Hyperbolic Approximation of Kinetic Equations Using Quadrature … · 2014. 8. 4. · Master Thesis Hyperbolic Approximation of Kinetic Equations Using Quadrature-Based Projection

Master Thesis

Hyperbolic Approximation of Kinetic EquationsUsing Quadrature-Based Projection Methods

Julian Kollermeier

RWTH Aachen University

Supervisor: Prof. Dr. Manuel TorrilhonCenter for Computational Engineering ScienceMathematics DivisionRWTH Aachen University

ii

Abstract

We derive hyperbolic PDE systems for the solution of the Boltzmann equation. First, wetransform the velocity in a highly non-linear way to allow for a physical adaptivity of themethod. The unknown distribution function is then approximated by the equilibriumMaxwellian times a series of orthogonal basis functions.

The standard continuous projection method for this approach yields a PDE systemfor the basis coefficients that is in general not hyperbolic. To overcome this problem, weapply quadrature-based projection methods which modify the structure of the systemin the desired way so that we end up with a hyperbolic system of equations.

With the help of a new abstract framework, we derive conditions such that theemerging system ist hyperbolic and give a proof of hyperbolicity for a Hermite ansatz inone dimension.

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iv

Acknowledgment

The following thesis was written between April and September 2013 at MathCCES,RWTH Aachen University in partial fulfillment of the requirements for the degree Masterof Science in Computational Engineering Science.

I would hereby like to thank all the people involved in the work on my thesis andduring my studies helping me to achieve the goals I pursued and to finish on time.

First of all, I want to thank my supervisor Manuel Torrilhon for helpful advicesearlier on during the past years as well as for his enduring support and encouragementsfrom the proposal of the topic until the final version of this thesis. The many fruitfuldiscussions and impulses helped a lot in focussing on the important parts of the workand still left the necessary freedom for own ideas and developments.

Many thanks also go to my colleagues at MathCCES, especially to my advisor RomanSchaerer who always patiently listened to my questions and deliberations and repliedwith useful hints and ideas for further investigations.

Thanks to Claudia, Marc and Marcus for proofreading my thesis. It must have beena tough job.

I do not want to miss my fellow students in the course of Computational EngineeringScience (CES) as well as my old and new friends in Aachen and elsewhere, who provedthemselves a valuable support during the years of my studies and especially in the pastfew months.

Special thanks I would like to give to my parents and my family, who always animatedme to go my own way and gave me a good deal of curiosity to take along that helps mewhichever topic I work on.

Last but not least, I convey grateful thanks to the Friedrich-Naumann Foundationfor Freedom, who greatly helped me during my studies with a fellowship of large valuefor myself. Apart from the financial support, it opened up new possibilities for me that Ihad never thought of and gave me the freedom and independence I appreciate so much.

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vi

Contents

Abstract iii

Acknowledgment v

Contents ix

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aims of the Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Boltzmann Transport Equation 52.1 Basics of Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Knudsen Number, Applications and Effects of Gas Rarefaction . . 52.1.2 Phase Space and Probability Density Function . . . . . . . . . . . 72.1.3 Macroscopic Quantities of the Flow Field . . . . . . . . . . . . . . 7

2.2 Properties of the Boltzmann Transport Equation . . . . . . . . . . . . . . 82.2.1 Equilibrium Distribution . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Direct Simulation Monte Carlo . . . . . . . . . . . . . . . . . . . . 102.3.2 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . . . . 122.3.4 Discrete Velocity Method . . . . . . . . . . . . . . . . . . . . . . . 12

3 Mathematical Properties 153.1 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Standard Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 Orthonormal Hermite Polynomials . . . . . . . . . . . . . . . . . . 17

3.2 Laguerre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 Generalized Laguerre Polynomials . . . . . . . . . . . . . . . . . . 18

3.3 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.1 Cartesian Spherical Harmonics . . . . . . . . . . . . . . . . . . . . 23

3.4 Jacobi Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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viii Table of contents

3.5 Gaussian Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5.1 Gauss-Hermite Quadrature . . . . . . . . . . . . . . . . . . . . . . 26

3.5.2 Gauss-Laguerre Quadrature . . . . . . . . . . . . . . . . . . . . . . 263.5.3 Generalized Gauss-Laguerre Quadrature . . . . . . . . . . . . . . . 27

3.5.4 Non-Classical Quadrature . . . . . . . . . . . . . . . . . . . . . . . 273.5.5 Interpolation Property and Aliasing . . . . . . . . . . . . . . . . . 29

4 Motivational Examples 31

4.1 One-Dimensional Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.1 Simple Kinetic Equation . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.2 Generalized Kinetic Equation c2 . . . . . . . . . . . . . . . . . . . 334.1.3 Generalized Kinetic Equation c+ c2 . . . . . . . . . . . . . . . . . 35

4.1.4 Relation between Quadrature Projection and DVM . . . . . . . . . 364.2 Multi-Dimensional Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Simple Kinetic Equation 3D . . . . . . . . . . . . . . . . . . . . . . 384.2.2 Generalized Kinetic Equation c2i 2D . . . . . . . . . . . . . . . . . 40

5 Theoretical Concepts 455.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.1 Variable Transformation for Physical Adaptivity . . . . . . . . . . 465.1.2 Derivation of Transformed Boltzmann Equation . . . . . . . . . . . 47

