 512210-410-310-210-11

10

Nc

Wallclocktime[s]

Nx=1024NB .

CPU SIMD SIMD (16x) GPU

Figure 5.2: The left plot shows the elapsed wall clock time vs. the number of Hessian com-putations for a quadrature grid with Nc = 642 nodes. The right plot displaysthe elapsed clock time for NB = 210 Hessian computations vs. the number ofquadrature points, Nc, where Ncx = Ncr .


Nc=162 Nc=322 Nc=642 Nc=1282 Nc=2562 Nc=51220%

25%

50%

75%

100%Relativewallclocktime

GPU Kernel Data Transfer CPU

Figure 5.3: Relative execution times of the GPU kernel, data transfer between host andGPU and CPU run time for NB = 210 Hessian computations using differentnumbers of quadrature points, Nc. The CPU time includes the execution timeof PopulateHessian and CUDA API calls.

8.61x 13.7x

93.9x 92.6x

0

25

50

75

100

Speed-up

8.69x 14.1x

118x

392x

0

100

200

300

400

Speed-up

SIMD CPU (16x) SIMD (16x) GPU

Figure 5.4: Strong speed-up measurements for NB = 210 Hessian computations withNc = 322 (left) and Nc = 1282 (right) quadrature points for different optimizedimplementations as compared to the serial CPU implementation.


162 322 642 1282 2562 512210%

20%

30%

40%

Nc

Efficiency

SIMD (16x) GPU

Figure 5.5: Efficiency of the Hessian computations vs. the number of quadrature points, Nc,for a batch-size of NB = 210.

10-2 10-1 1 10 102 103 104 1050.1

1

10

100

1000

Flop/Byte

GFlop/s

Roofline-Model

Nc →∞

Intel Xeon E5-2670 (16x)

NVIDIA Tesla K20C

NVIDIA Tesla K20C as accelerator

CPU

CPU (16x)

SIMD (16x)

GPU

Figure 5.6: Rooflines for the 16 Intel Xeon cores, the NVIDIA Tesla K20C and the systemof 16 Intel Xeon cores with the NVIDIA Tesla K20C used as an accelerator, forwhich the data is assumed to reside in the CPU memory. The Hessian ker-nels were run with NB = 210 and Nc = 162, 322, 642, 1282, 2562, 5122 quadraturepoints (from left to right).

6N U M E R I C A L E X A M P L E S

This chapter presents several numerical test examples for the 35-moment systemwith the maximum-entropy closure (ME35) in slab geometry to investigate theapproximation quality of the moment system, as well as the robustness and ef-ficiency of the numerical methods and implementations presented in Chapters 4

and 5.In Section 6.1 a simple homogeneous relaxation test problem is considered,

for which a convergence analysis of the time-integration schemes is presented,together with a study of the conservation errors in mass, momentum and energyas a function of the Newton error tolerance. A smooth density perturbation prob-lem is presented in Section 6.2, for which the influence of the velocity domainsize and the quadrature error in the ME35 system are investigated. Addition-ally, a convergence analysis of the LLF and LW schemes is shown. Numericalsolutions of the ME35 system for shock structure problems with Mach numbersMa = 2, 4, 8 are presented in Section 6.3. The ME35 system is compared with nu-merical solutions to the maximum-entropy based 14-moment system proposed in[90], as well as reference DVM results. Finally, Section 6.4 presents results to sev-eral two-beam Riemann problems. The reference DVM results for the two-beamproblem have been provided by Zhenning Cai.

Table 6.1 lists some of the numerical algorithms presented in Chapter 4 thatare used for the numerical test examples.

6.1 homogeneous relaxation test

Let us consider the evolution of the homogeneous BGK equation given by

∂t f = − f − E( f )Kn

, (6.1)

81

82 numerical examples

Numerical Method Description

ME35-EE-LF First-order explicit Euler time stepping scheme with theLax-Friedrichs flux.

ME35-EE-LLF First-order explicit Euler time stepping scheme with thelocal Lax-Friedrichs flux.

ME35-SIE-LLF Semi-implicit Euler time step scheme with the local Lax-Friedrichs flux.

ME35-HEUN-KF Second-order discretization with the Heun time steppingscheme in the moment variables and the kinetic based fluxusing the minmod limiter.

ME35-LW Lax-Wendroff type method of second-order in space andtime with no limiting.

Table 6.1: Numerical methods used for the test examples.

with specified initial data f (0) = f 0 in slab geometry. The corresponding mo-ment system is given by

∂tu = p(u) = −u− uEKn

, (6.2)

with initial condition u(0) = u0. The equilibrium state uE is determined fromthe initial condition by uE = 〈φE( f 0)〉.

Note that the moment system (6.2) does not depend on higher-order momentsand hence does not require a closure theory. Therefore, the moment system cantrack the evolution of the corresponding moments of the kinetic equation exactly.

The state vector and production terms of the 35-moment system generated bythe basis

φ =1ρ

(ρ, cx, C2

x, C3x, C4

x,C2

r2

, C4r , CxC2

r , C2xC2

r

)T

(6.3)

6.1 homogeneous relaxation test 83

are given by

u =

ρ

vx

θxx

Qxxx

Rxxxx

θrr

Rrrrr

Qxrr

Rxxrr

, p = − 1Kn

0

0

θxx − θ

Qxxx

Rxxxx − 3ρθ2

θrr − θ

Rrrrr − 8ρθ2

Qxrr

Rxxrr − 2ρθ2

, (6.4)

where θ = (θxx + 2θrr)/3 is the thermodynamic temperature. Note that thanksto the conservation of mass, momentum and energy, the terms ρ, vx, θ are con-stants, so that the equations in (6.2) are decoupled and the exact solution to thehomogeneous moment system is given by

u(t) = uE − exp(− t

Kn

)(uE − u0

). (6.5)

Let us consider the initial value problem with

u0 =

(1, 2,

32

,32

√32

,117

4,

34

,8116

,−38

√32

,458

)T

(6.6)

at t = 0. For t→ ∞, the state u0 converges to the equilibrium state

uE = (1, 2, 1, 0, 3, 1, 8, 1, 2)T . (6.7)

For the numerical simulation we use the following parameters: The Knudsennumber is Kn = 10−1 and the velocity space is set to Ωc = [−10, 10] × [0, 10]with a discretization given by Nb

cx= 2, Nb

cr= 1 and Ng

cx = Ngcr = 64. The Newton

tolerance is set to τ = 10−12, so that the numerical error is dominated by thediscretization error of the time-integration scheme.

Figure 6.1 displays the convergence for the numerical solution of Qxxx com-puted with the explicit Euler scheme ME35-EE-LLF and the Lax-Wendroff schemeME35-LW at time T = 0.1 in the numbers of time steps Nt. The empirical ratesof convergence match the theoretical rates of p = 1 and p = 2 for the ME35-EE-LLF and ME35-LW scheme, respectively. On the right, the evolution of theerror in Qxxx is shown for several schemes using Nt = 800 time steps over thetime interval t = [0, 1]. As expected, the numerical solutions of the explicit and


◼◼

◼◼

◼

80 160 320 640 1280

10-7

10-6

10-5

10-4

10-3

10-2

Nt

Rel.ErrorinQxxx

p=1

p=2

◼ ◼◼

◼◼

◼◼

◼◼

◼

0 0.2 0.4 0.6 0.8 110-8

10-7

10-6

10-5

10-4

10-3

10-2

t

ErrorinQxxx

◼ ME35-EE-LLF ME35-SIE-LLF ME35-LW ME35-HEUN-KF

Figure 6.1: The left plot shows a convergence analysis of the homogeneous relaxation prob-lem for the ME35-EE-LLF and ME35-LW schemes. On the right, the absoluteerror in Qxxx is shown for Nt = 800 time steps for several numerical schemes.Note that the error curves for the first-order schemes ME35-EE-LLF and ME35-SIE-LLF, as well as the second-order schemes ME35-LW and ME35-HEUN-KF,lie on top of each other.

semi-implicit Euler schemes are identical. For this simple problem, the leading-order error terms of the second-order methods ME35-HEUN-KF and ME35-LWare identical, so that both methods yield essentially the same numerical error inthis example.

Except for the ME35-HEUN-KF scheme, the numerical methods carry out thetime-update in the Lagrange multipliers, such that mass, momentum and en-ergy are not conserved exactly, but depend directly on the error tolerance of theNewton solver.

Figure 6.2 shows the error in the conserved quantities for the ME35-LW schemeusing the error tolerances τ = 10−4 and τ = 10−12. Clearly, the error in the con-served quantities is below the specified error tolerance. Thanks to the quadraticconvergence rate of the Newton iteration in the vicinity of the root, only one ortwo additional Newton steps are necessary to satisfy the smaller error toleranceτ = 10−12 in comparison to τ = 10−4.

In Figure 6.3, the evolution of several moments of the ME35 system are shownfor the homogeneous relaxation problem with the initial condition (6.6) and theKnudsen number Kn = 10−1. The converged moment solution was computedwith the ME35-LW scheme.

6.2 smooth density perturbation problem 85

◼

◼ ◼ ◼ ◼ ◼◼ ◼ ◼ ◼ ◼

◼

0 0.20.2 0.4 0.6 0.8 110-16

10-14

10-12

10-10

10-8

10-6

10-4

t

Error Tol. 10-4 Tol. 10-12

◼ ρ

vx

θ

ρ

vx

θ

Figure 6.2: Absolute errors in the mass density, ρ, velocity, vx , and temperature, θ, for thehomogeneous relaxation problem with the Newton error tolerances τ = 10−4

and τ = 10−12.

6.2 smooth density perturbation problem

Let us consider the initial value problem with data in local equilibrium given byf 0(c; x) = Eβ(x)(c) with

ρ0(x) = 1 +ρ√2πθ

exp(− x2

2θ

), v0(x) = 0, θ0(x) = 1. (6.8)

The analytical solution to the collisionless, linear kinetic transport equation withthe initial condition (6.8) can be evaluated easily. E.g., the first three moments ofthe exact solution are given by

ρ(x, t) = 1 +ρ√

2π(t2 + θ)exp

(− x2

2(t2 + θ)

),

ρ(x, t)v(x, t) =ρtx√

2π(t2 + θ)3exp

(− x2

2(t2 + θ)

),

ρ(x, t)e(x, t) =32+

ρ(2t4 + 3θ2 + t2(x2 + 5θ))

2√

2π(t2 + θ)5exp

(− x2

2(t2 + θ)

).

(6.9)

For this problem, the reference microscopic length scale is chosen as the mean-free path λ = aτ0

√θ0 with a = a2, see Eq. (2.49). Figure 6.4 shows the kinetic

transport solution, together with the numerical solution to the ME35 system forthe initial condition (6.8) with ρ = 4.7, θ = 122.7 at time T = 5. The momentsystem is solved with the ME35-LW scheme and the following parameters: The


0 0.2 0.4 0.6 0.8 10

1

2

3

4

t

vx

0 0.2 0.4 0.6 0.8 11.0

1.1

1.2

1.3

1.4

1.5

t

θxx

0 0.2 0.4 0.6 0.8 10.75

0.80

0.85

0.90

0.95

1.00

t

θrr

0 0.2 0.4 0.6 0.8 10.0

0.5

1.0

1.5

t

Qxxx

0 0.2 0.4 0.6 0.8 1

-0.4

-0.3

-0.2

-0.1

0.0

t

Qxrr

0 0.2 0.4 0.6 0.8 1

5

10

15

20

25

30

t

Rxxxx

0 0.2 0.4 0.6 0.8 12

3

4

5

t

Rxxrr

0 0.2 0.4 0.6 0.8 15.05.56.06.57.07.58.0

t

Rrrrr

Figure 6.3: Temporal evolution of several moments of the ME35 system for the homoge-neous relaxation problem. The black line is the exact solution to the relaxationproblem and the red circles show the converged numerical solution with theME35-LW method.


spatial domain Ωx = [−78, 78] is discretized with Nx = 256 cells. The velocitydomain bounds are set to cM

x = cMr = 6.27 and the velocity domain is discretized

with Ngcx = 64, Ng

cr = 32 Gauss-Legendre quadrature points and the number ofblocks is set to Nb

cx= Nb

cr= 1. Here, the fixed time step ∆t = 5 · 10−3 was used.

Even though this is a collisionless gas flow, the numerical solution to the ME35system turns out to be in excellent agreement with the kinetic solution.

-40 -20 0 20 401.00

1.05

1.10

1.15

x

ρ

-40 -20 0 20 40

-0.02

-0.01

0.00

0.01

0.02

x

v

-40 -20 0 20 40

0.996

0.998

1.000

1.002

x

θ

-40 -20 0 20 40-0.004

-0.002

0.000

0.002

0.004

x

qx

Kinetic Transport ME35

Figure 6.4: Profiles of mass density, ρ, velocity, v, temperature, θ, and heat flux, qx , forthe smooth density perturbation problem for both the exact kinetic transportsolution and the 35-moment system.

In Figure 6.5, the numerical solution to the 35-moment system is shown, to-gether with the heat flux predicted by Fourier’s law, given by

q(NSF)x = −κ ∂xθ, κ =

5 Kn2a

ρθ (6.10)

in non-dimensional form, where κ denotes the heat-conductivity and a = a2 isthe non-dimensional parameter in Eq. (2.49). Here, Fourier’s law is evaluated byusing the ME35 solution to the mass density and temperature fields in Eq. (6.10).

Clearly, for the case Kn = 10−2, the heat flux of the ME35 system is in excel-lent agreement with the heat flux field predicted by the law of Fourier. On the


-40 -20 0 20 40-1.5⨯10-5

-5⨯10-6

-10-5

0

5⨯10-610-5

1.5⨯10-5

x

qx

Kn=10-2

-40 -20 0 20 40-1.5⨯10-3

-5⨯10-4

-10-3

0

5⨯10-410-3

1.5⨯10-3

x

qx

Kn=1

ME35 Fourier Law

Figure 6.5: Profiles of the heat flux field of the ME35 system and Fourier’s law, evaluatedfrom the solution to the ME35 system, for the density perturbation problemwith Kn = 10−2 and Kn = 1.

other hand, in the transition regime with Kn = 1, the heat flux deviates fromFourier’s law, suggesting that the non-equilibrium effects of this gas flow cannotbe predicted accurately with the NSF equations.

To study the quadrature and the truncation error of the bounded velocity do-main for the smooth density perturbation problem, several numerical solutionsfor varying velocity domain sizes cM = cM

x = cMr and average grid spacings

∆c = cM/Ncx = cM/Ncr are compared to a reference solution to the ME35 sys-tem that has converged in the parameters cM → ∞ and ∆c→ 0. For this analysis,the Knudsen number is set to Kn = 1 and the solution is computed from t = 0 tothe final time T = 5. The numerical solutions are computed with the ME35-EE-LLF method using the following parameters: The spatial domain Ωx = [−78, 78]with open boundary conditions is discretized with Nx = 1024 cells and the timestep is set to ∆t = 1.25 · 10−2.

Figure 6.6 shows the relative L1 errors

e(ρ) =||ρ− ρ(ref)||L1

||ρ(ref)||L1

, e(qx) =||qx − q(ref)

x ||L1

||q(ref)x ||L1

(6.11)

at the final time T = 5, where ρ(ref), q(ref)x denote the reference solutions to the

mass density and heat flux. The reference solution is computed with a largevelocity space cM = cM

x = cMr ≈ 12.5 and ∆c ≈ 0.049. As expected, the error

decreases spectrally in ∆c for the Gauss-Legendre quadrature until the truncationerror of the bounded velocity domain dominates the error.


0.0 0.2 0.4 0.6 0.8

10-14

10-11

10-8

10-5

0.01

10

Δc

Rel.L1Error

Error in ρ

cM=3.13 cM=4.70 cM=6.27

0.0 0.2 0.4 0.6 0.8

10-14

10-11

10-8

10-5

0.01

10

Δc

Rel.L1Error

Error in qx

cM=3.13 cM=4.70 cM=6.27

Figure 6.6: Relative errors in the L1 norm for the mass density, ρ, and heat flux, qx , forthe smooth perturbation problem, using various domain sizes of the underlyingvelocity grid cM = cM

x = cMr and velocity grid spacings ∆c = cM/Ncx = cM/Ncr .

Figure 6.7 shows a convergence analysis in the spatial and temporal discretiza-tion for both the ME35-EE-LLF and ME35-LW schemes. The error in the momentfields is computed at the final time T = 5 in the relative L1 norm for the massdensity, velocity and heat flux fields. The parameters of the velocity grid arecM

x = cMr = 6.27 with Ncx = 64, Ncr = 32 Gauss-Legendre quadrature nodes

and Nbcx

= Nbcr

= 1. The time step is set according to ∆t/∆x = 6.4 · 10−2. Therelative L1 error is computed with respect to a reference solution that has beencomputed with the ME35-LW scheme using Nx = 2048 cells in the spatial do-main. The empirical convergence rates match the expected convergence rates ofp = 1 and p = 2 for the ME35-EE-LLF and ME35-LW scheme respectively.

Figure 6.8 shows a comparison of the numerical solutions on a coarse gridwith Nx = 64 cells for the ME35-EE-LF, ME35-EE-LLF and ME35-LW schemes,together with a converged reference solution. As expected, the Lax-Friedrichsflux is the most diffusive numerical flux. While the ME35-EE-LLF scheme yields amore accurate result, the ME35-LW method, being a second-order scheme, yieldsthe most accurate result.

6.2.1 Time measurements

Figure 6.9 shows the average wall-clock time for one time step for both the ME35-EE-LLF and ME35-SIE-LLF schemes applied to the smooth perturbation problem.Note that since the first time step includes the execution of some additional ini-tialization routines it has been excluded from the time measurements. The fig-ures also include a break down of the execution time in the most time-consumingtasks of the time-stepping routine listed in Table 6.2.


64 128 256 512 1024

1

10-1

10-2

10-3

10-4

10-5

10-6

Nx

Rel.L1Error

Convergence Analysis LLF

p=1

ρ vx qx

64 128 256 512 1024

1

10-1

10-2

10-3

10-4

10-5

10-6

Nx

Rel.L1Error

Convergence Analysis LW

p=2

ρ vx qx

Figure 6.7: Convergence analysis of the ME35-EE-LLF method (left) and the ME35-LWscheme (right).

Abbreviation Description

HESS Computation of the Hessian matrices Du, Df.

BGK Computation of the production terms p and the Jacobian Dp.

LS Direct matrix factorizations and solutions to the linear systems.

EV Eigenvalue computations.

SM Filling of sparse matrix.

Table 6.2: Description of the most time-consuming tasks in the time-stepping routines.

As before, the equations are integrated from t = 0 to T = 5. The parametersfor the numerical simulation are set to Nx = 210, cM

x = cMr = 6.27, Ncx = 64,

Ncr = 32, Nbcx= Nb

cr= 1 and the Newton tolerance is 10−8.

Here we consider the SIMD, SIMD (16x) and GPU implementations that havebeen presented in Chapter 5. In the GPU implementation, the Jacobians Du, Dfare computed simultaneously on the GPU in batches of size NB = Nx by com-puting the unique terms of Du and Df, analogously to the Hessian computationdiscussed in Chapter 5. The Hessian computations in the Newton iteration for theLagrange parameters of the equilibrium distribution are evaluated with an op-timized implementation that uses AVX instructions and is parallelized with theIntel’s TBB library. In contrast to the Newton method used in the time-steppingmethods, the source terms are computed for each cell independently, using aNewton solver with backtracking. All of the time-consuming tasks, such as theeigenvalue computation, are parallelized with Intel’s TBB library in the SIMD(16x) and GPU implementations.


-60 -40 -20 0 20 40 601.00

1.05

1.10

1.15

x

ρ

Ref. LF LLF LW

-60 -40 -20 0 20 40 60

-0.02

-0.01

0.00

0.01

0.02

x

vx

Ref. LF LLF LW

-60 -40 -20 0 20 40 60

0.996

0.998

1.000

1.002

x

θ

Ref. LF LLF LW

-60 -40 -20 0 20 40 60

-0.0010

-0.0005

0.0000

0.0005

0.0010

x

qx

Ref. LF LLF LW

Figure 6.8: Mass density, velocity, thermodynamic temperature and heat flux fields for thesmooth density perturbation problem using the Lax-Friedrichs (LF), local Lax-Friedrichs (LLF) and Lax-Wendroff (LW) numerical fluxes on a coarse meshwith Nx = 64.

Interestingly, for the ME35-SIE-LLF method over 50% of the execution time isconsumed by tasks unrelated to quadrature computations in the fully optimizedGPU implementation, so that replacing the expensive quadrature computationwith an explicit expression for the moments yields at most a speed-up of 2x.

Table 6.3 shows several run time measurements of the explicit and semi-implicitschemes using the GPU implementation. For this problem, the number of New-ton steps turns out to very insensitive to the time step size.

In Figure 6.10, the elapsed wall-clock time for the ME35-EE-LLF and ME35-LW schemes are shown, highlighting the surprising observation that the second-order ME35-LW scheme is slightly faster than the first-order ME35-EE-LLF scheme.This is due to the fact that most of the higher-order derivative required by theLax-Wendroff method have to be computed also for the LLF method, e.g. Du andDf are used for the computation of the spectral radius. The additional evaluationof Dp is relatively inexpensive.


0 0.05 0.1 0.15 0.2 0.25

SIMD

SIMD (16x)

GPU

Wall clock time [s]

Explicit LLF Time Step

HESS BGK LS EV Other

0% 25% 50% 75% 100%

SIMD

SIMD (16x)

GPU

Relative wall clock time

Explicit LLF Time Step

HESS BGK LS EV Other

0 0.05 0.1 0.15 0.2 0.25

SIMD

SIMD (16x)

GPU

Wall clock time [s]

Semi-Implicit LLF Time Step

HESS BGK LS EV SM Other

0% 25% 50% 75% 100%

SIMD

SIMD (16x)

GPU

Relative wall clock time

Semi-Implicit LLF Time Step

HESS BGK LS EV SM Other

Figure 6.9: Averaged execution times for one time step with Nx = 1024 cells and Nc = 2048quadrature points, using the explicit and semi-implicit Euler schemes with thelocal Lax-Friedrichs flux. The colored segments refer to the most time consum-ing tasks described in Table 6.2.

In comparison to the Lax-Wendroff type scheme, a two-stage explicit Runge-Kutta method with linear reconstruction in space is expected to have a muchhigher run time, since such a scheme requires the evaluation of an intermedi-ate stage and additional quadrature computations at the interfaces for the fluxevaluations of the reconstructed values.

0 0.0025 0.005 0.0075 0.01

LLF

LW

Wall clock time [s]

Explicit Time Step

HESS BGK LS EV Flux Other

Figure 6.10: Average wall-clock time of one time step for both the ME35-LW and ME35-EE-LLF schemes.

6.3 shock structure problems 93

M35-EE-LLF M35-SIE-LLF

Time step size T/128 T/128 T/32 T/8 T/2

Avg. number of Newton steps 2 2 2 3 3

Avg. run time per time step [s] 0.01 0.042 0.041 0.056 0.056

Total run time [s] 1.3 5.4 1.3 0.44 0.11

Table 6.3: Average number of Newton steps, average run time per time step and total runtime of both the M35-EE-LLF and M35-SIE-LLF schemes for different fixed timestep sizes.

6.3 shock structure problems

Shock structure problems have been used as standard test cases to study theaccuracy of moment approximations, see e.g. [53, 90, 111, 112, 124]. In the shockstructure problem a traveling wave connects two equilibrium states

u0 := limx→−∞

u(x), u1 := limx→∞

u(x) with limx→±∞

u′(x) = 0,

by the Rankine-Hugoniot conditions of the Euler equations, where u0 denotesthe state before the shock in the upstream region and correspondingly, u1 thestate after the shock in the downstream region. The relevant scaling parameter isthe Mach number Ma = v0/c0, which is defined as the ratio of the macroscopicvelocity and the speed of sound c0 =

√γθ0 at x → −∞ in the upstream region.

For monatomic gases in a three-dimensional velocity space the adiabatic index isγ = 5/3.

The Rankine-Hugoniot conditions for a hyperbolic system of the form

∂tu + ∂xf = 0 (6.12)

are given by

s[u] = [f(u)], (6.13)

where s is the velocity of the shock wave and [·] denotes the jump across theshock. In a frame co-moving with the shock wave, these conditions simplify to[f(u)] = 0. The ME35 system reduces to the Euler sub-system in the limit Kn→ 0.Thus, in the limit of small Knudsen numbers, the shock structure is a disconti-nuity that connects u0 with u1 by the Rankine-Hugoniot conditions.


For the Euler equations the Rankine-Hugoniot conditions are given by

ρ1v1 − ρ0v0 = 0,

ρ1(v21 + θ1)− ρ0(v2

0 + θ0) = 0,

ρ1v1(e1 + θ1)− ρ0v0(e0 + θ0) = 0.

(6.14)

The state behind the shock can be expressed in terms of the state in front of theshock by

ρ1 =4 Ma2

3 + Ma2 ρ0, v1 =Ma2 +34 Ma2 v0, θ1 =

(3 + Ma2)(5 Ma2−1)16 Ma2 θ0, (6.15)

where

Ma =v0√γθ0

(6.16)

denotes the Mach number in front of the shock. The speed of the shock wave isgiven by

s =√

γθ03 + Ma2

Ma. (6.17)

From Eqs. (6.15) it is clear that the state after the shock satisfies

ρ1 > ρ0, v1 < v0, θ1 > θ0 (6.18)

for Ma > 1.

6.3.1 Numerical Results

In the following we consider shock waves with Ma = 2, 4, 8. To estimate the accu-racy of the ME35 system, the numerical solutions are compared to the referenceBGK equation and the closed-form 14-moment model proposed by McDonald etal. in [90]. The stead-state solution to the shock structure problem is computedwith the time-stepping methods introduced in Chapter 4.

The initial condition is given by u0(x) = 〈φ f 0(x)〉Q with

f 0(x, c) = w(x)EβL(c) + (1− w(x))EβR

(c), (6.19)


where EβLand EβR

are equilibrium distributions determined by the boundaryconditions and w is a smooth weight function given by

w(x) =

1 x ≤ −s,g(1−y(x))

g(y(x))+g(1−y(x)) −s < x < s,

0 x ≥ s,

(6.20)

where

y(x) =x + s

2s, g(x) = exp

(− 1

x

). (6.21)

To obtain a steady-state shock structure solution, the left boundary conditionsare set to

ρL = 1, vL =

√53

Maa

, θL = 1, (6.22)

where a = a2 in Eq. 2.49. The boundary conditions on the right can be deter-mined from Eqs. (6.14).

In the following, the reference mean free path is set to λ0 = a√

θ0 with a = a2,see Eq. (2.49). The numerical solutions to the ME35 system are computed using avelocity domain with bounds cM

x = cMr = 15.67 and a velocity discretization with

Nbcx

= 2, Ngcx = 64, Nb

cr= 1, Ng

cx = 64, yielding a total of Nc = 8192 quadraturenodes. In order to reduce the run time, the optimized GPU implementation ofthe ME35-SIE-LLF scheme was used on a coarse mesh to find an approximatesolution to the shock structure. After convergence, the numerical solution wasused as initial condition for the second-order ME35-HEUN-KF scheme to com-pute an accurate solution on a fine mesh with the grid spacing ∆x ≈ 0.05 andthe time step

∆t = 0.9(

cMx γmax

∆x(2 + Θ)

2+

1Kn

)−1

. (6.23)

The stopping criteria of the Newton solver were set to τ = 10−6, εγ = 0.1 andΘ = 1.5 for the linear reconstruction of the kinetic Riemann solver.

The profiles of the solutions to the ME14 and the reference BGK equation,shown in Figures 6.11 and 6.12, have been extracted from the publication [90].The ME35 system shows a discontinuity in the otherwise smooth shock structuresolution. This so-called sub-shock is especially significant for the reduced heatflux q?x = qx/(ρθ3/2). For the Ma = 4 and Ma = 8 shock structures the occur-rence of this sub-shock is expected, since it was shown by Ruggeri et al. in [14]


Mach 2 Shock-Structure

-15 -10 -5 0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

x

ρ

-15 -10 -5 0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

x

θ

-15 -10 -5 0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

1.2

x

θxx

-15 -10 -5 0 5 10 15-0.5

-0.4

-0.3

-0.2

-0.1

0.0

x

qx

Kinetic BGK ME35 ME14


-20 -10 0 10 200.0

0.2

0.4

0.6

0.8

1.0

x

ρ

-20 -10 0 10 200.0

0.2

0.4

0.6

0.8

1.0

x

θ

-20 -10 0 10 200.0

0.2

0.4

0.6

0.8

1.0

1.2

x

θxx

-20 -10 0 10 20

-2.0

-1.5

-1.0

-0.5

0.0

x

qx


Figure 6.11: Profiles of normalized mass density, ρ, thermodynamic temperature, θ, direc-tional temperature, θxx , and reduced heat flux, qx , through a Mach 2 and aMach 4 shock structure.



-60 -40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

x

ρ

-60 -40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

x

θ

-60 -40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x

θxx

-60 -40 -20 0 20 40-6

-5

-4

-3

-2

-1

0

x

qx


Figure 6.12: Profiles of normalized mass density, ρ, thermodynamic temperature, θ, direc-tional temperature, θxx , and reduced heat flux, qx , through a Mach 8 shockstructure.

that solutions to hyperbolic moment equations endowed with an entropy law donot admit smooth solutions for shock structure problems if the shock speed sexceeds the largest characteristic speed λ

(max)0 of the moment system evaluated

in front of the shock. Equivalently, no smooth shock structure solutions existif the Mach number of the shock wave is greater than a critical Mach numberMa∗, which for the ME35 evaluates to Ma∗ ≈ 2.2, see e.g. [7]. Interestingly, thesub-shock decreases in size for larger Mach numbers, allowing a surprisinglyaccurate description of the Ma = 8 shock structure.

The existence of smooth solutions is unclear in the converse case of shockstructures with Ma < Ma∗ [96]. The numerical solution to the Ma = 2 < Ma∗

shock structure of the ME35 system presented here shows the formation of a sub-shock. In the context of Extended Thermodynamics, the formation of sub-shockswas investigated by Au et al. in [7] for higher-order moment theories, includingthe 35-moment theory. In [7], the authors observed the formation of a sub-shockof the 35-moment theory already at Ma = 1.8.


In contrast to the ME35 system, the closed-form 14-moment system shows novisible sub-shock. While the ME35 system considered here has a bounded veloc-ity domain, the velocity domain for the 14-moment system is unbounded and thesystem allows for arbitrarily high characteristic speeds in the vicinity of the equi-librium state [90], effectively rendering the moment system non-hyperbolic. Thisfeature of the 14-moment system allows to suppress the sub-shock. Analogously,it can be expected that the sub-shock for the ME35 decreases as the underlyingvelocity domains is expanded, since the wave-speeds become unbounded in thelimit of |Ωc| → ∞ in the neighborhood of the equilibrium state, see also Chapter7. The dependence of the solution on the velocity domain size is studied in thenext section for the Mach 4 shock structure problem.

6.3.2 Efficient Solvers applied to the Shock Structure Problem

In this section, the highly optimized GPU implementations of the ME35-EE-LLFand ME35-SIE-LLF schemes are used to solve the Mach 4 shock structure prob-lem. The software is run on one node with 16 Intel Xeon E5-2670 cores and oneNVIDIA K20C graphics card, see Chapter 5.

For the ME35-EE-LLF, the time step is chosen according to the CFL conditionin Eq. (4.106) with a safety factor of cs = 0.9. If the Newton iteration fails toconverge, the time step is discarded and re-computed with a new time step givenby the old time step multiplied by the factor β f = 0.1. For the ME35-SIE-LLFmethod, the time step is determined according to the adaptive method describedin Section 4.6 with the parameters

βi = 1.2, βd = 0.8, β f = 0.5,

minOptNewtIter = 3, maxOptNewtIter = 8.

The spatial domain Ωx = [−47, 47] is first discretized with Nx = 512 cells andthe steady-state is computed for both the explicit and the semi-implicit methodswith the initial condition (6.19) and smoothing length s = |Ωx|/4 in Eqs. (6.20)-(6.21). The converged solutions are then used as an initial condition to computethe numerical solution on a refined mesh with Nx = 2048 cells. Finally, the steadystate solutions on Nx = 2048 cells are used as an initial condition for a numericalsimulation on a very fine mesh with Nx = 8192 cells.

Figure 6.13 shows several measurement results for the ME35-EE-LLF and ME35-SIE-LLF methods applied to the Ma = 4 shock structure problem using the gridrefinements described above. The total wall-clock time and the total number oftime steps reported in Figure 6.13 include also the measurements of the coarsergrids.


◼ ◼◼

512 2048 81921

2

4

8

15

30

60

Nx

Totalwallclocktime[m

]

◼ ◼ ◼

512 2048 8192

103

104

105

Nx

Totalno.oftimesteps

◼◼

◼

512 2048 8192

1

2

3

4

5

Nx

Avg.Newtonsteps

◼

◼

◼

512 2048 8192

10-3

10-2

10-1

1

Nx

Avg.timestepsize

ME35-EE-LLF ◼ ME35-SIE-LLF

Figure 6.13: Total elapsed wall-clock time in minutes, total number of time steps, averagenumber of Newton steps per time step and average time step size for theMa = 4 shock structure calculation.

Note that the semi-implicit method converges on the finest grid in a total runtime that is about 15x smaller than the explicit method.

The convergence of the heat flux profile in the number of spatial cells Nx isdisplayed in Figure 6.14 for the ME35-SIE-LLF method, together with a refer-ence numerical solution to the ME35 system computed with the second-orderME35-HEUN-KF method. Since the numerical method has only a first-order con-vergence rate, the convergence in the number of spatial cells is slow, such thateven for the highly refined mesh with Nx = 8192 cells the heat flux field has notfully converged.

In Figure 6.15, the scaled mass density, scaled thermodynamic temperature,scaled directional temperature and physical heat flux profiles are shown forthe ME35-SIE-LLF scheme with Nx = 8192 cells, together with the referencemaximum-entropy solution obtained with the ME35-HEUN-KF method and areference DVM solution, which has been provided by Zhenning Cai. The solu-tion of the ME35 system is in good agreement with the DVM solution, especiallyconsidering that the model has only N = 9 degrees of freedoms in the two-dimensional velocity space. For the shown profiles, the largest deviations are


-20 -10 0 10 20-15

-10

-5

0

x

qx

ME35-SIE-LLF (Nx=512) ME35-SIE-LLF (Nx=2048)

ME35-SIE-LLF (Nx=8192) ME35-HEUN-KF (Nx=1600)

Figure 6.14: Numerical solutions to the heat flux for the semi-implicit method with the localLax-Friedrichs flux for several spatial resolutions, together with the convergedreference solution using the ME35-HEUN-KF scheme.

-20 -10 0 10 200.0

0.2

0.4

0.6

0.8

1.0

x

ρ

-20 -10 0 10 200.0

0.2

0.4

0.6

0.8

1.0

x

θ

-20 -10 0 10 200.0

0.2

0.4

0.6

0.8

1.0

1.2

x

θxx

-20 -10 0 10 20-15

-10

-5

0

x

qx

DVM ME35-HEUN-KF ME35-SIE-LLF

Figure 6.15: Scaled mass density, ρ, thermodynamic temperature, θ, directional tempera-ture, θxx , and physical heat flux, qx , through a Ma = 4 shock structure forthe reference DVM and the moment approximations with the schemes ME35-HEUN-KF and ME35-SIE-LLF.

6.4 two-beam collision problem 101

-20 -10 0 10 20-15

-10

-5

0

x

qx

|Ωc|→∞

-20 -15 -10 -5

-10

-1

-0.1

-0.01

x

qx

cxM→∞

DVM ME35-SIE-LLF (cxM=11.75)

ME35-SIE-LLF (cxM=15.67) ME35-SIE-LLF (cx

M=23.5)

Figure 6.16: Heat flux profiles of the ME35 system with the ME35-SIE-LLF method for dif-ferent underlying velocity domain sizes, together with the reference DVM so-lution. The plot on the right shows a close-up of the sub-shock in the upstreamregion of the shock structure profile on a logarithmic scale.

visible in the upstream region of the heat flux field, due to the formation of asub-shock.

Figure 6.16 shows the numerical solution to the heat flux profile for the ME35-SIE-LLF method with Nx = 8192 cells for different velocity domain bounds cM

x .While the numerical solutions are identical in the down-stream region, increasingthe velocity domain shifts the location of the sub-shock further upstream anddecreases its strength. This effect has been observed also in [111, 112] for boththe shock structure problem and the two-beam Riemann problem. See also [90,112] for a discussion of the sub-shock phenomenon for entropy-based closures.

6.4 two-beam collision problem

Let us consider the Riemann initial value problem

f 0(x, t, c) =

Eρ0,v0,θ0 (c), x ≥ 0,

Eρ0,−v0,θ0 (c), x < 0,(6.24)

which models the collision of two thermalized beams. For this problem we usethe scaling presented in Section 2.10 with a1 = 1 in Eq. (2.49), so that the meanfree path is given by λ0 =

√θ0τ0.


Test case Kn ρ0 θ0 v0 Ωx T

1 10−3 1 1 1 [−1.5, 1.5] 0.6

2 10−2 1 1 1 [−1.5, 1.5] 0.6

3 10−1 1 1 4 [−3, 3] 0.5

Table 6.4: Parameters used for the two-beam problems, where T denotes the end time ofthe numerical simulations.

In the limit Kn → 0, the kinetic BGK equation formally reduces to the Eulersystem. The solution to the Euler equations reads

u(x, t) =

uL, x ≤ −st,

uM, −st < x < st,

uR, st ≤ x

(6.25)

and consists of three constant states that are separated by two shock waves trav-eling at speed s, where

s =

√4(v0)2 + 15θ0 − v0

3. (6.26)

The left and right states read

ρL = ρR = ρ0, vL = −vR = v0, θL = θR = θ0 (6.27)

and the intermediate state uM satisfies

ρM =4 Ma2

3 + Ma2 ρ0, vM = 0, θM =(Ma2 +3)(5 Ma2−1)

16 Ma2 θ0. (6.28)

The Mach number of the shock waves is determined by the initial condition andgiven explicitly by

Ma =v0 + s√

γθ0=

2v0 +√

4(v0)2 + 15θ0√

15θ0. (6.29)

In the following we consider three different two-beam problems, for which theparameters are listed in Table 6.4. For all test cases the numerical solution tothe ME35 system have been computed with the ME35-HEUN-KF scheme. Thereference BGK results for the two-beam problem have been computed with asecond-order DVM by Zhenning Cai.


Two-Beam Problem (Kn = 10−3, v0 = 1)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.51.0

1.2

1.4

1.6

1.8

x

ρ

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.0

-0.5

0.0

0.5

1.0

x

vx

Reference BGK 35-Moment System Euler

Figure 6.17: Profiles of mass density and velocity for the numerical solutions to the refer-ence BGK equation and the ME35 moment approximation, together with theexact solution to the Euler system.

Figure 6.17 shows the numerical solution to the ME35 system, the referencekinetic BGK solution and the exact solution to the Euler system for the test case1 in Table 6.4. For the ME35 system, the spatial domain is discretized with Nx =2000 cells and the velocity domain bounds are set to cM

x = cMr = 10. The velocity

grid is highly resolved with the parameters Nbcx

= Nbcr

= 2 and Ngcx = Ng

cr =64. The second-order DVM used to compute the reference BGK solution has auniform spatial discretization with Nx = 1000 cells and a velocity domain withbounds cM

x = cMr = 5 and Ncx = 500, Ncr = 100 grid points in the velocity

domain. Clearly, both the reference BGK solution and the ME35 solution are invery good agreement with the Euler solution.

In Figure 6.18 the numerical solutions are shown for the test case 2 in Table6.4. The spatial and velocity domain discretizations for the moment method andthe DVM are identical to test case 1. Clearly, the numerical solution to the ME35system is in good agreement with the reference BGK solution. Both the overshootin the directional temperature θxx and the heat flux profile are captured correctly.However, in the solution to the ME35 moment system a small hump is visible infront of the shock wave that is not predicted by the BGK equation.

Finally, Figure 6.19 shows the numerical solutions for the third test case. Forthis problem, both the DVM and the ME35 system use a velocity domain withbounds cM

x = cMr = 10. The velocity grid of the moment system is discretized

with Nbcx= 2, Nb

cr= 1, Ng

cx = Ngcr = 16.

Even though the ME35 system shows some deviations from the reference so-lution, it can capture the strong non-equilibrium effects, such as the overshoot inthe directional temperature θxx or the heat flux correctly. Clearly, the profiles of


the ME35 system show the occurrence of two sub-shocks in front of the wavestraveling to the left and right.

In Figure 6.20, the profiles of the directional temperature θxx and heat flux areshown for the third test case of the two-beam problem for different underlyingvelocity domain sizes given by cM

x = cMr = 10, cM

x = cMr = 15 and cM

x = cMr = 20.

The velocity grid is discretized with Nbcx

= 2, Nbcr

= 1 blocks and Ngcx = cM

x ,Ng

cr = cMr Gauss-Legendre quadrature points per block. The spatial domain Ωx =

[−3, 3] is discretized with Nx = 1000 grid cells and the time-step is set accordingEq. (6.23).

As the velocity domain size is enlarged, the strength of the shock waves de-creases, leading to a better agreement with the reference BGK solution. The nu-merical examples indicate that the shock strength tends to zero in the limit of anunbounded velocity domain.

The convergence in the velocity grid resolution is shown in Figure 6.21 for testcase 3. The velocity domain size is fixed to cM

x = cMr = 10 and the quadrature

grid uses Nbcx

= 2, Nbcr

= 1 blocks and different numbers of Gauss-Legendrequadrature points per block. Clearly, both θxx and qx are converged in the num-ber of Gauss-Legendre quadrature points for Ng

cx = Ngcr = 16, which corresponds

to a total of Nc = 512 quadrature points.



-1.5 -1.0 -0.5 0.0 0.5 1.0 1.51.0

1.2

1.4

1.6

1.8

x

ρ

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.0

-0.5

0.0

0.5

1.0

x

vx

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.51.0

1.1

1.2

1.3

1.4

1.5

1.6

x

θ

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.51.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

x

θxx

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.51.0

1.1

1.2

1.3

1.4

1.5

1.6

x

θrr

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-0.4

-0.2

0.0

0.2

0.4

x

qx

DVM ME35

Figure 6.18: Profiles of mass density, ρ, velocity, vx , thermodynamic temperature, θ, di-rectional temperatures, θxx , θrr , and heat flux, qx , of the numerical DVM andME35 solutions to the two-beam problem with Kn = 10−2 and v0 = 1 at timeT = 0.6.



-3 -2 -1 0 1 2 31.0

1.5

2.0

2.5

3.0

x

ρ

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

x

vx

-3 -2 -1 0 1 2 31

2

3

4

5

6

7

x

θ

-3 -2 -1 0 1 2 3

2

4

6

8

x

θxx

-3 -2 -1 0 1 2 31

2

3

4

5

6

x

θrr

-3 -2 -1 0 1 2 3

-20

-10

0

10

20

x

qx

DVM ME35

Figure 6.19: Profiles of mass density, ρ, velocity, vx , thermodynamic temperature, θ, direc-tional temperatures, θxx , θrr , and heat flux, qx , of the numerical DVM and ME35solutions to the two-beam problem with Kn = 10−1 and v0 = 4 at T = 0.5.


Two-Beam Problem: Convergence in |Ωc| → ∞

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2

4

6

8

x

θxx

cxM→ ∞

0.0 0.5 1.0 1.5 2.0 2.5 3.00

5

10

15

20

x

qx

cxM→∞

cxM=cr

M=10 cxM=cr

M=15 cxM=cr

M=20

Figure 6.20: Profiles of the directional temperature and heat flux of the ME35 system fordifferent velocity domain sizes.

Two-Beam Problem: Convergence in Nc → ∞

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2

4

6

8

x

θxx

0.0 0.5 1.0 1.5 2.0 2.5 3.00

5

10

15

20

x

qx

Ncxg =Ncr

g=8 Ncxg =Ncr

g=16 Ncxg =Ncr

g=32

Figure 6.21: Profiles of the directional temperature and heat flux of the ME35 system fordifferent velocity grid resolutions.

7R E G U L A R I Z E D S I N G U L A R C L O S U R E S

The ME35 system introduced in Chapter 3 is able to capture non-equilibriumeffects in moderately rarefied gas flows, such as the shock structure problem forhigh Mach number shock waves as shown in Chapter 6. However, the numericalsolutions to the ME35 system show an unphysical sub-shock, i.e. a discontinu-ity in the smooth shock structure profile, which deteriorates the approximationquality of the 35-moment system.

The formation of sub-shocks is not unique to the ME35 system, but a generaldeficiency of hyperbolic moment approximations. Ruggeri analyzed in [104] theproblem of the sub-shock in the context of modern extended thermodynamics[96], which lead to a series of papers on the subject of sub-shocks [6, 7, 127]. In[14] Boillat and Ruggeri provided an answer to the question of when continuoussolutions to generic balance laws endowed with a convex entropy can occur byproving the non-existence of C1 solutions to the shock wave structure problemif the shock velocity exceeds the largest characteristic velocity evaluated in theequilibrium state in front of the shock.

Several approaches to the sub-shock problem have been considered in the lit-erature. As shown by Boillat and Ruggeri [13] in the framework of extendedthermodynamics, the largest characteristic speed evaluated in the equilibriumstate is bounded from below by a monotonically increasing function of the high-est tensorial degree of the moment system. Thus one possible approach to findsmooth solutions is to consider systems with a higher number of moment equa-tions, see e.g. [6]. However, this approach turns out to be rather impractical asthe required number of moment equations would have to be increased drasti-cally even for moderate Mach numbers, see [7]. A completely different approachhas been considered in [116], where a regularization of Grad’s equations by theinclusion of parabolic terms was proposed. The resulting R13 equations allowsmooth shock structure solutions as shown in [124].

McDonald and Torrilhon considered in [90] a closed-form closure with a singu-larity in the closing flux similar to the maximum-entropy closure. The numericalsolutions to several continuous shock structure problems presented in [90] show

109

110 regularized singular closures

smooth solutions to hypersonic shock wave problems, which are furthermore invery good agreement with the reference BGK solution.

To gain insight into the singular closure proposed in [90], an explicit closurethat removes the singularity with a regularization parameter, β > 0, is investi-gated in this chapter.

As shown in subsequent sections, this strongly nonlinear closure allows tomitigate the problem of the emerging sub-shock in the shock structure problemby decreasing its strength as a function of the regularization parameter β. Themoment systems considered in this chapter are based on a simplified kineticdescription of the gas with a one-dimensional velocity space. Consequently, theclosure models discussed are not intended to rival fully three-dimensional fluiddynamic equations, but rather highlight interesting properties of singular clo-sures.

The rest of this chapter is organized as follows: Section 7.1 presents the 5-moment approximation for general moment equations, together with the well-known closure theories of Grad and maximum-entropy. An explicit closure, basedon the 5-moment closure proposed in [90], is introduced in Section 7.2 with a dis-cussion of how this closure is related to the Grad and maximum-entropy systems.Properties of Riemann solutions to the 5-moment system with the explicit closureare discussed in Section 7.3. Section 7.4 contains a description of the numericalmethods used in Sections 7.5 and 7.6, where numerical solutions to the continu-ous shock structure problem and two Riemann problems are presented. The finalsection summarizes the main findings of this chapter.