5.1.3 Expansion Using Basis Functions . . . . . . . . . . . . . . . . . . . 485.1.4 Compatibility Conditions . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.4.1 1D Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.1.4.2 3D Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1.4.3 Coupling of Compatibility Conditions to PDE System . . 515.2 Theoretical Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2.1 Generalized Kinetic Equation . . . . . . . . . . . . . . . . . . . . . 545.2.1.1 Discrete Velocity Method . . . . . . . . . . . . . . . . . . 54

5.2.1.2 Quadrature-based Projection . . . . . . . . . . . . . . . . 555.2.2 Shifted Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . 57

5.2.2.1 Discrete Velocity Method . . . . . . . . . . . . . . . . . . 575.2.2.2 Quadrature-Based Projection . . . . . . . . . . . . . . . . 59

5.2.2.3 Hyperbolicity of Shifted BTE and Hermite Ansatz . . . . 625.2.3 Fully Transformed Boltzmann Equation . . . . . . . . . . . . . . . 65

5.2.3.1 Discrete Velocity Method . . . . . . . . . . . . . . . . . . 655.2.3.2 Quadrature-Based Projection . . . . . . . . . . . . . . . . 675.2.3.3 Hyperbolicity of Transformed BTE and Hermite Ansatz . 69

5.2.4 Relation to the Conservation Laws . . . . . . . . . . . . . . . . . . 73

5.2.5 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Conclusion 77

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Table of contents ix

A Appendix: Compatibility Conditions 3D 79A.1 Hermite Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79A.2 Spherical Harmonics Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . 80

References 81

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Chapter 1

Introduction

1.1 Motivation

Kinetic equations are the basis for many different applications and are widely used inindustrial and scientific fields. Especially for rarefied flows they provide an accuratesetting for the successful solution of important numerical simulations. In Chapter 2 wewill see that there are more or less distinct regions of the flow in which the applicationof standard fluid dynamic models like the Euler or Navier-Stokes equations is notappropriate for a physical solution. One then has to apply more advanced kinetic modelsthat are motivated directly by kinetic equations.

A standard method proposed by Grad in [10] derives equations for the macro-scopic flow variables like density, velocity and temperature of the flow by expandingthe unknown distribution function of the Boltzmann equation in a Hermite series. Thedrawback of this rather simple method is that the resulting system of partial differentialequations (PDEs) can loose hyperbolicity for certain values of higher moments. The lossof global hyperbolicity is a serious problem, because hyperbolicity is needed for physicalsolutions and stability of the solution in particular. The admissible region of variables forhyperbolicity of the system in fact becomes smaller for higher accuracy of the methods,as shown by Cai in [5].

There are some methods for which it can be shown under certain conditions thatthey are hyperbolic in special cases like one-dimensional flows. One of those is basedon multi-variate Pearson-IV-Distributions and was proposed by Torrilhon in [23].Another method to achieve hyperbolic equations has been published by Levermore in[18], but this method is unfortunately not given in analytical form.

In [5] Cai et al. have successfully performed a regularization of Grad’s momentsystem in one dimension that is globally hyperbolic. They essentially derived the char-acteristic polynomial of the corresponding matrix analytically and used this informationto set certain variables or entries in the matrix to zero so that the new characteristicpolynomial has real roots and the system becomes hyperbolic for all values of thevariables involved.

The approach by Cai et al. gives only limited insight into the underlying theoretical

1

2 Introduction

foundations of this regularization and it is not really clear how to generalize the procedureto similar problems.

Another important question is the possibility of an efficient numerical simulation.The velocity can usually attain very large values leading to the necessity of a very finediscretization in the velocity space with many unknowns. Recent developments by Kaufin [17] show a way to circumvent these problems at the expense of a more difficult PDEinvolving additional terms.

Our concrete question for this thesis is therefore:

Is it possible to set up a general framework for the derivation of efficient, yet stableand hyperbolic systems of PDEs for the solution of kinetic equations such as the

Boltzmann equation?

1.2 Aims of the Project

As specified above, the main part of this thesis is concerned with the setup of a generalframework to derive hyperbolic PDEs for the solution of the Boltzmann equation. Withthe help of this framework it should be feasible to decide about the hyperbolicity of theemerging system a priori before inserting a special ansatz and performing projections ofthe equation, just by the specific choice of the ansatz and the projection method.

We want to investigate the use of quadrature-based projection methods in particularand analyze the application of those methods with respect to the effects on the structureof the equations as well as on the eigenvalues of the system matrix, which is closelyrelated to the hyperbolicity of the system.

The framework should include these quadrature-based projections and give concreteconditions under which the system will be hyperbolic.

After the framework has been set up, application to some choices of important classesof functions for the expansion together with related quadrature methods should yieldresulting systems that are hyperbolic.

1.3 Overview

After this short introduction, Chapter 2 is dedicated to basic properties of the Boltzmannequation and the kinetic approach in contrast to other existing methods. The probabilitydensity function f is also introduced and we show how f and the Boltzmann equationare related to macroscopic quantities of the flow field.

In Chapter 3 the most important mathematical preliminaries that are used in thefollowing sections are explained. This includes normalized versions of Hermite andLaguerre polynomials as well as spherical harmonics. A large part of the chapter coversthe foundations of quadrature methods, especially Gauss-quadrature for the involvedpolynomials.

Some early investigations are summarized in a chapter about motivational examples,see Chapter 4. We here apply quadrature-based projections to simple kinetic equations

Introduction 3

and explain the difference to exact projections. 1D as well as multi-dimensional examplesshow the desirable properties of the emerging systems for the basis coefficients and helpfor a better understanding and the developments in the next chapter.

The main part of this thesis is presented in Chapter 5, where we first derive theformulation of the Boltzmann equation under a non-linear transformation of the velocityvariable that allows for efficient simulations. Next is the development of the conceptualframework for the derivation of conditions to achieve hyperbolicity. This part coversdifferent kinetic equations and draws an analogy between the discrete velocity methodand the quadrature-based projections. Using the mathematical properties and theframework developed before, we also give here a proof for hyperbolicity of the regularizedsystem in the case of the transformed one-dimensional Boltzmann equation with Hermiteansatz and Gauss-Hermite quadrature.

We summarize the results in Chapter 6 and discuss future work on the project.

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Chapter 2

The Boltzmann TransportEquation

Before we turn our attention towards the detailed mathematical discussion and thedifferent models that we want to develop, we should first explain the Boltzmann equationitself and the context it is used in. We therefore describe the kinetic setting, importantproperties of the Boltzmann equation and the most common solution methods togetherwith their benefits and drawbacks.

2.1 Basics of Kinetic Theory

In standard fluid dynamics the fluid is modeled as a continuum meaning that the atomsand molecules or particles as we will call them from now on in general are in constantcontact with the other particles. This is obviously valid for a fluid, which usually alsohas a relatively large density. When it comes to rarefied gases, particles do only interactrarely and are in free flight for the most of the time. This fact can be related to lowdensity, for example for low ambient pressures. At this point, the flow behavior is moreand more influenced by binary collisions of the particles. It is therefore required tomodel the interactions between individual particles in a different way than the standardcontinuum approach. In the following chapter we want to explain the viewpoint of kinetictheory and briefly explain the most important properties and basic terms related to thispoint of view. For more information about kinetic theory, we recommend the textbook[22] by Struchtrup. An approach from the engineering viewpoint can be found in thebook by Heinz [12].

2.1.1 Knudsen Number, Applications and Effects of Gas Rarefaction

Many flow problems can be characterized by small or moderate velocities and ambientconditions. However, the advanced technical capabilities made it possible to reach moreextreme values for all parameters involved. In order to further distinguish different flowregimes, the Knudsen number Kn was introduced. It is the quotient of the mean free

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6 Boltzmann Transport Equation

path length λ, e.g. the average distance a particle travels between two collisions, and acharacteristic length L of the flow problem, e.g. the size of a plane or the diameter of apipe. The definition of the Knudsen number Kn reads

Kn =λ

L. (2.1)

As a dimensionless flow parameter, the Knudsen number is an important quantitythat influences the behavior of the flow. Standard models like the Navier-Stokes equa-tions or the Euler equations are only valid for very small Knudsen numbers, becausethey rely on the assumption of a continuum in the so-called equilibrium.

According to Kauf [17] and Struchtrup [22] , the Knudsen number can be usedto roughly divide the flow field into different regimes as follows:

• Kn ≤ 0.01: equilibrium or hydrodynamic regime, which is accurately described bythe Navier-Stokes equations;

• 0.01 ≤ Kn ≤ 0.1: slip flow regime, where the Navier-Stokes equations needadditional slip boundary conditions to be still valid;

• 0.1 ≤ Kn ≤ 1: transition regime, in which the Navier-Stokes equations are notvalid, Boltzmann equation or advanced models are needed;

• 1 ≤ Kn ≤ 10: kinetic regime, here the Boltzmann equation is also valid, but adirect simulation is expensive;

• 10 ≤ Kn: free flight regime, where direct simulations start to become efficient;

The specific flow regime therefore suggests a corresponding model for the flow andis closely related to the numerical solution approach to solve the flow problem. For therarefied gases with Kn ≥ 0.1, one is interested in efficient and accurate methods to solvethe Boltzmann equation, which is the topic of this thesis.

In the literature (see e.g. [17], [6]), there are many relevant applications that arecovered by the kinetic regime of a rarefied gas. Among those are:

• Reentry flights of spacecrafts at very high altitude: Gas pressure and density arevery low, leading to large mean free paths and in turn to a large Knudsen number,even for large characteristic lengths like spacecrafts. The correct prediction of theheat flux close to the thermal shield is crucial in this example.

• Shock waves at very high speed: The velocity jumps from super- to sub-sonicyields sharp gradients over only a very small distance. The shocks also influencethe behavior of the flow further downstream.

• Microscopic channel flows: At very small length scales, the Knudsen number willbecome large even for ambient conditions. Examples are porous media or ionchannels in membranes.

Boltzmann Transport Equation 7

When dealing with applications above, it is absolutely necessary to use extendedmodels and methods to correctly predict the various effects of gas rarefaction. The useof standard methods like Navier-Stokes, for example, may lead to wrong predictions ofthe relevant macroscopic quantities. The method therefore fails to correctly simulate thetypical effects for large Knudsen numbers, the so-called kinetic effects (see also [17]) likethe Knudsen paradox. It says that the mass flow through a tube is decreasing with thediameter of the tube only until the diameter reaches a certain value of the order of themean free path λ. From then on, the mass flow increases again. This is not consistentwith the Navier-Stokes equations and only one example for the need of better models.

Another example that is often cited, is the very small Knudsen pump (see [16]). Ithas no moving parts and works only with temperature differences along the wall. Due tothat, the gas inside moves from the cold end to the warm end. This allows for a preciseand reliable control of the gas flow.