7.1 moment equations

Let us consider a simplified one-dimensional setting, for which both the spatialand velocity domain are one-dimensional. In this simplified geometry, the 35-moment system reduces to a 5-moment system generated by the polynomialbasis

φ = (1, c, c2, c3, c4)T . (7.1)

Note that the 5-moment system is the smallest admissible moment system be-yond the Euler equations in this simplified geometry, which incorporates anequation for the heat flux.

Let f = f (x, t, c) denote the one-particle distribution function and M theMaxwellian distribution over a one-dimensional velocity space. The 5-momentapproximation to the kinetic BGK equation is given by

∂t〈ci f 〉+ ∂x〈ci+1 f 〉 = −〈ci( f −M( f ))〉

τ, for i = 0, 1, . . . , 4, (7.2)

7.1 moment equations 111

where τ = τ0 is a constant relaxation time. Let t0,√

θ0, ρ0 denote a characteristictime, velocity and mass density, then we can readily define the nondimensionalquantities

t =tt0

, θ =θ

θ0, ρ =

ρ

ρ0,

x =xl0

, v =v√θ0

, c =c√θ0

, f =

√θ0

n0f ,

(7.3)

where l0 =√

θ0t0 is a characteristic length scale and n0 = ρ0/m a referencenumber density. The moment system (7.2) is given in nondimensional form by

∂t〈ci f 〉+ ∂x〈ci+1 f 〉 = −〈ci( f − fM)〉

Knfor i = 0, 1, . . . , 4, (7.4)

where, for better readability, all bars have been removed. If not stated otherwise,all quantities are assumed to be dimensionless in the rest of this chapter and allbars are omitted.

7.1.1 Reduced Moments

For ρ > 0 and θ > 0 a reduced distribution function f can be defined by

f (x, t, c) =√

θ

ρf (x, t,

√θc + v)

⇔ f (x, t, c) =ρ√θ

f (x, t, ξ) with ξ =c− v√

θ,

(7.5)

such that 〈 f 〉 = 〈ξ2 f 〉 = 1 and 〈ξ f 〉 = 0. Let ui := 〈ξ i f 〉, then we can define thesymbols

Q := u3, R := u4, S := u5, (7.6)

for the higher-order moments of f . In the following it is always assumed thatρ, θ > 0, so that the mapping (7.5) is well defined.

7.1.2 The 5-moment System

Let the state vector of the 5-moment system be given by

u = (ρ, v, θ, Q, R)T . (7.7)


Introducing the vector fields E = (Ei), F = (Fi), G = (Gi), defined by

Ei(u) := 〈ci f 〉, Fi(u) := 〈ci+1 f 〉, Gi(u) := −〈ci( f − fM)〉

Kn, (7.8)

allows the moment system (7.4) to be written in compact form as

∂tE(u) + ∂xF(u) = G(u). (7.9)

The convective moment vector can be explicitly written in terms of the primitiveand reduced moments as

E(u) =

ρ

ρv

ρv2 + ρθ

ρv3 + 3ρvθ + Qρθ32

ρv4 + 6ρv2θ + 4vQρθ32 + Rρθ2.

(7.10)

Similarly, the production terms are given by

G(u) = − 1Kn

0

0

0

Qρθ32

Rρθ2 + 4vQρθ32 − 3ρθ2

. (7.11)

The flux vector reads Fi(u) = Ei+1(u) for i = 0, . . . , 3 and

F4(u) = ρv5 + 10ρv3θ + 10v2Qρθ32 + 5vRρθ2 + Sρθ

52 . (7.12)

In the collisional limit of small Knudsen numbers (Kn → 0) the moment system(7.9) reduces to the one-dimensional Euler equations,

∂tEi(u) + ∂xFi(u) = 0 for i = 0, 1, 2 (7.13)

with (Q, R) = (0, 3). Thus the moment system (7.9) includes the one-dimensionalEuler equations as its equilibrium sub-system. For a more detailed discussion ofequilibrium sub-systems, see e.g. [14].


The pressure, physical heat flux and total energy density can be easily obtainedfrom the field variables by

p = ρθ, q =ρθ3/2Q

2, ρe =

ρv2 + p2

. (7.14)

The reduced heat flux variable, Q, is a measure for the skewness of the distribu-tion function. Similarly, the field variables θ and R are related to the variance σ2

and excess kurtosis ∆ of f by

σ2 =〈 f C2〉〈 f 〉 = θ, ∆ =

〈 f 〉〈 f C4〉〈 f C2〉2 − 3 = R− 3.

7.1.3 Closure Problem

To close the System (7.9) the general distribution f is replaced by a model distri-bution function f (Model) that is uniquely determined by the constraints

ui = 〈ξ i f (Model)〉 for i = 0, 1, . . . , 4, (7.15)

where ui = 〈ξ i f 〉 is a known vector of moments. The closing flux moment is thengiven by S = S(Q, R) = 〈ξ5 f (Model)〉. Symmetry considerations show that S is anodd function of the normalized heat flux Q.

7.1.4 Hyperbolicity

The moment system (7.9) is hyperbolic if and only if the flux Jacobian

J(E) :=∂F∂E

(E) (7.16)

is diagonalizable with real eigenvalues λ1, . . . , λ5. The explicit structure of theflux Jacobian is given in Appendix B.7 for general 5-moment closures.

7.1.5 Grad’s Closure Theory

The classical closure ansatz by Grad [50] yields

f (Grad)(x, t, ξ) = M(ξ)N−1

∑n=0

αn(x, t)Hen(ξ), (7.17)


where Hen denotes the nth Hermite polynomial satisfying the orthogonality con-dition

〈Hem, Hen〉w :=∫

RHem(ξ)Hen(ξ)w(ξ) dξ = δmnn! (7.18)

for all m ∈N0, where the weight function is the standard normal distribution

w(ξ) = M(ξ) :=1√2π

exp(− ξ2

2

). (7.19)

The coefficients αi of the Grad distribution function f (Grad) are uniquely deter-mined by the moment constraints

〈ξ i f (Grad)〉 = ui for i = 0, 1, . . . , N − 1. (7.20)

In case of the 5-moment system, the unique solution is given by

f (Grad5)(ξ) = M(ξ)4

∑n=0

αn(x, t)Hen(ξ),

= M(ξ)

(1 +

Q6

He3(ξ) +R− 3

24He4(ξ)

),

(7.21)

from which the closing flux is determined to be

S = 〈ξ5 f (Grad5)〉 = 10Q. (7.22)

It is evident from (7.21) that Grad’s distribution fails to be non-negative even inthe neighborhood of the equilibrium state. However, the major drawback of thisclosure is its local domain of hyperbolicity depicted in Figure 7.1, which restrictsthe range of applications to weakly non-equilibrium flows.

7.1.6 The Maximum-entropy Hierarchy

Closures based on entropy maximization [75, 96] yield hyperbolic systems of par-tial differential equations compatible with an entropy law, as discussed in Chap-ter 3. The maximum-entropy density f (ME) is determined by the constrainedoptimization problem

f (ME) = arg minf

h( f ) s.t. 〈ξ i f (ME)〉 = ui for i = 0, 1, . . . , N − 1,

(7.23)


Figure 7.1: The local hyperbolicity domain H(Grad5) of Grad’s 5-moment closure projectedon the (Q, R) phase-plane. The projected equilibrium states correspond to thepoint E = (0, 3).

where h denotes an entropy functional. The classical Boltzmann entropy func-tional yields the ansatz distribution

f (ME)(ξ) = exp

(N−1

∑i=0

αiξi

), (7.24)

where the Lagrange parameters αi are determined by the dual minimizationproblem 3.33.

A necessary condition for the minimization problem to have a solution is therealizability of the vector u. In the next section the realizability problem is dis-cussed both for unbounded and bounded domains on the real line.

7.1.7 Realizability Conditions for the 5-moment System

The question of existence and uniqueness of a positive Borel measure µ satisfying∫

Ωc

cidµ(c) = ui, i = 0, . . . , N (7.25)

for a given set of moments u = (u0, . . . , uN)T is known as the classical truncatedmoment problem, see e.g. [1, 2]. If µ is absolutely continuous with respect to theLebesgue measure we can write

∫

Ωc

ci f dc = ui, i = 0, . . . , N, (7.26)

where f = f (c) is a Lebesgue integrable distribution function.


In the following sections explicit characterizations of the existence and unique-ness of solutions to the 5-moment problem are discussed.

If ρ, θ > 0, then it suffices to consider the realizability in the reduced variables,for which the truncated moment problem (7.26) is given by

∫

Ωξ

ξ i f dξ = ui, i = 0, . . . , N, (7.27)

such that f only depends on the higher moments Q, R, . . ..

7.1.8 Realizability Conditions in 1D

Let us consider a one-dimensional velocity space on the real line and let

A(k) := (ui+j)0≤i,j≤N ∈ R(k+1)×(k+1) (7.28)

denote a Hankel matrix of the reduced moment vector u. The following existenceresult was shown by Curto and Fialkow in [29]:

Theorem 1 (Curto, Fialkow). A necessary and sufficient condition for (7.25) to have asolution is

A(k) is positive semidefinite and rank A(k) = rank u, (7.29)

where rank u is defined by k + 1 if A(k) is nonsingular; else if the Hankel matrix A(k)is singular then it is the smallest integer 1 ≤ i ≤ k, such that ai ∈ spana0, . . . , ai−1,where ak denotes the kth column of A(k).

Applying Theorem (1) to the 5-moment system yields the following character-ization of the realizability domain:

Corollary 1. If k = 2, then Eq. (7.25) has a solution for ρ, θ > 0 if and only ifR ≥ 1 + Q2.

Proof. Since ρ, θ > 0, the mapping (7.5) is well defined. By applying Sylvester’scriterion for positive semidefinite matrices on the Hankel matrix A in the reducedmoments u we find

A(2) =

1 0 1

0 1 Q

1 Q R

0 ⇔ R ≥ 1 + Q2, (7.30)

where A(2) 0 denotes that A(2) is positive semidefinite. If R > 1 + Q2, thenA(2) is non-singular and rank u = 3 = rank A(2). If on the other hand R =1 + Q2, then A(2) is singular and rank u = 2 = rank A(2).


In [29] Curto and Fialkow have shown the following uniqueness result:

Theorem 2 (Curto, Fialkow). If A(k) 0 and rank A(k) = rank u, then there existsa representing measure according to Thm. (1). If furthermore rank u ≤ k, then thereexists a unique representing measure of u.

For the special case k = 2 we have rank u = k + 1 for R > 1+ Q2. On the otherhand, for R = 1 + Q2 we have rank u = 2 ≤ k, such that there exists a uniquemeasure of the form

µ(ξ) = w1δ(ξ − s1) + w2δ(ξ − s2), (7.31)

where δ are Dirac delta distributions. The coefficients can be determined to begiven by

s1(Q) =12

(Q−

√Q2 + 4

), s2(Q) =

12

(Q +

√Q2 + 4

),

w1(Q) =12

(1 +

Q√Q2 + 4

), w2(Q) =

12

(1− Q√

Q2 + 4

).

(7.32)

Note that the unique distribution (7.31)-(7.31) immediately yields the closing flux

S∣∣∂R = Q(2 + Q2) (7.33)

on the realizability boundary.

Junk showed in [64] that the maximum-entropy closure does not allow for a so-lution on the half-line J = (0, R); R > 3, such that the domain of definitionD = R\J of the maximum-entropy closure is strictly smaller than the realizabil-ity domain R. The half-line J will be referred to as Junk half-line in the follow-ing. Furthermore, as shown by Junk [64], the closing flux S(ME5) = 〈ξ5 f (ME5)〉 issingular on J and satisfies the following lower bound:

Proposition 1 (Junk, 1998). For all (Q, R) ∈ D the maximum-entropy closure S(ME5)

satisfies

S(ME5)Q ≥ (R− 3)(R− 1) + 4Q2, (7.34)

such that for all tuples (Q, R) ∈ D with R > 3 the lower bound

|S(ME5)| ≥ S(ME5)lb =

∣∣∣∣(R− 3)(R− 1)

Q+ 4Q

∣∣∣∣ (7.35)

holds.


Junk’s lower bound implies that the closing flux S(ME5) is singular on the half-

line J . Interestingly, S(ME5) is singular also in the equilibrium state, as e.g. S(ME5)lb

diverges on the curve (Q, R) = (Q, 3 + Q1/2) as Q→ 0.It was shown in [35] that a formal linearization of S(ME5) around the equi-

librium state yields Grad’s closure S(Grad5)(Q, R) = 10Q. Thus Grad’s closurecan be seen as a linearized maximum-entropy closure. Note, however, that thislinearization is purely formal and neglects the presence of the singularity of theclosing moment S(ME5) at the equilibrium state.

The singularity on the Junk half-line is not a unique feature of the maximum-entropy closure. In Section A.1 it is shown that also the bi-Gaussian ansatz issingular on J , similarly to the closure by McDonald presented in [90]. Since theJunk half-line consists of all distributions with an excess kurtosis ∆ > 0, it isnatural to consider velocity distributions that decay slower than exponentially atthe tails. In [120] moment systems based on the Pearson-IV distribution, whichallows for nonexponential tails, have been considered. As another example of adistribution with nonexponential decay, Section A.3 shows that the κ-Distributionis realizable on J with algebraically decaying tails for a suitably chosen κ pa-rameter.

-3 -2 -1 0 1 2 30

2

4

6

8

10

Q

R

ℰ

∂ℛ

Figure 7.2: Realizability boundary ∂R = 1 + Q2 of the 5-moment system with an un-bounded velocity domain, together with the equilibrium state E at (0, 3) andthe Junk half-line J .

7.1.9 Realizability Conditions on Bounded Velocity Domains

The truncated moment problem on a bounded velocity domain Ω = [a, b] withbounds a ≤ b is called the truncated Hausdorff moment problem. The follow-ing theorem by Krein and Nudel’man in [69] provides necessary and sufficientconditions for the existence of a measure µ with suppµ ⊆ [a, b]:


Theorem 3 (Krein, Nudel’man). There exists a measure µ with suppµ ⊆ [a, b] satis-fying (7.25) if and only if

A(k) 0 and (a + b)B(k− 1)− abA(k− 1)− C(k− 1) 0, (7.36)

where

B(k) := (ui+j+1)0≤i,j≤k, C(k− 1) := (ui+j)1≤i,j≤k. (7.37)

Applying Theorem 3 to the 5-moment system with a bounded velocity domaingiven by Ωξ = [−ξM, ξM] in the reduced velocity variable ξ yields the followingcharacterization of the realizability domain:

Corollary 2. For ρ, θ > 0, there exists a measure µ supported on the interval [−ξM, ξM]for ξM ≥ 0 satisfying (7.25) if and only if the conditions

ξM > 1, 1 + Q2 ≤ R ≤ ξ2M −

Q2

ξ2M − 1

(7.38)

or

ξM = 1, Q = 0, R = 1 (7.39)

are fulfilled.

Proof. The condition R ≥ 1 + Q2 results from the requirement A(2) 0 as in theHamburger problem. The second condition simplifies to

ξ2MA(1)− C(1) 0 ⇔

(ξ2

M − 1 −Q

−Q ξ2M − R

) 0. (7.40)

Sylvester’s criterion yields the conditions

ξM ≥ 1, R ≤ ξ2M, ξ4

M − ξ2M(1 + R) + R− Q2 ≥ 0, (7.41)

which, together with the condition R ≥ 1 + Q2, are equivalent to the conditionsin the corollary.

Thus the realizability boundary of the 5-moment system is given by

∂R = ∂RL ∪ ∂RU , (7.42)


where

∂RL =(

Q, R = 1 + Q2)

; −(ξ2M − 1) ≤ ξMQ ≤ ξ2

M − 1

,

∂RU =

(Q, R = ξ2

M −Q2

ξ2M − 1

); −(ξ2

M − 1) ≤ ξMQ ≤ ξ2M − 1

.

(7.43)

On the lower boundary ∂RL the unique representing measure is given by (7.31).However, on the upper boundary ∂RU the matrix A is nonsingular and theuniqueness theorem given in [29] cannot be applied. Nevertheless, a representingmeasure on ∂RU can be constructed easily with the ansatz

µ = w1δ(ξ + ξM) + w2δ

(c− Q

ξ2M − 1

)+ w3δ(ξ − ξM). (7.44)

The weights are given by

w1 =1 + ξM(Q− ξM)

2ξM(ξM − ξ3M + Q)

,

w2 =(ξ2

M − 1)3

ξ6M − 2ξ4

M + ξ2M − Q2

,

w3 = 1− w1 − w2.

For Q → ±√

R− 1 the measure converges to the unique representing measureon ∂RL given in Eq. (7.31), such that both representations coincide on the inter-section ∂RL ∩ ∂RU .

In Figure 7.3 the realizability domain for the 5-moment system is shown for abounded velocity domain [−ξM, ξM] for various values of ξM.

Figure 7.4 shows the maximum-entropy distribution of the 5-moment systemon the bounded velocity domain ξ ∈ [−10, 10] for Q = 0 and different valuesof R. In the left plot, the maximum-entropy distributions are shown for severalvalues of R, starting from R = 3, which corresponds to the Maxwellian distribu-tion if the velocity domain unbounded, to the lower boundary R = 1. Clearly,the distribution converges to the sum of Dirac distributions in agreement withthe representation (7.31). In the right plot, the maximum-entropy distributionsare shown for increasing values of R approaching the upper boundary R = ξ2

M,for which the distribution function converges to the representation (7.44).


-10 -5 0 5 100

20

40

60

80

100

Q

R

∂ℛL

∂ℛU

ξM=2

ξM=4

ξM=6

ξM=8

ξM=10

Figure 7.3: Realizability boundaries of the 5-moment system for different velocity bounds.

-6 -4 -2 0 2 4 610-8

10-6

10-4

0.01

1

ξ

f

R→ 1

-10 -5 0 5 1010-8

10-6

10-4

0.01

1

ξ

f

R→ 100

Figure 7.4: Reduced maximum-entropy distributions for Q = 0, R = 3, 2, 1.1, 1.05, 1.01 (left)and Q = 0, R = 3, 10, 50, 90, 99.75 (right).

7.1.10 Regularized Maximum-entropy Closures

Several methods have been considered in the literature to regularize the maximum-entropy closure, see e.g. [5, 6, 89]. In order to remove the singularity on Jwe consider the bounded velocity domain Ωc = [v0 − cM, v0 + cM] with fixedvalues v0 and cM > 0. The velocity domain Ωc is discretized into intervalsIk = [ck−1/2, ck+1/2] for k = 1, . . . , M with constant length ∆c, over which thedistribution function is assumed to be constant.

Figure 7.5 shows the normalized heat flux Q of the 5-moment equations usingthe discrete maximum-entropy closure described above, together with the refer-ence BGK solution for a Mach 4 shock structure problem, where the upstreamstate is given by ρL = 1, vL = 4

√3, θL = 1. The velocity v0 was set to the

downstream equilibrium velocity vR ≈ 3.68. The figure shows several numerical


Figure 7.5: Numerical solutions to the 5-moment equations for a Ma = 4 shock structureusing the discrete maximum-entropy closure with cM = 15, 20, 25, 35 (red lines),together with the reference BGK solution (black, dotted line).

solutions to the 5-moment equations for increasing values of cM and fixed ve-locity cell lengths ∆c = 0.02. While all solutions show an unphysical sub-shock,increasing values of cM decrease the shock strength of the sub-shock and moveits location further upstream. Unfortunately, the computational complexity of themaximum-entropy closure renders this approach very difficult for further inves-tigations. Instead we will consider an explicit, closed-form singular closure in thenext section.

7.2 a new closed-form closure

In order to reduce the excessive computational demand of the maximum-entropyclosure McDonald et al. put forward an affordable, closed-form closure based onthe maximum-entropy hierarchy in [90]. Here we discuss a regularized variantof this closure that was proposed in [110].

In order to define the explicit closure we make use of the mapping

σ : R → [0, 1], (Q, R) 7→√(

R− 34

)2

+Q2

2− R− 3

4, (7.45)

which defines parabolas for the level curves σ(Q, R) = c with 0 < c ≤ 1, see [90].In the special case σ(Q, R) = 1, the implicitly defined parabola R = 1 + Q2

coincides with the realizability boundary ∂R; whereas in the limit c → 0, the

7.2 a new closed-form closure 123

-3 -2 -1 0 1 2 30

3

6

9

Q

R

σ = 1

Figure 7.6: Level curves of the σ mapping defined in Eq. (7.45) for σ = 0.1, 0.3, 0.5, 0.7, 0.9,together with the half-line J and the realizability boundary of the 5-momentsystem.

parabola degenerates to the Junk half-line. Let σ = σ(Q, R), then we can writethe closed-form closure as

Sβ(Q, R) = Q(

10− 8√

σ +(2σ + R− 3)(1 + β(1− σ))

σ + β(1− σ)

), (7.46)

where β is a non-negative regularization parameter. This closure is designed tomimic the behavior of the maximum-entropy flux in the limit β → 0. We willrefer to this auxiliary closure as β-closure.

A series expansion of (7.46) around Q = 0 for R > 3 and β > 0 yields

Sβ(Q, R) = Q

(10− 8Q√

R− 3+

(β + 1)(R− 3)2

Q2(1− β) + β(R− 3)

)+ O(Q3). (7.47)

In the special case β = 0 the closure reduces to

S0(Q, R) = Q(

10− 8√

σ +(2σ + R− 3)

σ

),

= Q

(10− 8

√σ +

Q2

σ2

),

(7.48)

for which a series expansion around Q = 0 for R > 3 yields

S0(Q, R) =(R− 3)2

Q+ 14Q− 8Q2

√R− 3

+ O(Q3). (7.49)


It follows from (7.47) and (7.49) that for R > 3 we have

limQ→0±

Sβ(Q, R) =

±∞ if β = 0,

0 if β > 0.(7.50)

The behavior of S0 at (Q, R) = (0, 3) is a bit more subtle. Let C(η) = (η, 3 + ηα)with η ∈ [0, ∞) denote a parametrized curve for some α > 0 in the normalizedphase-plane with C(0) = (0, 3), then limη→0 S0(C(η)) diverges for 0 < α < 1

2 .On the other hand, if β > 0, then Sβ is differentiable at the equilibrium state withthe partial derivatives given by

∂Sβ

∂Q= 10,

∂Sβ

∂R= 0. (7.51)

On the realizability boundary we find Sβ = Q(2 + Q2), which agrees with theclosing flux of the unique solution to the truncated moment problem for stateson the realizability boundary. Furthermore, we find the partial derivatives of Sβ

on the realizability boundary ∂R to coincide with those of the maximum-entropyclosure, see [111]. A careful numerical investigation shows the hyperbolicity re-gion of the β-closure to at least contain the set (Q, R)|1 + Q2 < R ≤ 103 for0 < β . 0.7, thus allowing the simulation of strongly non-equilibrium gas pro-cesses.

Figure 7.7 shows a qualitative comparison between Grad’s closure and the β-closure around the equilibrium state: While for larger values of β the β-closure issimilar to Grad’s linear closure, smaller values clearly increase the nonlinearityaround the half-line J .

7.2.1 Characteristic Wave Speeds

For Q = 0, the symmetry of the moment system allows to write the characteristicvelocities as

λ1 = v−√

θc f , λ2 = v−√

θcs,

λ3 = v, λ4 = v +√

θcs, λ5 = v +√

θc f ,(7.52)


-0.1 -0.05 0 0.05 0.12.9

3

3.1

3.2

3.3

Q

R

Grad

S

-0.8

-0.4

0

0.4

0.8

-0.1 -0.05 0 0.05 0.12.9

3

3.1

3.2

3.3

Q

R

β=10-1

S

-0.8

-0.4

0

0.4

0.8

-0.1 -0.05 0 0.05 0.12.9

3

3.1

3.2

3.3

Q

R

β=10-3

S

-3

-2

-1

0

1

2

3

-0.1 -0.05 0 0.05 0.12.9

3

3.1

3.2

3.3

Q

R

β=10-4

S

-7.5

-5.0

-2.5

0

2.5

5.0

7.5

Figure 7.7: Closing flux, S, of both the Grad and β-closure for β = 10−1, 10−3, 10−4 in thevicinity of the equilibrium state (Q, R) = (0, 3).


where cs, c f denote the slow and fast normalized characteristic speeds respec-tively, which can be stated explicitly by

c f ,s =

√5− 2

√6− 2R±

√k1(R), if 1 ≤ R ≤ 3,

√√√√ R− 3 + 7β + Rβ±√

k2(R, β)

2β, else if R ≥ 3,

(7.53)

with

k1(R) = 19− 8√

6− 2R− 3R,

k2(R, β) = (R− 3)2 + 2(R− 3)(R + 1)β + (R(22 + R)− 35)β2.(7.54)

Proposition 2. If Q = 0 and 0 < β < 32 , then the characteristic speeds cs, c f are real

valued for all R ∈ [1, ∞) .

Proof. First we consider the case 1 ≤ R ≤ 3 : Clearly, we have 6 − 2R ≥ 0,such that k1(R) ∈ R. Furthermore, k1(1) = 0 and k′1 = 8/

√6− 2R− 3 ≥ 1 for

1 ≤ R < 3, so that k1(R) ≥ 0. Next we need to check whether

f (R) := c2s (R) = 5− 2

√6− 2R−

√k1(R), for 1 ≤ R ≤ 3 (7.55)

is non-negative for all R ∈ [1, 3]. Since the continuous function f (R) has thecomplex roots R1,2 = (1± 2i

√6)6/25 and f (1) = 1 is positive we conclude that

f (R) > 0 for all R ∈ [1, 3].Let now R ≥ 3. Then it is easy to verify that k2(R, β) ≥ 0 holds for all R ≥ 3

and β > 0. Next we have to consider

g(R, β) := 2β c2s (R) = R− 3 + 7β + Rβ−

√k2(R, β) (7.56)

for R ≥ 3. The only solution of g(R, β) = 0 is given by R0(β) = (9− 21β)/(3−2β). We have R0(0) = 3 and R′0 < 0 for all β ∈ R\ 3

2, such that R0 < 3 for0 < β < 3

2 . Since g(3, β) = 2β(5−√

10) > 0 for β > 0, we can infer that g(R, β)is positive for R ≥ 3 and 0 < β < 3

2 .Since cs is real valued on the half-line (0, R) with R ∈ [1, ∞) and 0 < β < 3

2 ,we can infer from the wave speeds cs, c f given in (7.53) that this property alsoholds for c f , which concludes the proof.


Figure 7.8: The closing flux, S, for both the maximum-entropy closure (black, dashed line)and the β-closure with β = 10−1, 10−2, 10−3 and β = 0 (red lines) is shown inthe left plot. On the right, the eigenvalues of the flux Jacobian are shown for themaximum-entropy closure (black, dashed lines) and the β-closure (red lines)with β = 10−3.

In the equilibrium states u(eq) = (ρ, v, θ, 0, 3)T the normalized wave speeds(7.53) simplify to

c f ,s =

√5±√

10 (7.57)

or approximately c f ≈ 2.86 and cs ≈ 1.36. Note that these wave speeds arethe same as those of the Grad closure. This property follows from the fact thatboth closure theories coincide in their closing flux and partial derivatives at theequilibrium point for β > 0, as can be seen by inspection of the coefficients ofthe general, normalized Jacobian matrix given in Appendix B.7.

A series expansion for R > 3 of (7.53) yields the following simple expressionsfor the wave speeds

c f =

√R(

1 +1β

)+ O

(R−

12

), cs =

√5

1 + β− 2 + O

(R−1

). (7.58)

Interestingly, only the fast wave speed c f diverges in the limit R → ∞, whereascs converges to a finite asymptotic value for large R.

Figure 7.8 shows a comparison between the maximum-entropy closure and theβ-closure along a line in the two-dimensional phase-plane (Q, R) for constantR = 5. Note that both closures coincide in the closing flux S and the eigenvaluesof the flux Jacobian on the realizability boundary. On the other hand, away fromthe realizability boundary, the two closures only share some qualitative features,such as the divergence of the closing flux for β = 0. Therefore, we cannot expect


the β-closure to match the numerical solutions of the moment system using themaximum-entropy closure in a quantitative sense, but rather illustrate the con-sequences of the nonlinearity of the closure around the Junk half-line for smallvalues of the regularization parameter β.

Summarizing, the β-closure has the following set of properties:

1. In the special case β = 0 the closure is singular on the set J , mimickingthe singularity of the maximum-entropy closure.

2. The closing flux moment and its partial derivatives are in agreement withthe linearized maximum-entropy closure in the equilibrium state for anyβ > 0.

3. The β-closure is in agreement with the maximum-entropy closure for anyβ ≥ 0 on the realizability boundary.

4. The closure yields a hyperbolic set of moment equations if 0 ≤ β . 0.7over a large range of non-equilibrium states.

5. The closure is Galilean invariant since the normalized closing flux is inde-pendent of the velocity v.

7.3 riemann solutions

We study the system (7.9) in the collisionless limit Kn → ∞ with the Riemanninitial data

u(x, 0) =

uL, x < 0,

uR, x > 0.(7.59)

The solution u to this problem is self-similar, i.e. u(x, t) = u(ηx, ηt) for η ∈ [0, ∞)with five traveling waves, each of which is either a shock, a contact discontinuityor a rarefaction fan. Let s denote the speed of a shock wave, then for u to be aweak solution to (7.9), the Rankine-Hugoniot conditions s[E(u)] = [F(u)] haveto be satisfied, where [·] = [·]+ − [·]− denotes the difference of the right andleft limits of the states at the shock location. In order to simplify the analysis,we consider the Rankine-Hugoniot conditions in a co-moving reference frame,such that [F(u)] = 0. Let u0 and u1 denote the states before and after a jump

7.3 riemann solutions 129

discontinuity respectively. Given the state vector u0, then u1 and the closing fluxS1 can be parametrized by ρ1 as

v1(ρ1; u0) =ρ0v0ρ1

,

θ1(ρ1; u0) =ρ0ρ1

(θ0 +

v20(ρ1 − ρ0)

ρ1

),

Q1(ρ1; u0) = Q0 +ρ0v0(ρ1 − ρ0)

ρ1

(3θ0 −

v20(2ρ0 − ρ1)

ρ1

),

R1(ρ1; u0) = R0 +v0(ρ1 − ρ0)

ρ1

(6ρ0v0θ0(ρ1 − ρ0)

ρ1+ 4Q0

+ρ0v3

0(3ρ20 − 3ρ0ρ1 + ρ2

1)

ρ21

)

(7.60)

and

S1(ρ1; u0) = S0 +v0(ρ1 − ρ0)

ρ1

(10v2

0θ0ρ0(ρ1 − ρ0)2

ρ21

+10Q0v0(ρ1 − ρ0)

ρ1

+ 5R0 −v4

0ρ0(2ρ0 − ρ1)(2ρ20 − 2ρ0ρ1 + ρ2

1)

ρ31

), (7.61)

where

Q = ρθ3/2Q = 〈C3 f 〉, R = ρθ2R = 〈C4 f 〉, S = ρθ5/2S = 〈C5 f 〉 (7.62)

denote central moments of f . Additionally, S1 has to fulfill the relation

S1 = S(Q1, R1)ρ1θ5/21 (7.63)

for a given closing flux S. Plugging the relation (7.63) into (7.61) allows to deter-mine the state u1 by solving the system (7.60)-(7.61).

7.3.1 Linear Degenerate Waves

Let us consider a wave in the co-moving frame of reference, so that s = 0. In thespecial case v0 = 0 and ρ0, θ0 > 0 the equations (7.60) to (7.61) simplify to

ρ0θ0 = ρ1θ1, v0 = v1 = 0, Q0 = Q1, R0 = R1, S0 = S1. (7.64)


If additionally Q0 = 0, then λ3(u−) = 0 holds for both Grad’s closure and theβ-closure with β > 0. The corresponding eigenvector of the flux Jacobian J isgiven by

r(3) = e(1) =(

1 0 0 0 0)T

. (7.65)

Since the eigenvalues of J do not depend on ρ, we have ∇λ3(u) · r(3)(u) = 0 andthus the field r(3) is linearly degenerate for Q = v = 0. In this case, the wave ofthe third characteristic family is a contact discontinuity and the states on the leftand right of the discontinuity are related by

θ1 =ρ0ρ1

θ0, R1 =ρ1ρ0

R0, v1 = v0 = Q0 = Q1 = S0 = S1 = 0. (7.66)

7.3.2 Hugoniot Locus from the Equilibrium State

In this section we study the Hugoniot locus from the equilibrium state for theshock wave of the first characteristic family.

7.3.2.1 Grad’s Closure

For (Q0, R0) = (0, 3) we find

S(Grad5)(Q1, R1)ρ1θ5/21 − S1(ρ1; u0) = 0

⇔ v0ρ0(ρ1 − ρ0)p(ρ1) = 0,(7.67)

where p(ρ) = a0 + ρa1 + ρ2a2 + ρ3a3 with coefficients

a0 = 24v40ρ3

0, a1 = −12v20ρ2

0(3v20 + 5θ0),

a2 = 2ρ0(7v40 + 30v2

0θ0 + 15θ20), a3 = −(15θ2

0 + 10v20θ0 + v4

0).(7.68)

The asymptotic limits of large inflow Mach numbers for the density and veloc-ity ratios across the shock wave of the first characteristic family are given by

limv0→∞

ρ1(v0)

ρ0= 2(3−

√6) ≈ 1.1, lim

v0→∞

v1(v0)

v0=

12+

1√6≈ 0.91. (7.69)

Similarly, we find

limv0→∞

θ1(v0)

v20

=1

12and lim

v0→∞(Q1, R1)(v0) = (−2

√2, 9) ∈ ∂R. (7.70)

7.3 riemann solutions 131

Figure 7.9: Density and velocity jump ratios across a shock wave of the first characteristicfamily versus the inflow Mach number for both Grad’s closure and the β-closurewith β = 1

10 , 120 , 1

40 , 1100 .

7.3.2.2 β-Closure

We compute the Hugoniot locus from the equilibrium state of the first charac-teristic family for the β-closure by solving the Rankine-Hugoniot conditions nu-merically for different values of β. In Figure 7.9 we show the density and velocityratios across the shock wave over a range of inflow Mach numbers M0 = s/

√3θ0

for both the Grad system and the β-closure. Interestingly, in case of the β-closureboth ratios approach 1 for β → 0. In contrast to Grad’s closure, the density andvelocity ratios are not monotone functions of the inflow Mach number, but havean extremum at a finite Mach number.

The left plot in Figure 7.10 displays the Hugoniot loci from the equilibriumstate

u = (ρ, v, θ, Q, R)T = (1, 0, 1, 0, 3)T (7.71)

for both closures in the (Q, R) plane. The plot on the right shows the total shockstrength ||[u]|| in the 2-norm over the Mach number of the shock wave for bothclosures and various values of the β parameter. These results show that the 1-shock of the β-closure has a diminishing total shock strength for decreasing val-ues of β. As shown in the subsequent sections, this property allows to effectivelycontrol the strength of the emerging sub-shock in the shock structure problem.


Figure 7.10: The left plot shows the Hugoniot loci of the first characteristic family from theequilibrium state in the phase-plane for decreasing values of β = 10−i withi = 1, . . . , 5. The right plot shows the shock strength ||[u]|| in the 2-norm forGrad’s closure and the β-closure with β = 10−i and i = 2, . . . , 7.

7.4 numerical methods

The 5-moment system (7.9) is discretized in space with a uniform mesh using thefinite-volume method introduced in Chapter 4, which leads to the semi-discretesystem

∂tEi +1

∆x

(Fi+ 1

2− Fi− 1

2

)= Gi, (7.72)

where Fi−1/2 and Fi+1/2 denote the numerical fluxes over the cell interfaces atxi−1/2 and xi+1/2 respectively. The vectors Ei and Gi denote the spatial averagesof the state variables and the source terms over the interval [xi−1/2, xi+1/2]. TheRiemann problems at the cell interfaces are solved with the approximate HLLRiemann solver, see e.g. [59]. The fluxes at the cell interfaces are reconstructedby linear functions and the Minmod limiter is used to prevent unphysical os-cillations near shocks. Here we use the explicit 2nd order Heun method for thediscretization of the time variable. Overall, the scheme is of second order in thespatial and temporal variables.

7.4.1 Reference BGK Solver

In order to assess the approximation quality of the 5-moment systems we com-pute reference BGK solutions to the Boltzmann equation with a second-orderdiscrete velocity method, see Section 4.7. In this chapter the finite phase-space[xL, xR]× [cL, cR] ⊂ R2 of the BGK equation is discretized with a regular meshwith Nx cells in the spatial and Nc cells in the velocity direction. Note that the

7.5 the continuous shock structure problem 133

Figure 7.11: Numerical solutions to a Ma = 1.8 shock structure for both Grad’s closure(gray, dotted line) and the β-closure with β = 10−6 (red, dashed line), togetherwith the reference BGK solution (black line).

number of degrees of freedoms in the kinetic velocity for the DVM are usuallychosen to be at least of the order 102, whereas the moment methods studied inthis chapter have only 5 degrees of freedoms in the velocity space.

7.5 the continuous shock structure problem

Consider the problem of finding a traveling shock wave solution u(φ) withφ = x − st, where s denotes the shock speed, to the moment system (7.9) withboundary conditions

limφ→−∞

u(φ) = uL, limφ→+∞

u(φ) = uR, limφ→±∞

dudφ

= 0. (7.73)

The states uL, uR are in local thermodynamic equilibrium and related by theRankine-Hugoniot conditions of the Euler equations, see [104].

For systems of N-balance laws, equipped with a convex entropy function, Boil-lat and Ruggeri have shown in [14] the non-existence of smooth C1(R) solutionsto the shock structure problem if the shock velocity, s, is larger than the maxi-mum characteristic speed of the equilibrium state, uL, in front of the shock. As aconsequence, moment solutions to the shock structure problem show unphysicalsub-shocks if the shock velocity exceeds a critical value.

The largest characteristic velocities µ(Max) of the Euler equations and λ(Max)

of the β-closure with β > 0, evaluated in the equilibrium state uL in front of theshock wave, are given by

µ(Max)(uL) = vL +√

3θL, λ(Max)(uL) = vL +

√(5 +

√10)θL. (7.74)


Figure 7.12: Numerical solutions to a Ma = 1.8 shock structure for both Grad’s closure(gray, dashed line) and the β-closure (red lines) with β = 10−i and i = 1, . . . , 4,together with the reference BGK solution (black line) on a logarithmic scale.

The range of Mach numbers Ma = s/√

3θL for which smooth solutions may existis thus bounded by the critical Mach number

Ma? =

√5 +√

103

≈ 1.65. (7.75)

Figure 7.11 shows the numerical solutions to the 5-moment system using bothGrad’s closure and the β-closure, together with the reference BGK solution, fora Ma = 1.8 > Ma? shock wave profile, where the upstream state is given byuL = (1, 1.8 ·

√3, 1, 0, 3). We use the mean free path λ(MFP) = 16τ/5

√θL/(2π)

as suggested in [90] as a reference length scale and thus set l0 = t0√

θ0 = λ(MFP).The left plot in Figure 7.11 shows the normalized density with ρ(xL) = 0 andρ(xR) = 1, where xL, xR denote the left and right boundaries of the computa-tional domain. The moment solutions were computed on a highly refined meshwith Nx = 5000 cells over the domain [xL, xR] = [−78, 78] and a CFL num-ber of approximately 0.5. The reference BGK solution was computed with Nx =4000 and Nc = 400 cells over the computational domain [xL, xR] × [cL × cR] =[−78, 78]× [−5, 9].

In Figure 7.11, the solution to the 5-moment system with Grad’s closure showsa strong sub-shock, that connects the equilibrium state uL with a non-equilibriumstate at x ≈ −5. However, the β-closure with β = 10−6 shows a smooth curvein the normalized density ρ and Q akin to the results presented in [90]. We notethat further decreasing the β-parameter does not affect the numerical solutionplotted in Figure 7.11. However, as shown in Figure 7.12, the moment solutionswith the β-closure have a sub-shock connecting the equilibrium upstream statewith a non-equilibrium state. This shock belongs to the first characteristic familyand is thus strongly influenced by the β-parameter as shown in Figure 7.10.

7.6 further numerical examples 135

Clearly, using smaller values of the β parameter leads to the generation of smallersub-shocks, so that the β parameter can be used to control the strength of theemerging sub-shock.

7.5.1 Vanishing Regularization Limit

The numerical study presented in this chapter suggests that in the limit β → 0the sub-shock in the solution for the shock structure problem indeed disappears.We expect the maximum-entropy closure with a suitable regularization to have asimilar behavior.

7.6 further numerical examples

In this section we present numerical results for both the BGK and a Fokker-Planck equation with Riemann initial conditions given by

f (x, t = 0, c) =

MρL ,vL ,θL (c), x < 0

MρR ,vR ,θR (c), x > 0, (7.76)

where

Mρ,v,θ(c) =ρ√2πθ

exp(− (c− v)2

2θ

)(7.77)

denotes a Maxwellian distribution.

7.6.1 Symmetric Two-Beam Riemann Problem

Let us consider the one-dimensional Boltzmann equation with the BGK collisionoperator and the initial condition (7.76) with

ρL = ρR = ρ0, vL = −vR = v0 ≥ 0, θL = θR = θ0, (7.78)

describing the collision of two Maxwellian distributed particle beams.


7.6.1.1 Continuum Limit

In the limit Kn → 0, the 5-moment system reduces to the Euler equations withthe self-similar solution

ρ(η) =

ρ0, |η| > s

ρM, |η| < s, v(η) =

vL, η < −s

0, |η| < s

vR, η > s

,

θ(η) =

θ0, |η| > s

θM, |η| < s,

(7.79)

where η = x/t and s denotes the shock speed of the left and right travelingshock waves. Let Ma = vL/

√3θ0 denote the Mach number of the generated

shock waves, where vL = vL + s is the velocity of the upstream state in the frameof reference co-moving with the shock front traveling to the left. The intermediatestate is determined by the Rankine-Hugoniot conditions and reads

ρM = ρ02 Ma2

Ma2 +1, vM = 0, θM = θ0

(1 + Ma2)(3 Ma2−1)4 Ma2 . (7.80)

The shock speed of the left and right traveling shock waves is given by

s =√

3θ0Ma2 +1

2 Ma. (7.81)

7.6.1.2 Free-streaming Limit

In the free-streaming limit, for which Kn→ ∞, the BGK equation reduces to thelinear transport equation

∂t f + c∂x f = 0 (7.82)

with the explicit solution

f (x, t, c) = f (x− ct, 0, c) =

Mρ0,v0,θ0 (c), x− ct < 0

Mρ0,−v0,θ0 (c), x− ct > 0. (7.83)


In this case the density fields can be expressed in terms of the error function, e.g.we find the mass density to be given by

ρ(x, t) =ρ02

(1 + erf

(v0 +

xt√

2θ0

)+ erfc

(−v0 +xt√

2θ0

)), (7.84)

where

erf(z) =2√π

∫ z

0e−t2

dt (7.85)

is the error function and erfc(z) = 1− erf(z) the complementary error function.

7.6.1.3 Explicit Wave Construction

For the symmetric two-beam problem with v0 > 0 we find the solution to theβ-closure in the collisionless case to consist of a 1-shock, 2-rarefaction, 3-contactdiscontinuity, 4-rarefaction, and a 5-shock wave.

A weak solution u can be constructed by explicitly computing the Hugo-niot loci and integral curves. Let u1(γ) denote the Hugoniot locus of the firstcharacteristic family from the equilibrium state, such that u1(0) = uL. Fur-thermore, let u2(γ, u1(γ1)) denote the integral curve of the second family withu2(0, u1(γ1)) = u1(γ1)). If there exists a pair (γ1, γ2) such that u2(γ2, u1(γ1)) =(ρ2, v2, θ2, Q2, R2) with v2 = Q2 = 0, then

u(γ) =

u1(γ), 0 ≤ γ ≤ γ1

u2(γ, u1(γ1)), γ1 ≤ γ ≤ γ2

(7.86)

is a weak solution to the 5-moment system on the half-line (−∞, 0].Figure 7.13 shows the numerical solutions to the explicit wave construction

method for the β-closure with several β-values and initial velocities v0 for statesover x ∈ (−∞, 0]. We observe that decreasing values of β lead to smaller jumpsin the non-equilibrium moments. The right plot shows the solutions for a fixedvalue of β and increasing initial velocities v0, which lead to stronger deviationsfrom the equilibrium distribution.

7.6.1.4 Numerical Results

Let us first consider the two-beam problem with the initial condition (7.78) withv0 = 0.5 and ρ0 = θ0 = 1 at time t = 0 and open boundary conditions. Forthis problem, a natural choice for the characteristic time is t0 = T, such that theKnudsen number can be set to Kn = τ/T, where τ denotes the relaxation time.Figures 7.14 and 7.15 display the numerical solutions to the 5-moment system


Figure 7.13: The free-streaming solutions to the 5-moment system with the β-closure forstates over the spatial domain (−∞, 0]. The left plot displays the moment solu-tions for the parameters β = 10−i with i = 1, . . . , 4 and initial Riemann condi-tion (7.78) with v0 = 0.5. The dotted and full red lines represent the Hugoniotloci and the integral curves respectively. The right plot shows the solutions forβ = 10−3 and different initial velocities v0 = 0.5, 1, 2, 4, 8. The black curve isthe realizability boundary of the 5-moment equations.

with Grad’s closure and the β-closure, together with the reference BoltzmannBGK solution at the final time T = 0.1 for various Knudsen numbers.

The numerical solutions to the moment systems were computed on a highlyrefined mesh with Nx = 104 cells and the reference BGK solutions were com-puted on a mesh with Nx = 2000 and Nc = 600 cells over the computationaldomain Ωx ×Ωc = [−10, 10]× [−10, 10]. In both cases, the CFL number was setto roughly 0.5.

Figures 7.16 and 7.17 show the two-beam solution for the initial conditionv0 = 2 and ρ0 = θ0 = 1 at time T = 0.1, using the same numerical discretizationas above. The higher initial velocity results in much stronger deviations from theequilibrium state, such that the numerical solution to the Grad system breaksdown as the solution leaves the region of local hyperbolicity. We found that theplotted numerical solutions to the β-closure with β = 10−5 to be converged in β,such that a further decrease in the parameter does not affect the solution.

Figure 7.18 shows the relative L1 errors

||uk||L1,Rel. =||uk − u(BGK)

k ||L1

||u(BGK)k ||L1

, where || · ||L1 =∫

Ωx

| · | dx (7.87)

and Ωx = [−10, 10], for the velocity, pressure and normalized heat flux, betweenthe converged numerical solutions to the 5-moment equations and the referenceBGK solution for both the β-closure and Grad’s closure for the two-beam prob-lem with initial condition v0 = 0.5 and ρ0 = θ0 = 1.


Two-Beam Problem (v0 = 0.5)

β=- β=-

Figure 7.14: Pressure, p, and normalized heat flux, Q, profiles of the numerical solutions toGrad’s closure, the β-closure and the reference BGK equation for the two-beamRiemann problem with the initial condition (7.78) and ρ0 = θ0 = 1, v0 = 0.5for the collisionless case (top) and Kn = 1 (bottom).


Two-Beam Problem (v0 = 0.5)

β=- β=-

Figure 7.15: Pressure, p, and normalized heat flux, Q, profiles of the numerical solutions toGrad’s closure, the β-closure and the reference BGK equation for the two-beamRiemann problem with the initial condition (7.78) and ρ0 = θ0 = 1, v0 = 0.5for the Knudsen numbers Kn = 10−1 (top) and Kn = 10−2 (bottom).