2.1.2 Phase Space and Probability Density Function

When it comes to rarefied gases, one might think about a straightforward method thattracks the way of every single particle. The collisions between particles then couple theevolution of the particles’ positions. The problem is that this procedure leads to thesolution of a vast number of coupled partial differential equations, as even a rarefied gasstill consists of too many particles per volume. For each of those particles, one wouldhave to determine the corresponding three-dimensional positions x and velocities c atevery time t, leading to a seven-dimensional solution space, the so-called phase space.

In kinetic theory we introduce a probability density function (PDF) or distributionfunction f(x, c, t). The PDF f is related to the number of particles with velocities c atposition x and time t. The number of particles with velocities in [c, c+ dc] in a certaininterval [x,x+ dx] at time t is given as N = f(x, c, t)dxdc.

Instead of following each single particle, it is in principle enough to know the valueof f at all times, positions as well as velocities to have complete knowledge about thestate of the gas.

2.1.3 Macroscopic Quantities of the Flow Field

Assuming a given PDF f , it is important to recover the macroscopic quantities of theflow field, because we are usually interested in variables like the overall velocity of thegas or the temperature. These and other quantities are all computed with the help ofso-called moments of f .

The mass density ρ is simply the mass m of one particle times the integral of thePDF f over the velocity space R3:

ρ := m

∫R3

f dc := mn, (2.2)

with number density n :=∫R3

fdc.


The mean velocity v can be computed by means of the momentum density ρv

ρv = m

∫R3

c · f(t,x, c) dc, (2.3)

or in componentwise/tensor notation

ρvi = m

∫R3

ci · f(t,x, c) dc. (2.4)

Higher moments can be defined using the peculiar velocities Ci, where

Ci := ci − vi. (2.5)

With this definition the thermal energy (sometimes also called internal energy) u isgiven as

ρu =m

2

∫R3

C2ii · f(t,x, c) dc, (2.6)

where the notation is C2ii := C2

1 + C22 + C2

3 in three dimensions.For an ideal gas, the temperature T is closely related to the internal energy u by

u = 32kmT , where k is the Boltzmann constant. Writing the temperature in energy units,

we define a new variable θ = kmT .

Lastly, we give the definition of the pressure tensor pij and heat flux qi

pij = m

∫R3

CiCj · f(t,x, c)dc, qi =m

2

∫R3

C2jjCi · f(t,x, c)dc. (2.7)

Apart from these definitions, there are several thermodynamical laws connectingdifferent variables and giving restrictions on parameters that can be found in [14].

2.2 Properties of the Boltzmann Transport Equation

In order to calculate the moments mentioned in the section before, one has to solve theBoltzmann equation for the unknown PDF f . The Boltzmann equation is a partial-integro-differential equation for f with usually seven independent variables t,x, c:

∂

∂tf(t,x, c) + ci

∂

∂xif(t,x, c) +Gi

∂

∂cif(t,x, c) = S(f), (2.8)

here, the first term denotes the change in time, the second term is due to the convectivetransport with velocity ci and the third term on the left hand side denotes changes invelocity in the presence of external forces Gi, such as gravity. The operator S(f) on theright hand side is the so-called collision operator that models collisions of particles withother particles.

For most of this thesis, we will neglect external forces and consider only the convectivepart as the main difficulty.


2.2.1 Equilibrium Distribution

The right hand side operator S(f) in Equation (2.8) forces the process towards itsequilibrium state and is zero, if equilibrium is achieved. At equilibrium, the densityfunction f has the form of a local Maxwellian that is in d dimensions defined as follows

fM(t,x, c) =ρ(t,x)

m

1√2πθ(t,x)

dexp

(−(ci − vi(t,x))2

2θ(t,x)

). (2.9)

As density, mean velocity and temperature may vary with t and x, a local Maxwellianis assigned to each point in time and space. Proper definitions of the collision operatorS(f) have to ensure that S(fM) = 0.

In the non-equilibrium case, the density function f differs from a Maxwellian. Whenwe use a special ansatz for the form of the density function f later on, it is neverthelessimportant to have the Maxwellian in the solution space in order to give the right solutionin the equilibrium case. This will later justify some particular choices for a basis of theansatz space.

2.2.2 Collision Operator

The form of the collision operator has a huge impact on the behavior of the distributionfunction f . The specific choice of S(f) is in fact already part of the model.

A well-known model for the collision operator was proposed by Boltzmann itself andwas derived using the so-called Stosszahlansatz

S(Boltz)(f) =1

ρ

∫R5

gb(f(c)f(c1)− f(c)f(c1)

)db dε dc1, (2.10)

where the notation f indicates the PDF for the post-collision velocity. In principle itis possible to use this approach for numerical simulations, but its high dimensionalitymakes already the evaluation of S(Boltz) at discrete points in t and x very costly. Formore information and details about this approach, see for example [6].

With some simple assumptions of the collisions (see e.g. [6]), it is possible to derivea linearized collision operator from the Boltzmann operator that is known as the BGKmodel [2]:

S(BGK)(f) =1

τBGK(fM − f) . (2.11)

This ansatz basically represents a relaxation towards the equilibrium distribution fM(see (2.9)) with relaxation time τBGK. The BGK model is often chosen because of itssimplicity. Note that the Boltzmann equation (2.8) together with the BGK model (2.11)is not a linear equation, because the Maxwellian fM still includes exponentials of themacroscopic variables ρ, vi and θ, which are in turn integrals of the distribution functionaccording to Section 2.1.3.


There is also another approach, which assumes small velocity changes due to collisionsand results in a Fokker-Planck operator for the collisions

S(FP)(f) =∂

∂ci

(1

τFP(ci − vi)f

)+

∂2

∂ck∂ck

(2es

3τFPf

), (2.12)

with relaxation time τFP and sensible energy es = 32kmT in three spatial dimensions.

The derivation of this operator is shown in detail in [6]. An application using theFokker-Planck model is described in [13].

2.3 Solution Methods

Numerical methods for the solution of the Boltzmann equation have been under devel-opment since the late 1960s. During the past fifty years, different methods have beensuccessfully applied to various problems. We will now give a short overview about themost important classes of methods.

2.3.1 Direct Simulation Monte Carlo

It was Bird, who proposed the first method to actually solve rarefied gas flows usingthe so-called Direct Simulation Monte Carlo method (DSMC) [3]. His method was laterimproved and used for many problems emerging from real world applications, see also[4].

The key to the DSMC method is the different viewpoint: The behavior of the gas isactually not modeled by a PDE like the Boltzmann equation, but the gas is describedby a system of particles. Each of the particles has a position and a velocity at everytime. Note that in a real computation the number of numerical particles is substantiallysmaller than the real number of particles, which is of the order of 1020. So usually somehundreds of thousand particles are used to simulate the gas flow.

Now the particles’ positions and velocities evolve according to the following steps(see [6] for more details):

(1) a proper initialization is done by sampling velocities and positions from initialvalues.

(2) in each time step, the particles first have a free-flight phase, where they are movedaccording to their assigned velocities for the time interval ∆t.

(3) the free-flight phase is followed by a collision phase, where particles undergocollisions that are modeled by a collision probability. Binary collisions then changethe velocities of the involved particles.

The particles thus move and collide in every time step. The calculation is by definitionunsteady and steady results are obtained by asymptotic limits in time. Macroscopicquantities can later be derived from the particles’ velocities by averaging over small cellsof the flow field.


The benefit of this type of method is clearly its simplicity. It is straightforward toimplement, once the collision probabilities are modeled. On the other hand, the numberof particles needs to be sufficiently high to enable accurate solutions. This was especiallya problem during the development of the method. Furthermore, the number of requiredparticles increases with the density of the gas, making the method less suitable forproblems with moderate Knudsen numbers.

2.3.2 Method of Moments

A relatively new method is the so-called method of moments (MoM), in which equationsfor the macroscopic moments are directly derived from the Boltzmann equation. A goodsummary of the method and the problem of the closure below is given by Levermorein [18].

The general procedure can be demonstrated for a simple, one-dimensional equationlike (2.13)

∂

∂tf(t, x, c) + c

∂

∂xf(t, x, c) = 0. (2.13)

Now the integral operator In(·) (2.14) is applied to the PDE, where In multiplies theequation by cn and integrates over the velocity space

In(·) :=

∞∫−∞

· cndc. (2.14)

After the application of In(·), we can identify the so-called moments (2.15) in theequation

Mn(t, x) :=

∞∫−∞

f(t, x, c)cndc. (2.15)

The lower moments have a direct physical meaning. For example, we have M0 = ρ,M1 = v.

The one-dimensional Boltzmann equation now transforms to

∂

∂tMi(t, x) +

∂

∂xMi+1(t, x) = 0, i = 0, . . . , n. (2.16)

By this simple trick, it is possible to eliminate the velocity dependence. We willessentially use the same procedure for our quadrature-based projection methods lateron.

Note that the convective term leads to the appearance of the higher moment Mn+1.Now, one has n equations for n + 1 variables, as the last equation also contains Mn+1.The difficult part is now to find a so-called closure that is an additional relation to closethe system of equations.

The easiest way would be to simply set Mn+1 = 0. Unfortunately, this simpleapproach does not yield satisfactory results. It can lead to negative values for the density,


for example. There are more advanced methods to close the system that relate thehighest moment to the lower moments. One of these approaches is the maximum entropymethod. The value of the highest moment is chosen to maximize the mathematicalentropy, leading to physical solutions. A disadvantage is that the relation can no longerbe written down in closed form, but is the solution of an optimization procedure in everystep of a numerical method.

Moment methods are also often used in kinetic models for radiative transfer, wherethe three-dimensional velocity of the particles is written in terms of direction and energyof the particles, see [9] for an example.

2.3.3 Lattice Boltzmann Method

The Lattice Boltzmann method (LBM) is actually a modification of another particlemethod, the Lattice Gas Automata method (more information can be found in [7]) inwhich particles are only allowed to travel along a discrete lattice through the flow field.As soon as two particles meet somewhere on the discrete lattice, a collision event takesplace, changing the velocities similar to the DSMC method. As for the LBM method,the single particles have then been replaced by the particle density function in order toreduce statistical noise.

Most of the time LBMs uses a BGK model for the collisions, where a collision stepis followed by a free-flight step just like in case of the DSMC method. The importantdifference is that the velocity space is discretized and only allows for particle velocitiesalong the lattice to the next lattice grid point. One possible velocity space discretizationin two dimensions could be

c ∈ Vh = {(0, 0), (±1, 0), (0,±1), (±1,±1)}. (2.17)

This is called the D2Q9 discretization, as it is two-dimensional and consists of 9 discretevalues for the velocity lattice.