Two-Beam Problem (v0 = 2)

β=- β=-

Figure 7.16: Pressure, p, and normalized heat flux, Q, profiles of the numerical solutionsto the β-closure and the reference BGK equation for the two-beam Riemannproblem with the initial condition (7.78) and ρ0 = θ0 = 1, v0 = 2 for thecollisionless case (top) and Kn = 1 (bottom).


Two-Beam Problem (v0 = 2)

β=- β=-

Figure 7.17: Pressure, p, and normalized heat flux, Q, profiles of the numerical solutionsto the β-closure and the reference BGK equation for the two-beam Riemannproblem with the initial condition (7.78) and ρ0 = θ0 = 1, v0 = 2 for theKnudsen numbers Kn = 10−1 (top) and Kn = 10−2 (bottom).


0.1 1 10

14%

12%

1%

2%

4%

Kn

Rel.L

1Error

v

0.1 1 10

14%

12%

1%

2%

4%

KnRel.L

1Error

p

0.1 1 10

8%

16%

32%

64%

Kn

Rel.L

1Error

Q

β=- β=-

Figure 7.18: Relative approximation error in the L1 norm of Grad’s closure and the β-closure to the reference BGK solution for the two-beam problem with the initialvelocity v0 = 0.5 for different Knudsen numbers.

In Figure 7.19, the relative L1 error is shown between the β-closure and thereference BGK solution for the two-beam problem with v0 = 2 and ρ0 = θ0 =1. Clearly, the solution for β = 10−5 is in better overall agreement with thereference solution when compared to the solution for β = 10−1. Note that Grad’s5-moment system fails for these initial conditions, since the solution leaves theregion of hyperbolicity H(Grad5).

7.6.2 Fokker-Planck Equation

As a final test case, let us consider the one-dimensional Fokker-Planck equation

∂t f + c∂x f =∂c(c f + θ0∂c f )

τ, (7.88)

where τ is a relaxation time and θ0 a constant reference temperature in energyunits. Following [35], we write the Fokker-Planck equation in the dimensionlessvariables

t =tτ

, x =x

τ√

θ0, c =

c√θ0

, (7.89)

so that, after dropping the bars, the dimensionless Fokker-Planck equation reads

∂t f + c∂x f = ∂c (c f + ∂c f ) . (7.90)


0.1 1 10

14%

12%

1%

2%

4%

Kn

Rel.L

1Error

v

0.1 1 10

12%

1%

2%

4%

8%

KnRel.L

1Error

p

0.1 1 10

8%

16%

32%

64%

Kn

Rel.L

1Error

Q

β=- β=-

Figure 7.19: Relative approximation error in the L1 norm of the β-closure with β = 10−1

and β = 10−5 to the reference BGK solution for the two-beam problem withthe initial velocity v0 = 2 for different Knudsen numbers.

From the non-dimensional Fokker-Planck equation, the 5-moment system

∂tui + ∂xui+1 = i ((i− 1)ui−2 − ui) , i = 0, . . . , 4, (7.91)

where ui = 〈ci f 〉, is easily obtained. Here we consider the initial condition

f (x, 0, c) =Mρ0(x),v0=0,θ0=1(c), ρ0(x) =

1, x < 0

0.01, x > 0, (7.92)

for which an exact solution was derived in [35]. Figure 7.20 shows the exactsolution to the Fokker-Planck equation, together with the numerical results tothe 5-moment system with the β-closure for both β = 10−1 and β = 10−6 attime t = 1. The numerical results were computed with Nx = 6000 cells over thecomputational domain Ωx = [−4, 5].

Despite the strong initial density jump, the β-closure matches the analyticalsolution surprisingly well. The solution for β = 10−1 shows a strong shock at x ≈3.6, connecting an equilibrium state with a non-equilibrium state. As expected,the shock strength of this shock wave diminishes for decreasing values of β.However, the shocks located at x ≈ −0.1 and x ≈ 1 have larger shock strengthsfor smaller values of β. Since these shocks connect two states in non-equilibrium,they are not affected by the regularization parameter in the same way as theshock at x ≈ 3.6.

7.7 discussion 145

Fokker-Planck Problem

-

ρ

-

-

θ

- -

-

-

-

β=- β=-

Figure 7.20: Exact reference solution to the Fokker-Planck equation, together with numer-ical solutions to the 5-moment equations using the β-closure with β = 10−1

and β = 10−6.

7.7 discussion

Hyperbolic moment equations with a convex entropy extension generate unphys-ical sub-shocks in the solution to the continuous shock structure problem, whichcan lead to large deviations from the reference BGK equation.

However, by using the highly nonlinear β-closure, the strength of the sub-shockcan be decreased, such that in the limit of β→ 0 the sub-shock disappears. In thesingular limit the proof of Ruggeri and Boillat [14] does not apply, since the char-acteristic speeds in the vicinity of the equilibrium are not bounded, rendering thesystem non-hyperbolic.

The results shown in this chapter indicate that the singularity in the flux of themaximum-entropy closure might not be a shortcoming, as argued e.g. in [35], butrather a desirable feature in the sense that suitably regularized singular closuretheories can mitigate the problem of sub-shocks in the shock structure problem.

8M O M E N T A P P R O X I M AT I O N S O F P L A S M A S

In contrast to the rarefied gases considered in the previous chapters, plasmas aremulti-component collections of charged and neutral particles in gas phase. Plas-mas can behave quite differently than gases, since the movement of electricallycharged particles, such as electrons and ions, can induce collective motions of theparticles.

This chapter presents moment models for single- and multi-component plas-mas. Section 8.1 provides a short introduction to the kinetic theory of multi-component plasmas. A dimensional analysis of the kinetic equations in Section8.2 exhibits the multi-scale nature of the plasma. Moment approximations to thekinetic plasma equations are introduced in Section 8.3. A simple chemical modelis presented in Section 8.3.1 for ionization and recombination reactions. Elasticcollisions are modeled with a simplified kinetic ansatz in Section 8.4, which al-lows the derivation of explicit expressions for hard-sphere collisions and electro-static interactions. Here, the production terms are shown for the one-dimensionalEuler system in Section 8.5.

In Section 8.6, the Vlasov-Poisson-BGK plasma model is considered, which as-sumes a fully ionized plasma in the limit of a small electron to ion mass ratio.Numerical results for the Vlasov-Poisson-BGK system for a Maxwellian and anon-equilibrium bump-on-tail velocity distribution function show the capabili-ties of moment equations to model Landau damping and plasma instabilities.

8.1 kinetic theory of plasmas

Let us consider a gas mixture consisting of multiple components. The particletype is denoted by γ ∈ S , where S denotes the set of all particle species. In kinetictheory, the state of the gas mixture is described by |S| one-particle distributionfunctions

f γ : Ωx ×Ωt ×Ωc → R+, (x, t, c) 7→ f γ(x, t, c), (8.1)

147

148 moment approximations of plasmas

where Ωx ⊆ Rdx , Ωc ⊆ Rdc , Ωt ⊆ R≥0, and f γ denotes the number density ofcomponent γ in the phase space Ωx ×Ωc. The evolution of f γ is governed by theBoltzmann equation

∂t f γ + cj∂xj f γ + aγj ∂cj f γ = Cγ (8.2)

where aγj describes the acceleration of component γ due to long-range force

fields. Let Ei, Bi denote the electric and magnetic fields respectively, then theparticle acceleration due to electromagnetic force fields can be written as

aγi =

Fγi

mγ=

qγ

mγ

(Ei + εijkcγ

j Bk

), (8.3)

where Fγi denotes the Lorentz force, mγ the particle mass, and qγ the charge of

component γ.The collision operator Cγ describes the effect of all binary collisions on com-

ponent γ. It can be decomposed as

Cγ = ∑β∈SCγδ( f γ, f δ), (8.4)

where the collision operator Cγδ models the effect of binary collisions betweenspecies γ and δ on species γ.

Integration over the velocity space yields the number density

nγ(x, t) =∫

Ωc

f γ dc := 〈 f γ〉 (8.5)

of species γ, which is related to the mass density by ργ = nγmγ. The mixturenumber density, n, and macroscopic velocity, v, are defined by

n = ∑γ∈S

nγ, nvi = ∑γ∈S

nγvγi = ∑

γ∈S〈cγ

i f γ〉. (8.6)

The diffusive random velocity Cγi is given by

Cγi = cγ

i − vi = Cγi + uγ

i , (8.7)

where Cγi = cγ − vγ

i is the random velocity of γ-particles with respect to themacroscopic velocity of the γ-components and uγ

i = vγi − vi denotes the diffusion

velocity of species γ.


Writing the distribution function in the random velocity Cγ yields the evolu-tion equation

∂ f γ

∂t+ (vj + Cγ

j )∂ f γ

∂xj+

(aγ

j −Dvj

Dt

)∂ f γ

∂Cγj− Cγ

j∂ f γ

∂Cγk

∂vk∂xj

= Cγ, (8.8)

where f γ = f γ(x, t, Cγ).

8.2 dimensional analysis

Let us consider a three-species plasma consisting of neutral particles (γ = n),ions (γ = i) and electrons (γ = e). In the following, a dimensional analysis ofthe Boltzmann equation (8.8) is presented, which closely follows the dimensionalanalysis of Degond and Lucquin-Descreux in [31, 32] and Magin and Degrez in[80].

Following the hypothesis of [32], the species are assumed to have temperaturesand densities of the same order of magnitude, such that T0 and n0 denote acommon reference temperature and number density.

A reference random velocity, Cγ0 , and characteristic temperature, θ

γ0 , in energy

units are given by

Cγ0 =

√θ

γ0 =

√kBT0mγ

. (8.9)

Note that due to the large disparity in the masses, the reference random velocityof the neutral particles is by a factor

εm =

√me

mn 1 (8.10)

slower than the random velocity of the light particles.Let σ0 denote a reference effective cross sectional area. Then a common, char-

acteristic mean free path is given by

λ0 =1

n0σ0=

Cγ0

νγ0

= τγ0 Cγ

0 , (8.11)

where νγ0 denotes a reference collision frequency of species γ.

The characteristic macroscopic length and time scales are denoted by l0 and t0respectively. As for the dimensional analysis in Chapter 2, the Strouhal number isassumed to be unity, such that v0 = l0/t0 is a characteristic macroscopic velocity.


The macroscopic and microscopic scales are related by

Ma′ =v0Cn

0, Kn =

λ0l0

, Ma′ Kn =τn

0t0

, (8.12)

where Ma′ is related to the physical Mach number of the neutral particles byMa′ =

√5/3 Ma for a three-dimensional velocity space.

A characteristic scale for the electric field is given by

E0 =kBT0e l0

(8.13)

and a reference scale B0 for the magnetic field is determined by

βe =eB0me τe

0 , βi =eB0

mi τi0 =

√me

mi βe, (8.14)

where βe, βi denote dimensionless Hall numbers, see [80]. The electron Hall num-ber is related to the gyrofrequency of the electrons, ω

ge0 , by βe = ω

ge0 τe

0 .We assume that the microscopic velocities for a given species are of the same

order of magnitude and set

cγ0 = Cγ

0 = Cγ0 . (8.15)

With the non-dimensional quantities

t t0 = t, xi l0 = xi, Cγi Cγ

0 = Cγi , f f0 = f

n n0 = n, EE0 = E, BB0 = B, C( f )f0τ0

= C( f ), Zi =qie0

,(8.16)

where e0 denotes the fundamental charge, the non-dimensional Boltzmann equa-tions can be written as

∂ f n

∂t+

(vj +

Cnj

Ma′

)∂ f γ

∂xj− Cn

j∂ f γ

∂Cnk

∂vk∂xj−Ma′

Dvj

Dt∂ f γ

∂Cnj=

Cn

Ma′ Kn,

∂ f γ

∂t+

(vj +

Cγj

rγ

)∂ f γ

∂xj− Cγ

j∂ f γ

∂Cγk

∂vk∂xj− rγ

Dvj

Dt∂ f γ

∂Cγj+ Sγ =

Cγ

rγ Kn,

(8.17)

where rγ = Ma′ rγm with rγ

m =√

mγ/mn and all bars have been removed forbetter readability. The non-dimensional terms Sγ and Cγ are given by

Sγ =Zγ

rγ

(Ei + εijk

(rγvj + Cγ

j

) βγ

KnBk

)∂ f γ

∂Cγi

(8.18)


and

Cγ = ∑β∈SCγβ( f γ, f β). (8.19)

Equivalently, the non-dimensional Boltzmann equations can be written in con-vective form as

∂t f n +cn

j ∂xj f n

Ma′=

Cn

Ma′ Kn,

∂t f γ +cγ

i ∂xi f γ

rγ+

Zγ

rγ

(Ei +

βγ

KnεijkcjBk

)∂ f γ

∂cγi=Cγ

rγ Kn,

(8.20)

where f = f (x, t, c).

8.2.1 Diffusion Velocity and Mixture Quantities

Let uγ0 = Cγ

0 denote a characteristic diffusion velocity and let cγ0 , Cγ

0 denote ref-erence microscopic velocities. The dimensional scaling introduced above yieldsthe following expressions the for nondimensional diffusive random velocity, Cγ

i ,the random velocity, Cγ

i , and the diffusion velocity, uγi :

Cγi = cγ

i − virγ, Cγi = cγ

i − vγi rγ, uγ

i = (vγi − vi)rγ. (8.21)

Note that for ions with masses mi = mn − me, we have rim ≈ rn

m = 1, while forelectrons re

m = εm rnm.

The moments in the velocity variables Cγ and Cγ can be related explicitly.Using the relation Cγ

i = Cγi + uγ

i rγ, we find the general expression

〈Cγi1,...,in

f γ〉 =n

∑k=0

(nk

)⟨Cγ(i1,...,ik

f γ⟩

uγik+1,...,in)

, (8.22)

where Cγi1,...,in

= Πnk=1Cγ

ik, such that e.g. the temperatures

θγ =〈|Cγ|2 f γ〉

dcnγ, θγ =

〈|Cγ|2 f γ〉dcnγ

(8.23)

are related by

θγ = θγ +(uγ

i )2

dc. (8.24)


Let ρ0 = n0mn denote a reference mixture mass density, then the nondimen-sional mixture density and mass-averaged mixture velocity are given by

ρ = ∑γ(rγ

m)2nγ, vi =

∑γ(rγm)

2vγi nγ

∑δ(rδm)2nδ

= ∑γ

cγvγi , cγ

m =(rγ

m)2nγ

∑δ(rδm)2nδ

. (8.25)

Rearranging the expression for the mixture velocity yields

∑γ∈S

nγ(rγm)

2nγ(vγi − vi) =

1Ma′ ∑

γ∈Suγ

i nγrγm = 0, (8.26)

such that the mass weighted sum of diffusion velocities is zero.

8.3 moment equations

A projection of the kinetic Eqs. (8.20) on a polynomial basis vector φ yields themoment system

∂tun +1

Ma′∂xj f

nj =

pn

Ma′ Kn,

∂tuγ +1rγ

(∂xj f

γj + Zγ

(dγ

j Ej +βγ

Knεjkle

γjkBl

))=

pγ

rγ Kn,

(8.27)

in convective variables, where

uγ = 〈φ f γ〉, fγi = 〈ciφ f γ〉, pγ = 〈φCγ〉,

dγi = 〈(∂ci φ) f γ〉, eγ

ij = 〈(δijφ + cj∂ci φ) f γ〉. (8.28)

In slab geometry, the moment system (8.27) reduces to

∂tun +1

Ma′∂xfn

x =pn

Ma′ Kn,

∂tuγ +1rγ

(∂xfγ

x + Zγdγx Ex

)=

pγ

rγ Kn,

(8.29)

if the magnetic field is neglected. Let us consider in the following the Eulerapproximation generated by the basis φ = (1, cx, c2

x + c2r )

T , for which

uγ =

nγ

nγrγvγx

nγ(rγvγx )

2 + 3nγ θγ

, fγ

x =

nγrγvγx

nγ(rγvγx )

2 + nγ θγ

nγ(rγvγx )

3 + 5nγvγx θγ

and dγx = (0, nγ, 2nγrγvγ

x )T .


8.3.1 Ionization and Recombination

At high temperatures some of the particles have enough energy to ionize neutralparticles, i.e. an electron is stripped from an atom or molecule, due to a collisionwith another particle. The ionization-recombination process can be modeled bythe chemical reaction

n + δ e + i + δ, (8.30)

where δ ∈ S is a catalyst for the reaction. This reaction has been considered in[78, 79] for the numerical simulation of multi-component plasmas.

Here we consider a simple model based on the chemical reaction models usedin [78, 79]. Clearly, the chemical reactions lead to a source term in the continuityequation, i.e.

∂tnγ + ∂x(nγvγx ) = wγ. (8.31)

The source term is modeled by wγ = sγR, where sγ denotes a stoichiometriccoefficient with sn = 1 and si = se = −1 and R is the dimensionless chemicalreaction rate

R = ∑δ∈S

(−r f kδ

f nnnδ + rbkδbneninδ

)(8.32)

with reaction constants

kδf = (Tδ)3/2 exp

(− (1− Tδ)Tact

Tδ

), kδ

b = exp

(− (1− Tδ)(Tact − Teq)

Tδ

)

and

r f = c f T3/20 e−Tact

n0t0

N2A

, rb = cbe−(Tact−Teq)n2

0t0

N3A

, (8.33)

where NA denotes Avogadro’s constant and Tact, Teq denote the dimensionlessactivation and equilibrium temperatures respectively. The dimensional reactionparameters c f and cb = c f /ceq have to be determined experimentally. For exam-ple, Hoffert and Lien provide in [61] reaction parameters for an argon plasma.

Note that the chemical reaction model (8.32) satisfies

∑γ

mγwγ = R(mn −me −mi) = 0, (8.34)

such that the total mass is conserved.


8.3.1.1 Chemical Equilibrium

The equilibrium constant is given by

kδeq =

kδf

kbδ

= (Tδ)3/2 exp

(− (1− Tδ)Teq

Tδ

). (8.35)

In chemical equilibrium we have

keq =neni

nnrbr f

=neni

nnn0

keq,0, (8.36)

where

keq,0 = ceq (T0)3/2 exp

(−Teq

). (8.37)

Figure 8.1 shows the ionization degree d = ne/(nn + ne) for a homogeneous,neutral plasma (ne = ni) in chemical equilibrium over the Temperature T forthe dimensional parameters Teq = 2× 105 K, ceq = 1022 m−3 K−3/2 of an argonplasma [78] at constant Pressure p = 105 Pa.

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

T [103 K]

IonizationDegree

Figure 8.1: Ionization degree versus temperature at constant pressure.

8.3.1.2 Extension to Compressible Flows

For simplicity let us consider a homogeneous plasma. A simple extension of thechemical model to the Euler system is given by

∂tnγ = wγ,

∂t(nγrγvγi ) = viwγ,

∂t

(nγ|rγvγ|2 + 3nγ θγ

)= |v|2wγ,

(8.38)

8.4 kinetic modeling of collisions 155

where v denotes the mixture velocity. This model is based on the assumptionthat particles involved in the ionization-recombination reaction have an averagevelocity given by the mixture velocity, thus neglecting differences in the speciesvelocities vγ. This system conserves total mass, momentum and energy since∑γ∈S mγwγ = 0.

8.4 kinetic modeling of collisions

In general the Boltzmann collision integral can not be evaluated in closed-formfor arbitrary model distribution functions and differential cross sections. There-fore, further simplifications are required to derive explicit expressions for theproduction terms. By approximating the model distribution function in the col-lision kernel with a simpler Grad like distribution function, expanded around aMaxwellian in the diffusive velocity Cγ, it is possible to compute explicit expres-sions for the production terms, see e.g. [25].

Let φ denote the vector of generating basis functions of the moment system

and f (ME)α the reconstructed maximum-entropy distribution function. In the col-

lision operator the velocity distribution f (ME)α is replaced with a simpler model

given by

f γG(C

γ) =Mγ(Cγ)φγ(Cγ) · aγG, (8.39)

whereMγ is the Maxwellian

Mγ(Cγ) =nγ

(2πθγ)3/2 exp(−|C

γ|22θγ

)(8.40)

and aγG ∈ RN are coefficients determined by the moment constraints

〈φ f (ME)α 〉 = 〈φ f γ

G〉. (8.41)

The function f γG is a Grad like polynomial expansion about the MaxwellianMγ.

Note that the mean of Mγ is the mixture velocity v, whereas the number den-sity and temperature are species dependent. The approximation (8.39) allows asignificant simplification of the collision integral and the evaluation of explicitproduction terms, see e.g. [54–57].


8.5 production terms for the one-dimensional euler system

Let us consider a system of |S| interacting species, for which the distributionfunction of each species is a Maxwellian

Mγ(c) =nγ

(2πθγ)3/2 exp(−|c− vγ|2

2θγ

). (8.42)

A natural choice for the expansion is to consider φγ =(1, Cγ

x , |Cγ|2)T , for which

〈φMγ〉 =(nγ, nγuγ

x , 3nγθγ)T . (8.43)

The Grad5 approximation to the Maxwellian Mγ is given by

f γG5(C

γ) =Mγ(Cγ)

(1 +

uγx Cγ

xθγ

). (8.44)

Instead of the above Grad5 approximation, it is possible to use higher-ordermoment approximations to Mγ. Let us consider the Grad10 and Grad35 mo-ment theories, which include all monomials up to order two and order fourrespectively. For the one-dimensional gas process, the Grad approximations tothe Maxwellian Mγ(c) simplify to

f γG10(C

γ) =Mγ(Cγ)

(1 +

uγx Cγ

xθγ

+13

(uγ

x Cγx

θγ

)2

− 16

(uγ

x Cγr

θγ

)2)(8.45)

and

f γG35(C

γ) =Mγ(Cγ)

(1− 1

12(uγ

x )4

(θγ)2 +

(1 +

(uγx )

2

3θγ

)uγ

x Cγx

θγ+

(13+

5(uγx )

2

18θγ

)(uγ

x Cγx

θγ

)2

− (uγx )

2 + 3θγ

18θγ

(uγ

x Cγr

θγ

)2

− 16(uγ

x )3Cγ

x (Cγr )

2

(θγ)3

− 136

(uγ

x Cγx

θγ

)4

− 118

((uγ

x )2Cγ

x Cγr

(θγ)2

)2

+172

(uγ

x Cγr

θγ

)4). (8.46)

In Figure 8.2, the marginal density

〈·〉Ωr = 2π∫ ∞

0· cr dcr (8.47)

is shown for the Grad5, Grad10 and Grad35 approximations of the distributionM with nγ = θγ = 1 and a mixture velocity of vx = 0 for different diffusion

8.5 production terms for the one-dimensional euler system 157

velocities with uγx = vγ

x . As expected, for small diffusion velocities, the approxi-mations are in good agreement with the distribution function Mγ, whereas forlarger deviations, the approximation errors increase and higher-order approxi-mations may become necessary, depending on the accuracy requirements.

-4 -2 0 2 4-0.1

0.0

0.1

0.2

0.3

0.4

Cxγ

MarginalDensity

-4 -2 0 2 4-0.1

0.0

0.1

0.2

0.3

0.4

Cxγ

MarginalDensity

-4 -2 0 2 4-0.1

0.0

0.1

0.2

0.3

0.4

Cxγ

MarginalDensity

-4 -2 0 2 4-0.1

0.0

0.1

0.2

0.3

0.4

Cxγ

MarginalDensity

<Mγ>Ωcr <Mγ>Ωcr <fG5

γ >Ωcr <fG10γ >Ωcr <fG35

γ >Ωcr

Figure 8.2: Marginal densities of the reference distribution Mγ, the weight function Mγ

and different Grad like approximations for uγx = 0.25 (top left), uγ

x = 0.5 (topright), uγ

x = 1 (lower left), and uγx = 1.5 (lower right).

In the following, explicit expressions for binary collisions are presented for theGrad5 approximation of Mγ.

Let νγ0 = (τγ

0 )−1 denote a characteristic collision frequency of component γ,

then the dimensionless production term in the diffusive velocity Cγ can be writ-ten as

Pγδi1,...,in

=rγδ

ν

rγ

〈Cγi1,...,in

Cγδ〉Kn

, where rγδν =

νγδ0

νγ0

. (8.48)


The production terms of the Euler system in the convective moments are relatedto the diffusive production terms by

pγ

Kn rγ= (0,Px,Pii + 2Px(vγx rγ − uγ

x ))T . (8.49)

8.5.1 General Momentum and Energy Production Terms

A dimensionless collision frequency is given by

νγδ = nδΩγδ(1,1)

√θγµδ + θδµγ, (8.50)

where µγ = mγ/(mγ + mδ) and Ωγδ(1,1) is a standard Ω-collision integral, see [70],

given by

Ω(l,r) =∫ ∞

0Q(l)e−γ2

γ2r+3 dγ, Q(l) =∫ ∞

0(1− cosl(χ))b db. (8.51)

For the one-dimensional Euler equations, the general production term for themomentum density can be expressed as

Pγδi =

32√

π

3 Kn rγrγδ

ν µδnγνγδ(

uδi rγδ

m − uγi

), (8.52)

where rγδm =

√mγ/mδ and

rγδν =

n0

√θ

γδ0 Ωγδ

0

νγ0

=Ωγδ

0√µδσ0

, (8.53)

denotes a dimensionless scaling factor, where Ωγδ0 is a characteristic value of the

cross section Ωγδ(1,1) and θ

γδ0 = (θγ

0 + θδ0)/2.

The linearized energy production term is given by

12Pγ

ii =16√

π

Kn rγrγδ

v µγµδnγνγδ(

θδ − θγ)

. (8.54)

8.5 production terms for the one-dimensional euler system 159

8.5.2 Hard-sphere Collisions

Let dγ and dδ denote diameters of different hard sphere particles. The referencecollision integral is given by

Ωγδ0 = Ωγδ

(1,1) =d2

γδ

2, (8.55)

where dγδ = (dγ + dδ)/2, such that the dimensionless collision frequency is

νγδ = nδ(

θγµδ + θδµγ)1/2

, (8.56)

which for inter-species collisions simplifies to

νγγ = nγ (θγ)1/2 . (8.57)

8.5.3 Electrostatic Interaction

Let us consider the interaction between two charged particles. The differentialcross section for electrostatic interactions is given in [12] as

σEL =b2

04 sin

( χ2) , (8.58)

where

b0 =qγqδ

4πε0µg2 with µ =mγmδ

mγ + mδ. (8.59)

It is well known that the Ω-collision integrals diverge for the Coulomb potential.A standard procedure is to cut-off the Coulomb potential at the Debye lengthλD. Then the integrals for Q(l) evaluate to

Q(1) = b20 ln(1 + Λ2

γδ),

Q(2) = 2b20

(ln(1 + Λ2

γδ)−Λ2

γδ

1 + Λ2γδ

),

Q(3) = 3b20

(ln(1 + Λ3

γδ)−2Λ2

γδ(1 + 2Λ2γδ)

3(1 + Λ2γδ)

),

(8.60)


where

Λγδ =λD

0b0

=

∣∣∣∣∣12πn0(λ

D0 )3

ZγZδ

∣∣∣∣∣ with λD0 =

√ε0θe

0me

n0e2 (8.61)

is a dimensionless Debye length, see [12]. Here, we assume that Λγδ 1, whichcorresponds to a weakly coupled plasma. Finally, the Ω-collision integrals eval-uate to

Ωγδ(1,1) = Ωγδ

(1,2) =

(qγqδ

16πθε0µ

)2

ln(Λγδ),

Ωγδ(2,1) = Ωγδ

(2,2) =

(qγqδ

16πθε0µ

)2

(2 ln(Λγδ)− 1)

(8.62)

for Λγδ 1. The dimensionless collision frequency can be identified as

νγδ = nδ

(θγµδ + θδµγ

)−3/2(8.63)

and the reference collision integral is given by

Ωγδ0 =

(ZγZδ)2 log(Λγδ)

64π2n20(λ

D0 )4

. (8.64)

8.6 vlasov-poisson-bgk system

Let us consider a fully ionized plasma that consists of two species. Formally, thetwo-component kinetic plasma equations can be reduced to a single equation forthe electrons in the limit of a small mass ratio and short observation time. Thelimiting equation is given in non-dimensional form by

∂t f e + c∂x f e − E∂c f e = − f e − E( f e)

Kn,

−λ2∂xxΦ = Zini − ne, ne = 〈 f e〉,(8.65)

where E = −∂xΦ denotes the electric field and Φ is the electric potential. Theparameter λ = λD

0 /l0 is a scaled Debye length. The ion density ni is a homo-geneous background charge distribution. In the following we set Zini = 1 forsimplicity.

The Vlasov-Poisson system has become a standard test case for numericalmethods, since the evolution of the distribution function f e can develop fine-scale structures that are difficult to resolve. The Vlasov-Poisson system exhibits

8.6 vlasov-poisson-bgk system 161

a peculiar damping that is non-collisional. The simulation of this so-called Lan-dau damping with a finite Hermite expansion in velocity space only allows tocapture the damping phenomenon correctly over a finite time-interval, see [52].However, if a collision term is added, the approximation of the discretized equa-tions to the exact solution can be improved. Here we consider the simple BGKoperator. Other choices have been considered in the Literature, such as the Kirk-wood or Lenard and Bernstein model operators that approximate the Landaucollision operator, see [99].

In the following section we consider a spectral representation in Hermite poly-nomials for the linearized Vlasov-Poisson-BGK system in order to generate refer-ence solutions. Such spectral Fourier-Hermite velocity discretizations have beenconsidered by several authors, see e.g. [21, 52, 99].

8.6.1 Expansion in Hermite Polynomials

Let us consider a linearization about a state f0 given by f = f0 + ε f1, where theperturbation is given by f1 = ψ f0 with

ψ =N−1

∑k=0

αkHek(c). (8.66)

The evolution equation for ψ reads

(∂tψ + c∂xψ− E1) f0 = − f1 − E1( f0, f1)

Kn, (8.67)

where

E1( f0, f1) =He2(c)θ1 + 2

2f0, (8.68)

with

ρ1 = 〈 f1〉, v1 = 〈He1 f1〉, θ1 = 〈He2 f1〉. (8.69)

Projecting the evolution Eq. (8.67) on Hermite basis functions yields the system

∂tαj + ∂xαj−1 + (j + 1)∂xαj+1 − δi,1E1

+ Kn−1(

αi − δj,1v1 − δj,2θ1/2− δj,0ρ1

)= 0, (8.70)

for j = 0, . . . , N − 1.


Here we are interest in the temporal propagation of plane waves and set

f1 = exp (ı(ωt− kx)) , (8.71)

which yields

∂xE = −ıkE = − ρ1λ2 (8.72)

and

ω∂tαj +−kαj−1 +−k(j + 1)αj+1 −α0

λ2kδj,1

− ı Kn−1(

αj − δj,1v1 − δj,2θ12− δj,0ρ1

)= 0. (8.73)

This system can be written in matrix-vector form as

(ωI + A) α = 0, (8.74)

where

A =

0 −k 0 0 0 . . .

− 1λ2k − k 0 −k 0 0

0 −2k 0 −k 0

0 0 −3k − ıKn −k

0 0 0 −4k − ıKn

. . ....

. . .. . .

. (8.75)

The tridiagonal matrix A can be symmetrized with a similarity transformationas

A(sym) =

0√

1λ2 + k2 0 0 0 . . .√

1λ2 + k2 0

√2k 0 0

0√

2k 0√

3k 0

0 0√

3k − ıKn 2k

0 0 0 2k − ıKn

. . ....

. . .. . .

, (8.76)

see e.g. [101]. Thus for Kn → ∞ the matrix A has real eigenvalues, implyingthat the temporal evolution for a finite number of equations N does not allow


-4 -2 0 2 40.0

0.2

0.4

0.6

0.8

1.0

c

f

Maxwellian Bump-on-tail

Figure 8.3: Background Maxwellian and bump-on-tail velocity distribution functions.

for long term damping. In contrast, in the collisional case, imaginary eigenvaluescan occur for N > 3, allowing for exponential damping.

8.6.2 Numerical Examples

Let us consider both a standard normal and a bump-on-tail function for thebackground distribution f0. The bump-on-tail function is the maximum-entropy

distribution f (ME)α that is determined by the moment constraints

〈 f (ME)α 〉 = 1, 〈c f (ME)

α 〉 = 0, 〈c2 f (ME)α 〉 = 1,

〈c3 f (ME)α 〉 = 1.5, 〈c4 f (ME)

α 〉 = 4.(8.77)

The initial condition is given by

f 0(x, t) = (1 + a cos(kx)) f0(c), k =2π

L, (8.78)

where a > 0 denotes the amplitude of the perturbation, k is a wave number andL denotes the wave length. The spatial domain is given by x ∈ [−2π, 2π]. In thefollowing the wave number is set to k = 0.5.

In the limit a → 0, the evolution is described by the linearized solution de-rived above. In Figure (8.4), the temporal evolution of the normalized L2 norm ofthe electric field is shown for the initial condition of the perturbed Maxwellianvelocity distribution for different Knudsen numbers. The reference solution wasevaluated with N = 200 equations. In the collisionless case, the reference solu-tion satisfies the theoretical exponential decay rate of Landau’s analysis, given


by γ = −0.1533, see e.g. [103]. The Grad5 solution was generated using a smallperturbation with a = 10−3.

Clearly, in the collisionless case, the Grad5 system only yields a sensible ap-proximation over a very short time interval. While the Grad5 system yields aslightly improved approximation in the case Kn = 10, a further reduction toKn = 1 allows for long term damping of the Grad5 approximation, with, how-ever, a slightly too strong damping rate. Finally, for Kn = 0.1 the Grad5 approx-imation is in excellent agreement with the exact solution over the shown timeperiod.

The approximation quality can be improved by considering higher-order mo-ment systems. Let us consider the case Kn = 1 shown in Figure 8.5. While theGrad5 approximation yields a slightly wrong damping rate, the Grad7 systemclosely follows the exact solution.

Let us consider a strong perturbation with a = 0.2, giving rise to nonlineareffects. The kinetic equation is solved with the discrete velocity method presentedin Chapter 4 and the β-closure introduced in Chapter 7.

Figure 8.6 shows the damping of the electric field for the nonlinear case. Thenumerical solutions to the β-closure were computed with a second-order finite-volume method on the spatial domain Ωx = [−2π, 2π] with Nx = 1000 cellsand β = 10−3. The reference DVM solution were computed on the same spatialdomain with Nx = 200, Nc = 200 and a velocity domain given by Ωc = [−5, 5].

Similarly to the linear case, the β-closure shows a slightly too large dampingrate for Kn = 1, but agrees very well with the reference solution for Kn = 0.1 onthe plotted temporal range.

Let us now consider the bump-on-tail background velocity distribution with asmall perturbation given by a = 10−3. Figure 8.7 shows the numerical solutionscomputed with the discrete velocity method and the β-closure with the sameparameters used for the Maxwellian case. Similarly to the previous cases, theβ-closure is in excellent agreement with the reference solution for Kn = 0.1. ForKn = 1, the solution is qualitatively correct, but the damping rate of the β-closureis a bit too large. Furthermore, the numerical results shows a phase-shift errorfor the β-closure. For Kn = 10, the β-closure only agrees with the reference solu-tion for a very short time interval. The collisionless case shows a growth in theelectric field energy, until a saturation level is reached. Interestingly, even toughthe β-closure deviates from the reference kinetic solution, it is able to capture thegrowth and eventual saturation of the electric field energy qualitatively correct.


Maxwellian Problem

0 10 20 30 40 50

1

10-1

10-2

t

||E||L2/||E0 || L2

Kn=0.1

Exact Grad5

0 10 20 30 40 50

1

10-1

10-2

t

||E||L2/||E0 || L2

Kn=1

Exact Grad5

0 10 20 30 40 50

1

10-1

10-2

10-3

10-4

t

||E||L2/||E0 || L2

Kn=10

Exact Grad5

0 10 20 30 40 50

1

10-1

10-2

10-3

10-4

t

||E||L2/||E0 || L2

Kn → ∞

Exact Grad5

Figure 8.4: Evolution of the normalized L2 norm of the electric field for the perturbation ofa Maxwellian distribution in the linear case for different Knudsen numbers.


Maxwellian Problem

0 10 20 30 40 50

1

10-1

10-2

t

||E||L2/||E0 || L2

Kn=1

Exact Grad5 Grad7

Figure 8.5: Evolution of the normalized L2 norm of the electric field for the Grad5 andGrad7 systems, together with the reference solution.

Maxwellian Problem (nonlinear case)

0 10 20 30 40 50

1

10-1

10-2

t

||E||L2/||E0 || L2

Kn=0.1

DVM β-Closure

0 10 20 30 40 50

1

10-1

10-2

t

||E||L2/||E0 || L2

Kn=1

DVM β-Closure

Figure 8.6: Evolution of the normalized L2 norm of the electric field for a strong perturba-tion with a = 0.2 in Eq. (8.78).


Bump-on-Tail Problem

0 10 20 30 40 50

1

10-1

10-2

t

||E||L2/||E0 || L2

Kn=0.1

DVM β-Closure

0 10 20 30 40 50

1

10-1

10-2

t

||E||L2/||E0 || L2

Kn=1

DVM β-Closure

0 10 20 30 40 50

1

10-1

10-2

t

||E||L2/||E0 || L2

Kn=10

DVM β-Closure

0 10 20 30 40 50

102

10

1

10-1

t

||E||L2/||E0 || L2

Kn → ∞

DVM β-Closure

Figure 8.7: Evolution of the normalized L2 norm of the electric field for the bump-on-tailbackground distribution with a = 10−3 in Eq. (8.78) for different Knudsen num-bers.

9C O N C L U S I O N S

The numerical test cases of the ME35 system presented in Chapter 6 show verypromising results for gas flows in the transition regime. We have removed thesingularity of the closure by considering a bounded underlying velocity domain.As expected, a sub-shock occurs for the shock-structure problems. However, thestrength of the sub-shock can be decreased by considering larger velocity do-mains. In this thesis, the maximum-entropy closure has been applied to bulkflow problems only. In future research, the applicability of the maximum-entropyclosure to boundary value problems, such as the classical Poiseuille or Couetteflow problems should be studied.

The computational resources required to solve the maximum-entropy closurecan be excessive. In Chapter 5, efficient algorithms and implementations havebeen presented to reduce the run time. The semi-implicit time-stepping methodintroduced in Chapter 4 allows to find steady-state solutions to shock-structureproblems with a significant run time reduction, when compared to the explicitEuler time-stepping method. For smooth, time-dependent problems, the Lax-Wendroff scheme was shown to be very efficient, since it does not rely on higher-order reconstructions, which would require additional numerical solutions tothe dual minimization problem at the cell interfaces for the evaluation of thenumerical flux function. For non-smooth solutions, a hybrid method combiningthe Lax-Wendroff scheme with a first-order method could be used, see e.g. [126].

The high-performance GPU quadrature implementations presented in Chapter5 yield speed-ups of up to two orders of magnitude, when compared to a naive,serial CPU implementation. The use of a fixed quadrature grid in the velocityspace has the advantage of reduced memory and communication requirements.However, in order to capture the distribution function accurately, a fine meshmight be needed especially for low temperature flows, resulting in high compu-tational costs.

In Chapter 4, a comparison between the maximum-entropy closure and theDVM showed that the ME closure can dramatically reduce the excessive mem-ory resources required by the DVM. Since both the maximum-entropy moment

169

170 conclusions

closure and the DVM rely on the availability of a quadrature grid, a naturalextension is the combination of the DVM with the ME method, for which theME model is used as a memory compression tool in some parts of the physicaldomain.

Another interesting hybrid method between the maximum-entropy closureand the DVM is the partial moment method [36, 37, 42], which has been con-sidered in the context of radiative transport. The idea is to partition the velocitydomain into non-overlapping sub-domains. On each sub-domain, a moment pro-jection is applied, yielding equations for partial moments. A natural choice forthe local basis on each sub-domain are the collision invariants γ = (1, c, |c|2)T ,such that the resulting system contains the conservation laws. For this set of ba-sis functions, the maximum-entropy closure yields the equilibrium distributionE = exp (α · γ) on each velocity sub-domain. Since the equilibrium distributioncan be factorized, the computational complexity of the quadrature can be re-duced dramatically as shown in Chapter 4. Section A.2 presents a partial momentsystem in a simplified one-dimensional geometry with promising numerical re-sults that motivate future research of the partial moment system also for the fullythree-dimensional case.

AO T H E R C L O S U R E T H E O R I E S

a.1 multi-gaussian closures

Let us consider for simplicity a one-dimensional setting with Ωx, Ωc ⊆ R. Veloc-ity distributions based on a weighted sum of Dirac delta distributions

f (Delta)(c) =k

∑i=1

wiδ(c− ci) (A.1)

have been considered for the simulation of rarefied gas-particle flows [33, 40, 41,81]. While this closure yields explicit expressions for the closing fluxes, momentclosures based on the ansatz of sums of delta distributions have been shown tobe only weakly hyperbolic and suffer from the generation of unphysical deltashock-waves, see [24].

In order to improve the quadrature-based Ansatz (A.1), a multi-Gaussian clo-sure has been proposed in [23], which reduces in the one-dimensional setting tothe ansatz

f (MG)(ξ) =k

∑i=1

wiσM(

ξ − ξiσ

), σ > 0, (A.2)

where f (MG) denotes a reduced distribution, ξ = (c − v)/√

θ is a normalizedvelocity and M denotes the standard normal distribution, see Section 7.1.1. Notethat the ansatz reduces to the Dirac distribution in the limit of σ → 0. Thisclosure has been studied in [26] both analytically and numerically in the purelyone-dimensional geometry.

The Ansatz (A.2) has 2k + 1 degrees of freedoms. Let us apply the multi-Gaussian closure to the 5-moment system, for which the bi-Modal ansatz (k = 2)

171

172 other closure theories

yields the required degrees of freedoms. The first four moments of f (MG) arethen given by

w1 + w2 = 1,

ξ1w1 + ξ2w2 = 0,

ξ21w1 + ξ2

2w2 + (w1 + w2)σ2 = 1,

ξ1w1(ξ21 + 3σ2) + ξ2w2(ξ

22 + 3σ2) = Q.

(A.3)

Assuming σ < 1, these equations can be solved for the weights wi and abscissasξi, yielding

w1 =12

(1 +

Q√Q2 + 4z3

), w2 =

12

(1− Q√

Q2 + 4z3

),

ξ1 =Q−

√Q2 + 4z3

2z, ξ2 =

Q +√

Q2 + 4z3

2z,

(A.4)

where z = 1− σ2 ∈ (0, 1].For z = 0, i.e. σ = 1, the System (A.3) is only solvable for Q = 0, which yields

an infinite set of solutions satisfying

w1 + w2 = 1, ξ1 = ξ2 = 0. (A.5)

Since ξ1 = ξ2 = 0, the distribution f (MG) only depends on the sum w1 +w2, suchthat the solution w1 = w2 = 1/2 can be chosen without loss of generality, see also[26]. The resulting distribution corresponds to a Maxwellian f (MG)(ξ) = M(ξ).Thus, for the special case z = 0 and Q = 0, the resulting fourth moment is givenby R = 〈ξ4M〉 = 3.

Inserting the Solution (A.4) into the reduced fourth moment relation

〈ξ4 f (MG)〉 = R (A.6)

yields

R = 3 +Q2

z− 2z2 (A.7)

for z ∈ (0, 1). For the special case Q = 0, the unique solution in the interval (0, 1]is given by z =

√−∆/2, where ∆ = R− 3. For the case Q 6= 0, the function

r(z) = R− 3− Q2

z+ 2z2 (A.8)

A.1 multi-gaussian closures 173

is continuous for z ∈ (0, 1) with limz→0 r(z) = −∞ and r(1) ≥ 0 for realizablemoments (Q, R), such that there always exists a solution to Eq. (A.7). Further-more, since r′(z) = (Q/z)2 + 4z, the function r is monotonically increasing onz ∈ (0, 1], such that there exists a unique real root on (0, 1] for Q 6= 0, see also [26].

The unique real solution to the cubic polynomial in Eq. (A.8) can be stated ex-plicitly as

z =

61/3∆− k2/31

62/3k1/31

, ∆ ≥ 0,

Re(k2/32 )− 61/3∆

62/3 Re(k1/32 )

, ∆ < 0,

where

k1 =√

D− 9Q2, k2 =√

D + 9Q2 (A.9)

with

D = 6∆3 + 81Q4. (A.10)

The closing flux can be written in terms of Q and z as

S = Q

(10− 8z +

Q2

z2

). (A.11)

A series expansion about the Junk half-line yields

S =∆2

4Q+ O(Q), (A.12)

which shows that S is singular on the Junk sub-space. Furthermore, S is alsosingular in the equilibrium point: For z = Q(3+ε)/2 with ε > 0 we find

limQ→0

S(Q) = limQ→0

(Q−ε + 10Q− 8Q(5+ε)/2

)= ∞. (A.13)

On the other hand, a linearization of Eq. (A.11) around a state (Q, R) withR < 3 yields in the limit (Q, R)→ (0, 3) the expansion

S = 10Q + O(√

∆Q + Q2), (A.14)

which agrees with the classical Grad closure to first order.


In [26] a proof of the hyperbolicity of the bi-Gaussian system has been pre-sented for the near thermodynamic equilibrium limit. Due to the singularity ofthe bi-Gaussian closure on the Junk half-line, the closure can be expected to yieldqualitatively similar results as the β-closure for β → 0, which correspondents tothe closed-form closure by McDonald [90]. Some promising numerical results ofthe bi-Gaussian closure have been presented in [26].

a.2 partial moment approximations

Let us consider the kinetic equation

∂t f + c∂x f =C( f )Kn

(A.15)

in a one-dimensional setting with Ωx = [xL, xR] and Ωc = [cL, cR] for simplicity.

The velocity space Ωc is partitioned into non-overlapping subsets Ω(i)c , such that

Ωc = ∪Np

i=1Ω(i)c and Ω(i)

c ∩Ω(j)c = ∅ for i 6= j. Let φ denote a vector of polynomial

basis functions over Ωc and let

φ(i)(c) =

φ(c), c ∈ Ω(i)c

0, else(A.16)

denote the restriction of φ on the subdomain Ω(i)c for i = 1, . . . , Np.