Despite its simplicity, limitations of the LBM are flows with high Mach numbers,because the methods were originally developed for isothermal problems (see [7]).

2.3.4 Discrete Velocity Method

Another method that is in fact closely related to our quadrature-based projection meth-ods is the Discrete Velocity Method (DVM). A good description of the method canbe found in [6]. The DVM discretizes the Boltzmann equation in distinct points ofthe velocity space. The discretization points cn are called discrete velocities. In theone-dimensional case, this means we end up with a system of equations for each of thediscrete velocities

∂

∂tf(t, x, cn) + cn

∂

∂xf(t, x, cn) = 0. (2.18)

Note that this special discretization of the velocity space leads to the unknownsfn(t, x) := f(t, x, cn) that do only depend on t and x. The discretization of the right


hand side collision operator is more delicate, as the collision invariants and conservationproperties impose some restrictions for a meaningful discretization.

There is a relation between DVM and the MoM. In the case of MoM, the equationis multiplied by the function cn and then integrated over the velocity space. DVMevaluates the equation at certain velocities cn which is equivalent to a multiplicationwith the dirac function δ(c− cn) followed by integration over the velocity space. Thus,the DVM can be seen as another projection method, where the test functions (which areused for the multiplication of the equation) are simple dirac functions.

In this sense the DVM method can also be seen as a special ansatz for the distributionfunction. With the point evaluations fk = f(t,x, ck) this ansatz has the form

f(t,x, c) =n∑k=1

fkδ (c− ck) . (2.19)

Note that the regularity of this ansatz is very weak. The expansion in delta functions isnot differentiable and there is no meaningful interpretation of f for intermediate valuesc 6= ck as well as for the derivative of f at any point c.

14

Chapter 3

Mathematical Properties

This chapter is about the necessary mathematical elements needed for the formulationand the expansion of the transformed Boltzmann equation as well as the solution usingquadrature-based projection methods.

We first introduce the different types of basis functions along with their respectiveproperties. These functions will later be used as ansatz or test functions in the varioussettings and will determine the structure of the emerging PDE system. We derivenormalized versions of the respective functions and show important recursion resultsthat will be used in Chapters 4 and 5.

The last part of this chapter is about quadrature methods and Gaussian quadrature inparticular. Together with the different quadrature methods, we also give an introductionto non-classical quadrature and explain useful properties of quadrature approximations.

3.1 Hermite Polynomials

We explain the standard version of Hermite polynomials first, before defining an or-thonormal set of Hermite functions that we will later use for the expansion of theunknown distribution function of the Boltzmann equation. The interested reader isreferred to [1] for a more detailed summary of properties and formulas concerningHermite polynomials.

3.1.1 Standard Definition

There are actually two different ways of defining the standard Hermite polynomials,each of them leading to a scaled version of the other one. We will stick to the so-calledprobabilists’ Hermite polynomials Hn that are defined in the following way for n ≥ 0

Hn(ξ) = (−1)neξ2/2 d

n

dξne−ξ

2/2. (3.1)

15

16 Mathematical Properties

The other type of definition, leading to the so-called physicists’ Hermite polynomialsHen, is

Hen(ξ) = (−1)neξ2 dn

dξne−ξ

2. (3.2)

Note the missing factor in the exponent that leads to the conversion formula

Hn(ξ) = 2−n/2Hen

(ξ/√

2). (3.3)

In the context of the Boltzmann equation, the probabilists’ Hermite polynomialsare usually considered, because they are closely related to the Maxwellian that is theequilibrium distribution of the Boltzmann equation. This is the reason to use themas a set of basis functions for the expansion of the distribution function in one spatialdimension.

The first Hermite polynomials can be easily calculated as

H0(ξ)=1,

H1(ξ)=ξ,

H2(ξ)=ξ2 − 1,

H3(ξ)=ξ3 − 3ξ,

H4(ξ)=ξ4 − 6ξ2 + 3.

(3.4)

The corresponding polynomials of higher degree can be derived using the followingrecursion formula that holds because of the definition (3.1)

Hn+1(ξ) = ξHn(ξ)− n · Hn−1(ξ). (3.5)

Furthermore, the derivative of a Hermite polynomial can be expressed in terms ofthe lower order polynomial according to

H ′n(ξ) = n · Hn−1(ξ). (3.6)

It is easy to show that the Hermite polynomials are an orthogonal basis of thecorresponding space of polynomials with respect to the weighted scalar product

< φ,ψ >w=

+∞∫−∞

φ(ξ)ψ(ξ)w(ξ)dξ, (3.7)

using the weighting function w(ξ) = 1√2πe−ξ

2/2. According to that, we can compute for

m ∈ N< Hn, Hm >w= n!δnm, (3.8)

which shows that the Hermite polynomials are in fact orthogonal but not normalized.

Mathematical Properties 17

3.1.2 Orthonormal Hermite Polynomials

As we have seen in Equation (3.8), the standard version of the Hermite polynomials arenot normalized with respect to the weighted scalar product defined above. We thereforedefine a normalized Hermite polynomial as follows

Hn(ξ) :=1√n!Hn(ξ). (3.9)

Due to their definition, we directly conclude the orthonormality of these polynomials(compare Equation (3.8))

< Hn, Hm >w= δnm. (3.10)

Using the definition (3.9) and the properties from Section 3.1.1, we can deriverecurrence relations and the derivative of the normalized Hermite polynomial:

ξHn(ξ) =√n+ 1Hn+1(ξ) +

√nHn−1(ξ) (3.11)

and

H ′n(ξ) =√n ·Hn−1(ξ). (3.12)

Another useful property is

d

dξ(Hn(ξ)w(ξ))=w(ξ)

(H ′n(ξ)− ξHn(ξ)

)=− w(ξ)

√n+ 1Hn+1(ξ).