Projecting the kinetic equation on φ(i) yields the partial moment system

∂t〈φ(i) f 〉+ ∂x〈cφ(i) f 〉 = 〈φ(i)C( f )〉Kn

, i = 1, . . . , Np. (A.17)

With the partial moments ui = 〈φ(i) f 〉, fi = 〈cφ(i) f 〉 and p(i) = 〈φ(i)C( f )〉, themoment system can be written equivalently as

∂tu(i) + ∂xf(i)(u(i)) =p(i)(u)

Kn. (A.18)

Note that the full moments can be recovered simply by summing the contribu-tions from the partial moments as

u = 〈φ f 〉. (A.19)

A.2 partial moment approximations 175

a.2.1 Conservation laws

Since ∑i φ(i) = φ, we have

∑i

p(i)(u) = 〈φC〉, (A.20)

such that the partial moment system contains the conservation laws of mass,momentum and energy if φ includes the collision invariants of the Boltzmanncollision operator.

a.2.2 Entropy Dissipation and Hyperbolicity

Applying the maximum-entropy closure on each subdomain yields the ansatz

f (ME)α(i) = exp

(φ(i) · α(i)

), (A.21)

where the Lagrange multipliers are determined by the constraints

〈φ f (ME)α(i) 〉 = u(i). (A.22)

With the notation

αs =

(α(1)T

, . . . , α(Np)T)T

, φs =

(φ(1)T

, . . . , φ(Np)T)T

, (A.23)

the maximum-entropy distribution over Ωc can be written as

f (ME)αs = exp

(αT

s φs

). (A.24)

Similarly to the full moment system, the partial system can be written in potentialform as

∂t∂α(i) h(i)∗ + ∂x∂α(i) ϕ

(i)∗ =

p(i)(us)

Kn, (A.25)

where h(i)∗ = 〈φ(i) f (ME)α(i) 〉 and ϕ

(i)∗ = 〈cφ(i) f (ME)

α(i) 〉. With

h∗ = ∑i

h(i)∗ , ϕ∗ = ∑i

ϕ(i)∗ (A.26)

we find

∂t∂αs h∗ + ∂x∂αs ϕ∗ =〈φsC〉

Kn. (A.27)


This is equivalent to the potential form of the full moment system (3.35) anda multiplication of Eq. (A.27) with αT

s from the left yields an entropy law, seeSection 3.5.2. The system can be written as

∂αsαs h∗∂tα + ∂αsαs ϕ∗∂xαs =〈φsC〉

Kn, (A.28)

where the matrix A = ∂αsαs h∗ has the block-diagonal form

A = diag(⟨

φ(1)φ(1) f (ME)α(1)

⟩, . . . ,

⟨φ(Np)φ(Np) f (ME)

α(Np )

⟩)(A.29)

with symmetric positive definite blocks. Thus, the partial moment system is hy-perbolic.

a.2.3 Numerical Example

A natural choice for φ are the collision invariants γ = (1, c, c2)T . The resulting

ansatz distribution f (ME)α on each subdomain is given by

f (ME)α(i) = exp(α(i)0 + α

(i)1 c + α

(i)2 c2). (A.30)

Let us consider the collisionless case in the limit of large Knudsen numbers. Thenthe partial moment system decouples into Np systems.

Figure A.1 shows the numerical solutions to the partial moment system for thetwo-beam problem (7.78) in the collisionless case with the initial condition ρ0 =v0 = θ0 = 1 at time T = 0.1. The spatial domain Ωx = [−0.5, 0.5] is discretizedwith Nx = 1000 cells and the velocity domain Ωc = [−5, 5] is partitioned into

Np subdomains of equal size for Np = 2, 4, 8, 16. The ansatz distribution f (ME)α is

integrated using a Gauss-Legendre quadrature with Nc = 32 quadrature nodeson each subdomain.

The numerical results show very promising results for the partial moment ap-proximations in this simple one-dimensional setting. In Figure A.2 the partialmoment approximation with two subdomains is compared to the maximum-entropy closure with N = 5 moment equations, which was solved with the samespatial discretization as the partial moment system and a velocity domain dis-cretization using Nb

c = 2 blocks and Ngc = 64 quadrature nodes per block. For

this problem, the partial and full moment system yield numerical results with asimilar approximation quality.

A.2 partial moment approximations 177

-0.4 -0.2 0.0 0.2 0.41.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

x

ρ

-0.4 -0.2 0.0 0.2 0.4-1.0

-0.5

0.0

0.5

1.0

x

v

-0.4 -0.2 0.0 0.2 0.41.0

1.5

2.0

2.5

x

θ

-0.4 -0.2 0.0 0.2 0.41.0

1.5

2.0

2.5

3.0

3.5

4.0

x

p

Exact PME(2) PME(4) PME(8) PME(16)

Figure A.1: Partial moment approximations to the collisionless two-beam problem.

-0.4 -0.2 0.0 0.2 0.4-1.0

-0.5

0.0

0.5

1.0

x

v

Exact ME5 PME(2)

-0.4 -0.2 0.0 0.2 0.41.0

1.5

2.0

2.5

3.0

3.5

4.0

x

p

Exact ME5 PME(2)

Figure A.2: Full moment approximation ME5, together with the partial moment approxi-mation PME(2) and the exact solution to the collisionless two-beam problem.


a.3 kappa distributions

The Kappa velocity distribution for a one-dimensional velocity space is given by

fκr (c) =ρ√

πθ(2κ−1r − 1)

Γ(1 + κ−1r )

Γ(2−1 + κ−1r )

(1 +

ξ2

θ(2κ−1r − 1)

)−(κ−1r +1)

,

(A.31)

where 0 < κr is defined as the reciprocal of κ defined in [100]. The parametersρ, θ correspond to the mass density and temperature, such that the first threemoments are given by

〈 fκr 〉 = ρ, 〈c fκr 〉 = 0, 〈c2 fκr 〉 = ρθ (A.32)

for κr < 2. In the limit κr → 0, the Kappa distribution degenerates into theMaxwellian

fκr→0(c) =ρ√2πθ

exp(− c2

2θ

). (A.33)

The reduced fourth moment is given by

R(κr) = 1 +4

2− 3κr, for κr <

23

. (A.34)

Figure A.3 shows the reduced fourth moment of the Kappa distribution as afunction of κr. The function R is monotonically increasing from R(κr → 0) = 3to R(κr → 2/3)→ ∞, such that for every point on the Junk half-line, there existsa parameter κr, such that the Kappa distribution satisfies the given moments.

The Kappa-distribution serves as an example of distributions defined on theunbounded velocity domain Ωc = R that are realizable on the Junk half-line.In contrast to the bi-Gaussian distribution, the Kappa-distribution decays alge-braically for κr > 0, whereas the bi-Gaussian decays exponentially.

In order to use Kappa distributions as closure theories, e.g. for the 5-momentsystem, more degrees of freedoms have to be incorporated into the Ansatz (A.31),e.g. to allow for a non-zero heat flux.

A.3 kappa distributions 179

0.01 0.1 231

3

6

12

24

48

κr

R

0 10 20 30 40

1

10-2

10-4

10-6

10-8

c

fκr

R(κr) κr → 0 κr=2/7 κr=2/5 κr=2/3

Figure A.3: Reduced fourth moment R of the κ-distribution vs κr . Note that R diverges atκr = 2/3.

BN O TAT I O N S A N D G E N E R A L R E S U LT S

b.1 tensor index notation

We denote scalar values by lower-case symbols, e.g. a, b, c. Vectors are written inbold font, e.g. a, b, c. Elements of a vector a are denoted by ai. A collection ofvectors is denoted by ai for i ∈ I , where I is an index set. Matrices are writtenin capitalized bold font, such as A, B.

In tensor index notation, vectors, matrices or general tensors are representedby their components: ai represents the vector a, Aij represents the matrix A, andAi1,...,in denotes the nth-order tensor A.

Let us consider the linear system Ax = b. In tensor index notation, this systemis equivalently expressed as

d

∑j=1

Aijxj = bi. (B.1)

Einstein introduced in [38] a short-hand notation by dropping the summationsymbol, whenever the same index repeats on one side of the equation, so thatthe equation can be written as

Aijxj = bi. (B.2)

For a detailed introduction to tensor index notation, see e.g. [108].

b.1.1 Special Tensors

The Kronecker delta is a second-order tensor defined as

δij =

1, if i = j,

0, else.(B.3)

181

182 notations and general results

The Kronecker delta can used to compute the trace of a second-order tensor aswe have δij Aij = Aii = tr A.

The Levi-Civita tensor, given by

εijk =

1, if (i, j, k) = (1, 2, 3) or (2, 3, 1) or (3, 1, 2),

−1, if (i, j, k) = (3, 2, 1) or (2, 1, 3) or (1, 3, 2),

0, else,

(B.4)

allows to write the cross product a× b = c as εijkajbk = ci.

b.2 trace-free tensors

Let Aij denote a second-order tensor. Its trace is given by Aii = tr A. The sym-metric part of Aij is given by

A(ij) =Aij + Aji

2. (B.5)

Let Sij, with 1 ≤ i, j ≤ d, denote a symmetric tensor. The deviatoric, trace-freepart of Sij is given by

S〈ij〉 = S(ij) −δij

dSii. (B.6)

For more general expressions of trace-free tensors see [114].

b.3 tensor contractions

A tensor contraction results from summing over two equal indices of a tensor.Consider e.g. the 4th-degree tensor Aijkl . The tensor Bjk = Aiijk is a second-ordertensor, resulting from summing over the first two indices of Aijkl . The second-order tensor can be further contracted to the scalar Bjj = Aiijj by summing overits diagonal elements. In general, an nth-order tensor is reduced to a tensor ofdimension n− 2s after s contractions.

b.4 moment tensors

Table B.1 defines the symbols used to denote primitive moments of a distribu-tion f in dimensional form. The thermodynamic pressure and temperature aredenoted by p = pii/dc and θ = θii/dc respectively, where dc is the dimension of

B.5 orthogonal polynomials 183

Physical quantity Symbol Definition

Number density n 〈 f 〉Mass density ρ m〈 f 〉Velocity vector vi m〈ci f 〉/ρ

Pressure tensor pij m〈CiCj f 〉Temperature tensor θij m〈CiCj f 〉/ρ

Heat flux tensor Qijk m〈CiCjCk f 〉* Rijkl m〈CiCjCkCl f 〉* Sijklm m〈CiCjCkClCm f 〉

Table B.1: Primitive moments of a distribution f .

the velocity space. The stress tensor σij and physical heat flux vector qi are givenby

σij = p〈ij〉 = pij − δij p, qi =Qijj

2. (B.7)

b.5 orthogonal polynomials

Let p = (p0, . . . , pn−1)T denote an orthogonal basis of polynomials with respect

to the inner product

〈pi, pj〉 :=∫

Ωpi pjw dΩ = δij, (B.8)

where w ≥ 0 denotes a weight function. The polynomials satisfy a three-termrecurrence relation

pi+1(x) = (x− αi)pi − βi pi−1, i = 1, 2, . . . , (B.9)

where p−1 = 0 and p0 = 1, with the coefficients

αi =〈xpi, pi〉〈pi, pi〉

, βi =〈pi, pi〉〈pi−1, pi−1〉

, (B.10)


see e.g. [45]. Normalization of the polynomials pi yields an orthonormal basisp = ( p0, . . . , pn−1), which satisfies the three-term recurrence relation

pi+1(x) =x− αiβi+1

pi −βi

βi+1pi−1, (B.11)

where αi = αi and βi =√

βi ≥ 0 as shown in [45].

The free-term recurrence relation can be written in compact form as

Tij pj(x) = xpi(x)− δi,n√

βn−1 pn(x), 0 ≤ i, j ≤ n− 1, (B.12)

where

T =

α0√

β1 0 0 0√

β1 α1√

β2 0 0

0. . .

. . .. . . 0

0 0√

βn−2 αn−2√

βn−1

0 0 0√

βn−1 αn−1

. (B.13)

For classical orthogonal polynomials, such as the Gauss-Legendre, Gauss-Her-mite or the Gauss-Laguerre polynomials, the coefficients αi, βi can be expressedin explicit form.

b.6 gauss-quadrature

The Gauss-Quadrature rule approximates the integral

I( f ) :=∫ b

aw(x) f (x) dx (B.14)

by the quadrature rule

Q( f ) :=n

∑i=1

wi f (xi), (B.15)

where xi are roots of the orthonormal polynomial

pn(x) = cnΠni=1(x− xi), cn > 0. (B.16)

B.7 jacobian matrix of the 5-moment system 185

The Gauss-Quadrature allows to integrate polynomials of degree 2n− 1 or lessexactly, such that

I(pi) = Q(pi), for i = 0, . . . , 2n− 1. (B.17)

Evaluating (B.12) at the quadrature nodes yields Tp(xk) = xkp(xk), such thatthe nodes xk are the eigenvalues of T. The symmetry of T guarantees that alleigenvalues are real.

Let T = VΛV−1 with VTV = I denote an eigenvalue decomposition of T, thenthe weights can be computed by

wi = V21,j

∫ b

aw dx, (B.18)

where V1,j is the first entry of the normalized jth eigenvector of T as shown in[48].

b.7 jacobian matrix of the 5-moment system

The Jacobian J of the 5-moment system presented in Chapter 7 is given by

J =∂F∂u

=

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

a0 a1 a2 a3 a4

. (B.19)

Due to the special structure of J, the eigenvalues are given by the roots of thequintic polynomial

p(λ) =4

∑i=0

aiλi − λ5. (B.20)

The characteristic polynomial can by normalized by the transformation

p(λ√

θ + v)

θ52

= p(λ) =4

∑i=0

aiλi − λ5, (B.21)


where

a0 =12

(−3S + 2R

∂S∂R

+ Q∂S∂Q

), a1 = 5R− 4Q

∂S∂R− 3

∂S∂Q

,

a2 =12

(5S− 4R

∂S∂R− 3Q

∂S∂Q

), a3 =

∂S∂Q

, a4 =∂S∂R

,

(B.22)

are the coefficients of the reduced Jacobian

J =

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

a0 a1 a2 a3 a4

. (B.23)

The eigenvalues of J and J are related by λi = v +√

θλi.

B I B L I O G R A P H Y

[1] N. I. Akhiezer. The Classical Moment Problem and Some Related Questions inAnalysis. Edinburgh: Oliver & Boyd, 1965.

[2] N. I. Akhiezer and M. G. Krein. Some Questions in the Theory of Moments.Translations of Mathematical Monographs. American Math. Soc., 1962.

[3] G. W. Alldredge, C. D. Hauck, and A. L. Tits. “High-order Entropy-basedClosures for Linear Transport in Slab Geometry II: A Computational Studyof the Optimization Problem.” In: SIAM J. Sci. Comput. 34.4 (2012), B361–B391.doi: 10.1137/11084772X.

[4] G. W. Alldredge et al. “Adaptive change of basis in entropy-based mo-ment closures for linear kinetic equations.” In: J. Comput. Phys. 258 (2014),pp. 489–508.doi: 10.1016/j.jcp.2013.10.049.

[5] C.-G. Ambrozie. “Multivariate truncated moments problems and maxi-mum entropy.” In: Anal. Math. Phys. 3.2 (2013), pp. 145–161.doi: 10.1007/s13324-012-0052-3.

[6] J. Au. “Lösung nichtlinearer Probleme in der erweiterten Thermodynamik.”PhD thesis. Technische Universität Berlin, 2001.

[7] J. Au and W. Weiss. “Shock Wave Structure Calculations With ExtendedThermodynamics of Many Moments.” In: Waves and Stability in ContinuousMedia, Proceedings of the 10th Conference on WASCOM 99. Ed. by V. Ciancioet al. 2001.

[8] N. Balmforth. “BGK states from the bump-on-tail instability.” In: Com-mun. Nonlinear. Sci. Numer. Simulat. 17 (2012), pp. 1989–1997.doi: 10.1016/j.cnsns.2011.07.022.

[9] T. Barth. “On discontinuous Galerkin approximations of Boltzmann mo-ment systems with Levermore closure.” In: Comput. Methods in Appl. Mech.Eng. 195.25–28 (2006). Discontinuous Galerkin Methods, pp. 3311–3330.doi: 10.1016/j.cma.2005.06.016.

[10] P. L. Bhatnagar, E. P. Gross, and M. Krook. “A Model for Collision Pro-cesses in Gases. I. Small Amplitude Processes in Charged and NeutralOne-Component Systems.” In: Phys. Rev. 94.3 (1954), pp. 511–525.doi: 10.1103/PhysRev.94.511.

187

http://dx.doi.org/10.1137/11084772X

http://dx.doi.org/10.1016/j.jcp.2013.10.049

http://dx.doi.org/10.1007/s13324-012-0052-3

http://dx.doi.org/10.1016/j.cnsns.2011.07.022

http://dx.doi.org/10.1016/j.cma.2005.06.016

http://dx.doi.org/10.1103/PhysRev.94.511

188 Bibliography

[11] G. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. 2ndedition. Oxford Engineering Science Series. Clarendon Press, 1998.

[12] J. A. Bittencourt. Fundamentals of Plasma Physics. 3rd edition. Springer-Verlag New York, 2004.

[13] G. Boillat and T. Ruggeri. “Moment equations in the kinetic theory ofgases and wave velocities.” In: Continuum Mech. Thermodyn. 9.4 (1997),pp. 205–212.doi: 10.1007/s001610050066.

[14] G. Boillat and T. Ruggeri. “On the shock structure problem for hyperbolicsystem of balance laws and convex entropy.” In: Continuum Mech. Thermo-dyn. 10.5 (1998), pp. 285–292.doi: 10.1007/s001610050094.

[15] L. Boltzmann. “Weitere Studien über das Wärmegleichgewicht unter Gas-molekülen.” In: Sitz. Math.-Naturwiss. Cl. Akad. Wiss. Wien 66 (1872), pp. 275–370.

[16] S. Boyd and L. Vandenberghe. Convex Optimization. Seventh edition. Cam-bridge University Press, 2009.

[17] S. L. Brown, P. L. Roe, and C. P. T. Groth. “Numerical solution of a 10-moment model for nonequilibrium gasdynamics.” In: 12th AIAA Compu-tational Fluid Dynamics Conference. 1995.doi: 10.2514/6.1995-1677.

[18] Z. Cai, Y. Fan, and R. Li. “Globally hyperbolic regularization of Grad’smoment system in one-dimensional space.” In: Comm. Math. Sci. 11.2(2013), pp. 547–571.doi: 10.4310/CMS.2013.v11.n2.a12.

[19] Z. Cai, Y. Fan, and R. Li. “Globally Hyperbolic Regularization of Grad’sMoment System.” In: Comm. Pure Appl. Math. 67.3 (2014), pp. 464–518.doi: 10.1002/cpa.21472.

[20] Z. Cai and R. Li. “Numerical Regularized Moment Method of ArbitraryOrder for Boltzmann-BGK Equation.” In: SIAM J. Sci. Comput. 32.5 (2010),pp. 2875–2907.doi: 10.1137/100785466.

[21] E. Camporeale et al. “On the velocity space discretization for the Vlasov—Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods.” In: Comput. Phys. Commun. 198 (2016), pp. 47–58.doi: 10.1016/j.cpc.2015.09.002.

http://dx.doi.org/10.1007/s001610050066

http://dx.doi.org/10.1007/s001610050094

http://dx.doi.org/10.2514/6.1995-1677

http://dx.doi.org/10.4310/CMS.2013.v11.n2.a12

http://dx.doi.org/10.1002/cpa.21472

http://dx.doi.org/10.1137/100785466

http://dx.doi.org/10.1016/j.cpc.2015.09.002

Bibliography 189

[22] C. Cercignani, R. Illner, and M. Pulvirenti. The Mathematical Theory of Di-lute Gases. Vol. 106. Applied Mathematical Sciences. Springer-Verlag, NewYork, 2004.

[23] C. Chalons, R. O. Fox, and M. Massot. “A multi-Gaussian quadraturemethod of moments for gas-particle fows in a LES framework.” In: Study-ing Turbulence Using Numerical Simulation Data-bases, Proceedings of the Sum-mer Program. Stanford University: Center for Turbulence Research, 2010,pp. 347–358.

[24] C. Chalons, D. Kah, and M. Massot. “Beyond pressureless gas dynam-ics: Quadrature-based velocity moment models.” In: Comm. Math. Sci. 10

(2012), pp. 1241–1272.doi: 10.4310/CMS.2012.v10.n4.a11.

[25] S. Chapman and T. G. Cowling. The Mathematical Theory of Non-uniformGases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction andDiffusion in Gases. Cambridge University Press, 1970.

[26] Y. Cheng and J. A. Rossmanith. “A class of quadrature-based moment-closure methods with application to the Vlasov-Poisson-Fokker-Plancksystem in the high-field limit.” In: J. Comput. Appl. Math. 262 (2014), pp. 384–398.doi: 10.1016/j.cam.2013.10.041.

[27] C. S. Committee on Sustaining Growth in Computing Performance, D. o. E.Telecommunications Board, and N. R. C. Physical Sciences. The Future ofComputing Performance, Game Over or Next Level? Ed. by S. H. Fuller andL. I. Millett. National Academies Press, 2011.

[28] CUDA C Programming Guide. NVIDIA Corporation. 2015.

[29] R. E. Curto and L. A. Fialkow. “Recursiveness, positivity, and truncatedmoment problems.” In: Houston J. Math. 17.4 (1991), pp. 603–635.

[30] A. Danowitz et al. “CPU DB: Recording Microprocessor History.” In: Pro-cessors 10.4 (2012).doi: 10.1145/2181796.2181798.

[31] P. Degond and B. Lucquin-Desreux. “The Asymptotics of Collision Op-erators for Two Species of Particles of Disparate Masses.” In: Math. Mod.Meth. Appl. S. 6.3 (1996), pp. 405–436.doi: 10.1142/S0218202596000158.

[32] P. Degond and B. Lucquin-Desreux. “Transport coefficients of plasmasand disparate mass binary gases.” In: Transport Theor. Stat. 25.6 (1996),pp. 595–633.doi: 10.1080/00411459608222915.

http://dx.doi.org/10.4310/CMS.2012.v10.n4.a11

http://dx.doi.org/10.1016/j.cam.2013.10.041

http://dx.doi.org/10.1145/2181796.2181798

http://dx.doi.org/10.1142/S0218202596000158

http://dx.doi.org/10.1080/00411459608222915

190 Bibliography

[33] O. Desjardins, R. O. Fox, and P. Villedieu. “A quadrature-based momentmethod for dilute fluid-particle flows.” In: J. Comput. Phys. 227 (2008).doi: 10.1016/j.jcp.2007.10.026.

[34] W. Dreyer. “Maximisation of the entropy in non-equilibrium.” In: J. Phys.A: Math. Gen. 20.18 (1987), pp. 6505–6517.doi: 10.1088/0305-4470/20/18/047.

[35] W. Dreyer, M. Junk, and M. Kunik. “On the approximation of the Fokker-Planck equation by moment systems.” In: Nonlinearity 14.4 (2001).doi: 10.1088/0951-7715/14/4/314.

[36] B. Dubroca and A. Klar. “Half-Moment Closure for Radiative TransferEquations.” In: J. Comput. Phys. 180.2 (2002), pp. 584–596.doi: 10.1006/jcph.2002.7106.

[37] B. Dubroca et al. “A half space moment approximation to the radiativeheat transfer equations.” In: Z. Angew. Math. Mech. 83.12 (2003), pp. 853–858.doi: 10.1002/zamm.200310055.

[38] A. Einstein. “Die Grundlage der allgemeinen Relativitätstheorie.” In: Ann.der Physik 49 (1916), pp. 769–822.

[39] L. C. Evans. “A survey of entropy methods for partial differential equa-tions.” In: Bull. Amer. Math. Soc. 41 (2004), pp. 409–438.doi: 10.1090/S0273-0979-04-01032-8.

[40] R. O. Fox. Computational Models for Turbulent Reacting Flows. CambridgeUniversity Press, 2003.

[41] R. O. Fox. “A quadrature-based third-order moment method for dilutegas-particle flows.” In: J. Comput. Phys. 227 (2008), pp. 6313–6350.

[42] M. Frank, B. Dubroca, and A. Klar. “Partial moment entropy approxima-tion to radiative heat transfer.” In: J. Comput. Phys. 218.1 (2006), pp. 1–18.doi: 10.1016/j.jcp.2006.01.038.

[43] K. O. Friedrichs and P. D. Lax. “Systems of Conservation Equations witha Convex Extension.” In: Proc. Nat. Acad. Sci. USA 68.8 (1971), pp. 1686–1688.doi: 10.1073/pnas.68.8.1686.