(3.13)

Combining (3.11) and (3.13), we obtain the relation

ξd

dξ(Hn(ξ)w(ξ))=ξw(ξ)

(H ′n(ξ)− ξHn(ξ)

)=− w(ξ)

√n+ 1

(√n+ 2Hn+2(ξ) +

√n+ 1Hn(ξ)

).

(3.14)

These relations will play a major role when we use a Hermite ansatz for our distri-bution function in the Boltzmann equation (see Section 5.2.2.3).

3.2 Laguerre Polynomials

The Hermite polynomials are defined such that they fulfill an orthogonality relationwhen integrated over the whole ξ ∈ R. But in special cases it is necessary to havesimilar polynomials but with different weights and integration intervals. One differentapproach are the so-called Laguerre polynomials, which are defined as

Ln(ξ) =eξ

n!

dn

dξne−ξξn. (3.15)


The first Laguerre polynomials are

L0(ξ)=1,

L1(ξ)=− ξ + 1,

L2(ξ)=1

2

(ξ2 − 4ξ + 2

),

L3(ξ)=1

6

(−ξ3 + 9ξ2 − 18ξ + 6

),

L4(ξ)=1

24

(ξ4 − 16ξ3 + 72ξ2 − 96ξ + 24

).

(3.16)

The polynomials also follow a recursion rule for the computation of polynomials ofhigher degree (using the definition L−1(ξ) := 0)

(n+ 1)Ln+1(ξ) = (2n+ 1− ξ)Ln(ξ)− nLn−1(ξ). (3.17)

The recursion formula for the derivative looks rather different from the one for Hermitepolynomials (compare Equation (3.11)). Derivatives can be calculated using

L′n(ξ) = L′n−1(ξ)− Ln−1(ξ) (3.18)

or from

ξL′n(ξ) = nLn(ξ)− nLn−1(ξ). (3.19)

The Laguerre polynomials are already orthonormal as they are defined in (3.15) withrespect to the scalar product

< φ,ψ >wL=

+∞∫0

φ(ξ)ψ(ξ)w(ξ)dξ, (3.20)

with weighting function wL(ξ) = e−ξ when integrated over the positive domain ξ ∈ R+

Consequently, we have for m ∈ N

< Ln, Lm >w= δnm. (3.21)

The Laguerre polynomials therefore form a set of orthonormal basis functions. Itis possible to use them for the expansion of the unknown distribution function of thethree-dimensional Boltzmann equation in the radial velocity direction.

3.2.1 Generalized Laguerre Polynomials

The Laguerre polynomials introduced in (3.15) are orthogonal with respect to the weightedstandard scalar product (3.20). But for a transformation of variables, additional termsappear in the integrals due to the Jacobian of the transformation rule. For spherical


velocity coordinates basically the term r2 sin(θ) has to be considered for a proper defi-nition of spherical coordinates (r, θ, φ). In the radial velocity direction, we then have tocompute integrals of the following form

∞∫0

f(r)w(r)r2dr. (3.22)

We therefore use polynomials that are orthogonal in the proper sense. This can beachieved by so-called generalized Laguerre polynomials Lαn of degree n that are definedas follows

Lαn(ξ) =ξ−αeξ

n!

dn

dξne−ξξn+α, (3.23)

for a parameter α ∈ R.The case α = 0 gives back the traditional version of Laguerre polynomials defined in

(3.15).The first three generalized Laguerre polynomials are

Lα0 (ξ)=1,

Lα1 (ξ)=− ξ + α+ 1,

Lα2 (ξ)=ξ2

2− (α+ 2)ξ +

(α+ 2)(α+ 1)

2.

(3.24)

Similar to the standard Laguerre polynomials, there exist some recursion rules and aformula for derivatives. One important formula is the following shift of the parameter α:

Lαn(ξ) = Lα+1n (ξ)− Lα+1

n−1(ξ). (3.25)

The formula remains valid for n = 0, if we define Lαn(ξ) := 0 for all n ≤ 0.We will here now concentrate on the important orthogonality result: The generalized

Laguerre polynomials are orthogonal with respect to the scalar product

< φ,ψ >wLα==

+∞∫0

φ(ξ)ψ(ξ)wLα(ξ)dξ, (3.26)

with weighting function wLα(ξ) = e−ξξα, because we have

+∞∫0

Lαn(ξ)Lαm(ξ)wLα(ξ)dξ =Γ(n+ α+ 1)

n!, (3.27)

for Gamma-function Γ.Consequently, we can write down a normalized version

Lαn(ξ) =

√n!

Γ(n+ α+ 1)

ξ−αeξ

n!

dn

dξne−ξξn+α, (3.28)


that leads to< Lαn, L

αm >wLα= δnm. (3.29)

Corresponding to (3.25), we can also derive a recursion formula for the normalizedfunctions using the definition (3.28) as follows

Lαn(ξ) =√n+ α+ 1Lα+1

n (ξ)−√nLα+1

n−1(ξ). (3.30)

This formula can be used for the analytical computation of integrals emerging from aspecial ansatz in three spatial dimensions.