[44] C. K. Garrett, C. D. Hauck, and J. Hill. “Optimization and large scalecomputation of an entropy-based moment closure.” In: J. Comput. Phys.302 (2015), pp. 573–590.doi: 10.1016/j.jcp.2015.09.008.


http://dx.doi.org/10.1088/0305-4470/20/18/047

http://dx.doi.org/10.1088/0951-7715/14/4/314

http://dx.doi.org/10.1006/jcph.2002.7106

http://dx.doi.org/10.1002/zamm.200310055

http://dx.doi.org/10.1090/S0273-0979-04-01032-8


http://dx.doi.org/10.1073/pnas.68.8.1686


Bibliography 191

[45] W. Gautschi. “Construction of Gauss Christoffel quadrature formulas.”In: Math. Comp. 22 (1968), pp. 251–270.doi: 10.1090/S0025-5718-1968-0228171-0.

[46] S. K. Godunov. “A Difference Scheme for Numerical Solution of Discon-tinuous Solution of Hydrodynamic Equations.” In: Math. Sb. 47.3 (1959),pp. 271–306.

[47] F. Golse. “Boltzmann-Grad limit.” In: Scholarpedia 8.10 (2013), p. 9141.

[48] G. H. Golub and J. H. Welsch. “Calculation of Gauss Quadrature Rules.”In: Math. Comp. 23.106 (1969), pp. 221–230.doi: 10.2307/2004418.

[49] T. I. Gombosi. Gaskinetic theory. Ed. by J. T. Houghton, M. J. Rycroft, andA. J. Dessler. Cambridge atmospheric and space science series. CambridgeUniversity Press, 1994.

[50] H. Grad. “On the kinetic theory of rarefied gases.” In: Comm. Pure Appl.Math. 2.4 (1949), pp. 331–407.doi: 10.1002/cpa.3160020403.

[51] H. Grad. “The profile of a steady plane shock wave.” In: Comm. Pure Appl.Math. 5.3 (1952), pp. 257–300.doi: 10.1002/cpa.3160050304.

[52] F. C. Grant and M. R. Feix. “Fourir–Hermite Solutions of the Vlasov Equa-tions in the Linearized Limit.” In: Phys. Fluids 10 (1967), pp. 696–702.doi: 10.1063/1.1762177.

[53] C. P. T. Groth and J. G. McDonald. “Towards physically realizable and hy-perbolic moment closures for kinetic theory.” In: Continuum Mech. Ther-modyn. 21.6 (2009), pp. 467–493.doi: 10.1007/s00161-009-0125-1.

[54] V. K. Gupta, N. Sarna, and M. Torrilhon. “Grad’s moment equations forbinary hard sphere gas-mixtures.” In: 4th Micro and Nano Flows Conference.MNF2014 Proceedings, 2014.

[55] V. K. Gupta and M. Torrilhon. “Automated Boltzmann collision integralsfor moment equations.” In: 28th International Symposium on Rarefied GasDynamics. Vol. 1501. AIP Conference Proceedings, 2012, pp. 67–74.

[56] V. K. Gupta and M. Torrilhon. “Comparison of relaxation phenomena inbinary gas-mixtures of Maxwell molecules and hard spheres.” In: Comput.Math. Appl. 70 (2015), pp. 73–88.doi: 10.1016/j.camwa.2015.04.028.

http://dx.doi.org/10.1090/S0025-5718-1968-0228171-0

http://dx.doi.org/10.2307/2004418



http://dx.doi.org/10.1063/1.1762177

http://dx.doi.org/10.1007/s00161-009-0125-1

http://dx.doi.org/10.1016/j.camwa.2015.04.028

192 Bibliography

[57] V. K. Gupta and M. Torrilhon. “Higher order moment equations for rar-efied gas mixtures.” In: Proc. R. Soc. A 471 (2015).doi: 10.1098/rspa.2014.0754.

[58] M. Hanke-Bourgeois. Grundlagen der Numerischen Mathematik und des Wis-senschaftlichen Rechnens. Mathematische Leitfäden. B.G. Teubner Verlag,2006.

[59] A. Harten, P. D. Lax, and B. van Leer. “On Upstream Differencing andGodunov-Type Schemes for Hyperbolic Conservation Laws.” In: SIAMRev. 25.1 (1983), pp. 35–61.doi: 10.1137/1025002.

[60] C. D. Hauck, C. D. Levermore, and A. L. Tits. “Convex Duality andEntropy-Based Moment Closures: Characterizing Degenerate Densities.”In: SIAM J. Control Opt. 47.4 (2008), pp. 1977–2015.doi: 10.1137/070691139.

[61] M. I. Hoffert and H. Lien. “Quasi-One-Dimensional Nonequilibrium GasDynamics of Partially Ionized Two-Temperature Argon.” In: Phys. Fluids10 (1967).doi: 10.1063/1.1762356.

[62] L. H. Holway. “New statistical models for kinetic theory: Methods of con-struction.” In: Phys. Fluids 9 (1966), pp. 1658–1673.

[63] P. Jenny, M. Torrilhon, and S. Heinz. “A solution algorithm for the fluiddynamic equations based on a stochastic model for molecular motion.”In: J. Comput. Phys. 229.4 (2010), pp. 1077–1098.doi: 10.1016/j.jcp.2009.10.008.

[64] M. Junk. “Domain of Definition of Levermore’s Five-Moment System.” In:J. Stat. Phys. 93.5-6 (1998), pp. 1143–1167.doi: 10.1023/B:JOSS.0000033155.07331.d9.

[65] M. Junk. “Maximum entropy for reduced moment problems.” In: Math.Models Methods Appl. Sci. 10.7 (2000), pp. 1001–1025.doi: 10.1142/S0218202500000513.

[66] M. Junk and A. Unterreiter. “Maximum entropy moment systems andGalilean invariance.” In: Continuum Mech. Thermodyn. 14.6 (2002), pp. 563–576.doi: 10.1007/s00161-002-0096-y.

[67] P. Kauf. “Multi-scale approximation models for the Boltzmann equation.”PhD thesis. ETH Zürich, 2011.

http://dx.doi.org/10.1098/rspa.2014.0754

http://dx.doi.org/10.1137/1025002

http://dx.doi.org/10.1137/070691139

http://dx.doi.org/10.1063/1.1762356


http://dx.doi.org/10.1023/B:JOSS.0000033155.07331.d9

http://dx.doi.org/10.1142/S0218202500000513

http://dx.doi.org/10.1007/s00161-002-0096-y

Bibliography 193

[68] J. Koellermeier, R. P. Schaerer, and M. Torrilhon. “A framework for hy-perbolic approximation of kinetic equations using quadrature-based pro-jection methods.” In: Kinet. Relat. Mod. 7.3 (2014), pp. 531–549.doi: 10.3934/krm.2014.7.531.

[69] M. G. Krein and A. A. Nudel’man. The Markov Moment Problem and Ex-tremal Problems. Vol. 50. Transl. Math. Monographs. American Math. Soc.,1977.

[70] G. M. Kremer. An Introduction to the Boltzmann Equation and Transport Pro-cesses in Gases. Springer-Verlag Berlin Heidelberg, 2010.

[71] A. Kurganov and E. Tadmor. “Solution of two-dimensional Riemann prob-lems for gas dynamics without Riemann problem solvers.” In: Numer.Methods Partial Differential Equations 18.5 (2002), pp. 584–608.doi: 10.1002/num.10025.

[72] O. E. Lanford III. “Dynamical Systems.” In: ed. by J. Moser. Vol. 38. Lec-ture Notes in Physics. Berlin, Heidelberg: Springer Berlin Heidelberg,1975. Chap. Time evolution of large classical systems, pp. 1–111.doi: 10.1007/3-540-07171-7_1.

[73] P. Le Tallec and J.-P. Perlat. Numerical analysis of Levermore’s moment system.Rapport de recherche 3124. INRIA, 1997.

[74] R. J. LeVeque. Numerical Methods for Conservation Laws. Ed. by O. E. Lan-ford. Lectures in Mathematics. ETH Zürich. Birkhäuser, 1992.

[75] C. D. Levermore. “Moment closure hierarchies for kinetic theories.” In: J.Stat. Phys. 83.5 (1996), pp. 1021–1065.doi: 10.1007/BF02179552.

[76] E. Lindholm et al. “NVIDIA Tesla: A Unified Graphics and ComputingArchitecture.” In: IEEE Micro 28.2 (2008), pp. 39–55.doi: 10.1109/MM.2008.31.

[77] J. Liouville. “Note sur la Théorie de la Variation des constantes arbi-traires.” In: J. Math. Pures Appl. 3 (1838), pp. 342–349.

[78] A. Lonkar, F. Palacios, and R. MacCormack. “Multidimensional Simula-tion of Plasma in Argon through a Shock in Hypersonic Flow.” In: 43rdAIAA Thermophysics Conference. 2012.doi: 10.2514/6.2012-3105.

[79] R. W. MacCormack et al. “Plasmadynamic Simulations with Strong ShockWaves.” In: 42nd AIAA Plasmadynamics and Lasers Conference. 2011.

[80] T. E. Magin and G. Degrez. “Transport properties of partially ionized andunmagnetized plasmas.” In: Phys. Rev. E 70.4 (2004), p. 046412.doi: 10.1103/PhysRevE.70.046412.

http://dx.doi.org/10.3934/krm.2014.7.531

http://dx.doi.org/10.1002/num.10025

http://dx.doi.org/10.1007/3-540-07171-7_1

http://dx.doi.org/10.1007/BF02179552

http://dx.doi.org/10.1109/MM.2008.31

http://dx.doi.org/10.2514/6.2012-3105

http://dx.doi.org/10.1103/PhysRevE.70.046412

194 Bibliography

[81] D. Marchisio and R. Fox. “Solution of population balance equations usingthe direct quadrature method of moments.” In: J. Aerosol Sci. 36 (2005),pp. 43–73.

[82] J. C. Maxwell. “Illustrations of the dynamical theory of gases. Part I. Onthe motions and collisions of perfectly elastic spheres.” In: ed. by D. Brew-ster et al. Vol. 19. 4th series. Philosophical Magazine and Journal of Sci-ence, 1860, pp. 19–32.

[83] J. C. Maxwell. “Illustrations of the dynamical theory of gases. Part II. Onthe process of diffusion of two or more kinds of moving particles amongone another.” In: ed. by D. Brewster et al. Vol. 20. 4th series. PhilosophicalMagazine and Journal of Science, 1860, pp. 21–37.

[84] J. C. Maxwell. “On the Dynamical Theory of Gases.” In: Phil. Trans. R. Soc.Lond. 157 (1867), pp. 49–88.doi: 10.1098/rstl.1867.0004.

[85] J. C. Maxwell. “On stresses in rarefied gases arising from inequalities oftemperature.” In: Phil. Trans. R. Soc. (London) 170 (1879), pp. 231–256.

[86] J. G. McDonald. “Extended fluid-dynamic modelling for numerical solu-tion of micro-scale flows.” PhD thesis. University of Toronto, 2011.

[87] J. G. McDonald and C. P. T. Groth. “Numerical modeling of micron-scaleflows using the Gaussian moment closure.” In: 35th AIAA Fluid DynamicsConference and Exhibit. 2005.doi: 10.2514/6.2005-5035.

[88] J. G. McDonald and C. P. T. Groth. “Extended fluid-dynamic model formicron-scale flows based on Gaussian moment closure.” In: 46th AIAAAerospace Sciences Meeting and Exhibit. 2008.doi: 10.2514/6.2008-691.

[89] J. G. McDonald and C. P. T. Groth. “Towards realizable hyperbolic mo-ment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution.” In: Continuum Mech. Thermodyn. 25.5 (2013), pp. 573–603.doi: 10.1007/s00161-012-0252-y.

[90] J. G. McDonald and M. Torrilhon. “Affordable Robust Moment Closuresfor CFD Based on the Maximum-Entropy Hierarchy.” In: J. Comput. Phys.251 (2013), pp. 500–523.doi: 10.1016/j.jcp.2013.05.046.

[91] L. Mieussens. “Convergence of a discrete-velocity model for the Boltzmann-BGK equation.” In: Comput. Math. Appl. 41.1 (2001), pp. 83–96.doi: 10.1016/S0898-1221(01)85008-2.

http://dx.doi.org/10.1098/rstl.1867.0004

http://dx.doi.org/10.2514/6.2005-5035

http://dx.doi.org/10.2514/6.2008-691

http://dx.doi.org/10.1007/s00161-012-0252-y


http://dx.doi.org/10.1016/S0898-1221(01)85008-2

Bibliography 195

[92] L. Mieussens. “Discrete Velocity Model and Implicit Scheme for the BGKEquation of Rarefied Gas Dynamics.” In: Math. Mod. Meth. Appl. S. 10.8(2000), pp. 1121–1149.doi: 10.1142/S0218202500000562.

[93] L. Mieussens. “Discrete-Velocity Models and Numerical Schemes for theBoltzmann-BGK Equation in Plane and Axisymmetric Geometries.” In: J.Comput. Phys. 162.2 (2000), pp. 429–466.doi: 10.1006/jcph.2000.6548.

[94] G. E. Moore. “Cramming more components onto integrated circuits.” In:Electronics 38.8 (1965).

[95] I. Müller and T. Ruggeri. Rational Extended Thermodynamics. Vol. 37. SpringerTracts in Natural Philosophy. Springer-Verlag New York, 1993.

[96] I. Müller and T. Ruggeri. Rational Extended Thermodynamics. 2nd edition.Vol. 37. Springer Tracts in Natural Philosophy. Springer New York, 1998.doi: 10.1007/978-1-4612-2210-1.

[97] J. Nickolls et al. “Scalable Parallel Programming with CUDA.” In: Queue6.2 (2008), pp. 40–53.doi: 10.1145/1365490.1365500.

[98] E. Oran, C. Oh, and B. Cybyk. “Direct Simulation Monte Carlo: RecentAdvances and Applications.” In: Annu. Rev. Fluid Mech. 30.1 (1998), pp. 403–441.doi: 10.1146/annurev.fluid.30.1.403.

[99] J. T. Parker and P. J. Dellar. “Fourier—Hermite spectral representation forthe Vlasov—Poisson system in the weakly collisional limit.” In: J. PlasmaPhys. 81.2 (2015).doi: 10.1017/S0022377814001287.

[100] V. Pierrard and M. Lazar. “Kappa Distributions: Theory and Applicationsin Space Plasmas.” In: Sol. Phys. 267.1 (2010), pp. 153–174.doi: 10.1007/s11207-010-9640-2.

[101] A.-H. Qassem. “Tridiagonal matrices and the computation of Gaussianquadratures.” In: Int. J. Pure Appl. Math. 55.3 (2009), pp. 301–310.

[102] D. Rossinelli. “Multiresolution Flow Simulations on Multi/Many-CoreArchitectures.” PhD thesis. ETH Zürich, 2011.doi: 10.3929/ethz-a-006483930.

[103] J. A. Rossmanith and D. C. Seal. “A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equa-tions.” In: J. Comput. Phys. 230.16 (2011), pp. 6203–6232.doi: 10.1016/j.jcp.2011.04.018.

http://dx.doi.org/10.1142/S0218202500000562

http://dx.doi.org/10.1006/jcph.2000.6548

http://dx.doi.org/10.1007/978-1-4612-2210-1

http://dx.doi.org/10.1145/1365490.1365500

http://dx.doi.org/10.1146/annurev.fluid.30.1.403

http://dx.doi.org/10.1017/S0022377814001287

http://dx.doi.org/10.1007/s11207-010-9640-2

http://dx.doi.org/10.3929/ethz-a-006483930


196 Bibliography

[104] T. Ruggeri. “Breakdown of shock-wave-structure solutions.” In: Phys. Rev.E 47.6 (1993), pp. 4135–4140.doi: 10.1103/PhysRevE.47.4135.

[105] T. Ruggeri. “The Entropy Principle from Continuum Mechanics to Hyper-bolic Systems of Balance Laws: The Modern Theory of Extended Thermo-dynamics.” In: Entropy 10.3 (2008), pp. 319–333.doi: 10.3390/e10030319.

[106] S. Saini, J. Chang, and H. Jin. “High Performance Computing Systems.Performance Modeling, Benchmarking and Simulation: 4th InternationalWorkshop.” In: Springer International Publishing, 2014. Chap. Perfor-mance Evaluation of the Intel Sandy Bridge Based NASA Pleiades UsingScientific and Engineering Applications, pp. 25–51.doi: 10.1007/978-3-319-10214-6_2.

[107] N. Satish et al. “Can traditional programming bridge the Ninja perfor-mance gap for parallel computing applications?” In: Proceedings of the39th Annual International Symposium on Computer Architecture. Portland,OR: IEEE, 2012, pp. 440–451.doi: 10.1109/ISCA.2012.6237038.

[108] H. Schade and K. Neemann. Tensoranalysis. 3., überarbeitete Auflage. Wal-ter de Gruyter, 2009.

[109] R. P. Schaerer, P. Bansal, and M. Torrilhon. “Efficient algorithms and im-plementations of entropy-based moment closures for rarefied gases.” In:J. Comput. Phys. 340 (2017), pp. 138–159.doi: 10.1016/j.jcp.2017.02.064.

[110] R. P. Schaerer and M. Torrilhon. “A new closure mitigating the sub-shockphenomenon in the continuous shock-structure problem.” In: PAMM. Vol. 13.1. 2013, pp. 499–500.doi: 10.1002/pamm.201310242.

[111] R. P. Schaerer and M. Torrilhon. “On Singular Closures for the 5-MomentSystem in Kinetic Gas Theory.” In: Commun. Comput. Phys. 17.2 (2015),pp. 371–400.doi: 10.4208/cicp.201213.130814a.

[112] R. P. Schaerer and M. Torrilhon. “The 35-Moment System with the Maximum-Entropy Closure for Rarefied Gas Flows.” In: Eur. J. Mech. B-Fluid 64

(2017), pp. 30–40.doi: 10.1016/j.euromechflu.2017.01.003.

[113] E. M. Shakhov. “Generalization of the Krook kinetic relaxation equation.”In: Fluid Dyn. 3.5 (1968), pp. 95–96.doi: 10.1007/BF01029546.

http://dx.doi.org/10.1103/PhysRevE.47.4135

http://dx.doi.org/10.3390/e10030319

http://dx.doi.org/10.1007/978-3-319-10214-6_2

http://dx.doi.org/10.1109/ISCA.2012.6237038


http://dx.doi.org/10.1002/pamm.201310242

http://dx.doi.org/10.4208/cicp.201213.130814a

http://dx.doi.org/10.1016/j.euromechflu.2017.01.003

http://dx.doi.org/10.1007/BF01029546

Bibliography 197

[114] H. Struchtrup. Macroscopic Transport Equations for Rarefied Gas Flows: Ap-proximation Methods in Kinetic Theory. Springer Berlin Heidelberg, 2005.doi: 10.1007/3-540-32386-4.

[115] H. Struchtrup. “Maxwell boundary condition and velocity dependent ac-commodation coefficient.” In: Phys. Fluids 25.11 (2013).doi: 10.1063/1.4829907.

[116] H. Struchtrup and M. Torrilhon. “Regularization of Grad’s 13 momentequations: Derivation and linear analysis.” In: Phys. Fluids 15.9 (2003),pp. 2668–2680.doi: 10.1063/1.1597472.

[117] Y. Suzuki et al. “Computational Fluid Dynamics 2004: Proceedings of theThird International Conference on Computational Fluid Dynamics, IC-CFD3, Toronto, 12–16 July 2004.” In: ed. by C. Groth and D. W. Zingg.Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. Chap. Applicationof the 10-Moment Model to MEMS Flows, pp. 529–534.doi: 10.1007/3-540-31801-1_75.

[118] E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics: APractical Introduction. 3rd edition. Springer-Verlag Berlin Heidelberg, 2009.doi: 10.1007/978-3-540-49834-6.

[119] M. Torrilhon, J. Au, and H. Struchtrup. “Explicit fluxes and productionsfor large systems of the moment method based on extended thermody-namics.” In: Continuum Mech. Thermodyn. 15.1 (2003), pp. 97–111.doi: 10.1007/s00161-002-0107-z.

[120] M. Torrilhon. “Hyperbolic Moment Equations in Kinetic Gas Theory Basedon Multi-Variate Pearson-IV-Distributions.” In: Commun. Comput. Phys.7.4 (2010), pp. 639–673.doi: 10.4208/cicp.2009.09.049.

[121] M. Torrilhon. “Krylov–Riemann Solver for Large Hyperbolic Systems ofConservation Laws.” In: SIAM J. Sci. Comput. 34.4 (2012), A2072–A2091.doi: 10.1137/110840832.

[122] M. Torrilhon. “Convergence Study of Moment Approximations for Bound-ary Value Problems of the Boltzmann-BGK Equation.” In: Commun. Com-put. Phys. 18 (Sept. 2015), pp. 529–557.doi: 10.4208/cicp.061013.160215a.

[123] M. Torrilhon. “Modeling Nonequilibrium Gas Flow Based on MomentEquations.” In: Annual Review of Fluid Mechanics. Ed. by S. Davis and P.Moin. Vol. 48. Annual Reviews, USA, 2016, pp. 429–458.doi: 10.1146/annurev-fluid-122414-034259.

http://dx.doi.org/10.1007/3-540-32386-4

http://dx.doi.org/10.1063/1.4829907

http://dx.doi.org/10.1063/1.1597472

http://dx.doi.org/10.1007/3-540-31801-1_75

http://dx.doi.org/10.1007/978-3-540-49834-6

http://dx.doi.org/10.1007/s00161-002-0107-z

http://dx.doi.org/10.4208/cicp.2009.09.049

http://dx.doi.org/10.1137/110840832

http://dx.doi.org/10.4208/cicp.061013.160215a

http://dx.doi.org/10.1146/annurev-fluid-122414-034259

198 Bibliography

[124] M. Torrilhon and H. Struchtrup. “Regularized 13-Moment-Equations: ShockStructure Calculations and Comparison to Burnett Models.” In: J. FluidMech. 513 (2004), pp. 171–198.doi: 10.1017/S0022112004009917.

[125] M. Torrilhon and H. Struchtrup. “Boundary conditions for regularized13-moment-equations for micro-channel-flows.” In: J. Comput. Phys. 227.3(2008), pp. 1982–2011.doi: 10.1016/j.jcp.2007.10.006.

[126] F. D. Vuyst. “Stable and accurate hybrid finite volume methods based onpure convexity arguments for hyperbolic systems of conservation law.”In: J. Comput. Phys. 193.2 (2004), pp. 426–468.doi: 10.1016/j.jcp.2003.07.028.

[127] W. Weiss. “Continuous shock structure in extended Thermodynamics.”In: Phys. Rev. E 52.6 (1995), pp. 5760–5763.doi: 10.1103/PhysRevE.52.R5760.

[128] S. Williams, A. Waterman, and D. Patterson. “Roofline: An Insightful Vi-sual Performance Model for Multicore Architectures.” In: Commun. ACM52.4 (2009), pp. 65–76.doi: 10.1145/1498765.1498785.

http://dx.doi.org/10.1017/S0022112004009917



http://dx.doi.org/10.1103/PhysRevE.52.R5760

http://dx.doi.org/10.1145/1498765.1498785

Bibliography 199

C U R R I C U L U M V I TÆ

personal details

Full Name Roman Pascal Schärer

Date of Birth October 5, 1985 in Baden, Switzerland

Citizenship Swiss

education

2012–2016 Doctoral studies at RWTH Aachen University

2012 M.Sc. in Computational Sciene and Engineering,ETH Zürich

2011 Master thesis: Numerical Solutions of Kinetic Equations forPlasma Applications, RWTH Aachen University

2010 B.Sc. in Computational Science and Engineering,ETH Zürich

2010 Bachelor thesis: A Fast Poisson Solver for Complex Geome-tries, ETH Zürich

2006-2012 Studies in Computational Science and Engineering atETH Zürich

2006 Matura, Cantonal School, Baden, Switzerland

Entropy-based Moment Methods for Rarefied Gases and Plasmas · Second, the Hessian matrix of the Newton method used in the dual minimization problem for the Lagrange parameters is

Documents