Choosing α = 2 we can now compute integrals of the form (3.22).

3.3 Spherical Harmonics

The Hermite as well as the Laguerre polynomials do only depend on one variable,which makes them suitable for one-dimensional applications as well as e.g. for a one-dimensional part of a more complex application. We aim at a full three-dimensionalsetting of our simulations later in order to come close to real-world experiments.

Assuming functions of Laguerre type for the radial part, we also need ansatz functionsfor the angular portion of the solution. This is where the so-called spherical harmonics(SH) come into play. The spherical harmonics are polynomials in spherical coordinatesthat can be evaluated for every point on the unit sphere and return a single value. Weonly consider real SH, but there are also complex-valued SH functions.

We will now introduce a normalized version of the spherical harmonics, such that theupcoming integrals are easy to calculate and the system matrix will become symmetric.First, we need the so-called normalized associated Legendre polynomials Lml of degreel ∈ N and order m ∈ N, which can be calculated from the associated Legendrepolynomials Pml using the following formula

Lml :=

√(2l + 1)(l −m)!

2(l +m)!Pml (ξ) (3.31)

and

Pml :=(−1)m

2ll!

(1− x2

)m2dl+m

dxl+m(x2 − 1

)l. (3.32)

Together with the setting Lml := 0 for l > m, the normalized associated Legendrepolynomials satisfy a set of recursion relations that are very useful for computationswith the spherical harmonics later.

Now the real SH function of degree l and order m (−l ≤ m ≤ l) is defined as follows

Y 0l (θ, φ) :=Y0,l(θ, φ) :=

1√2πL0l (cos(θ)) (m = 0),

Y ml (θ, φ):=Y m

1,l (θ, φ) :=1√πLml (cos(θ)) cos(mφ) (m > 0),

Y ml (θ, φ):=Y

|m|2,l (θ, φ):=

1√πL|m|l (cos(θ)) sin(|m|φ)(m < 0).

(3.33)


Figure 3.1: SH degree l = 0: Y 00

At first sight it is important that Y 0l does not depend on φ and all the SH functions are

polynomials in trigonometric functions of θ and φ. Furthermore, we have 2l+1 functionsfor each degree l.

The first few SHs are:

Y 00 (θ, φ) =

1

2√π,

Y −11 (θ, φ)=− 1

2

√3

πsin(θ) sin(φ),

Y 01 (θ, φ) =

1

2

√3

πcos(θ),

Y 11 (θ, φ) =− 1

2

√3

πsin(θ) cos(φ),

Y −22 (θ, φ)=1

4

√15

πsin2(θ) sin(2φ),

Y −12 (θ, φ)=− 1

4

√15

πsin(2θ) sin(φ),

Y 02 (θ, φ) =

1

8

√5

π(3 cos(2θ) + 1),

Y 12 (θ, φ) =− 1

4

√15

πsin(2θ) cos(φ),

Y 22 (θ, φ) =

1

4

√15

πsin2(θ) cos(2φ).

(3.34)

In order to get an impression of how a spherical harmonics looks like, it is possible toplot Y m

l in the following way: we scale each vector that points from the origin to the unitsphere by the absolute value of Y m

l for this particular choice of θ and φ correspondingto the direction in which the vector points. The results are shown in Figures 3.1 to 3.3e.


(a) Y −11 (b) Y 0

1 (c) Y 11

Figure 3.2: SH degree l = 1

(a) Y −12 (b) Y 0

2 (c) Y 12

(d) Y −22 (e) Y 2

2

Figure 3.3: SH degree l = 2


By construction, the SH functions are orthonormal with respect to the scalar product

< f, g >=

2π∫0

π∫0

f(θ, φ)g(θ, φ) sin(θ)dθdφ, (3.35)

meaning that we simply have

< Y ml , Y m′

l′ >= δm,m′δl,l′ . (3.36)

The set of spherical harmonics is therefore an orthonormal basis for all functionsdefined on the unit sphere. It is possible to use them as part of the ansatz functions ina full three-dimensional setting.

3.3.1 Cartesian Spherical Harmonics

As we have seen in the previous section, the spherical harmonics are part of a basis ofall polynomial functions in x, y, z in the three-dimensional space. But they are naturallyformulated in the spherical coordinates r, θ, φ together with a radial part that dependssolely on the radius r. Consistently, one would have to calculate the emerging integralsusing spherical integration, too. Depending on the context, this can be inefficient orinflexible.

In [15] a cartesian version of the solid spherical harmonics is defined. The solidspherical harmonics Nm

l (r, θ, φ) are related to the spherical harmonics Y ml (θ, φ) by an

additional factor rl:

Nl,m(r) = rlY ml (θ, φ). (3.37)

The additional factor rl enables the representation of the solid spherical harmonicsin cartesian coordinates x, y, z. The cartesian solid spherical harmonics are first definedas follows

N+l,m =

{rl

(1+|m|)!P|m|l (cos(θ) cos(|m|φ)) if m ≥ 0

−N+l,m if m < 0

, (3.38)

N−l,m =rl

(l + |m|)!P|m|l (cos(θ) sin(|m|φ)), form ∈ Z. (3.39)

This is already an equivalent basis of the space spanned by the orthonormal solidspherical harmonics rlY m

l (θ, φ). The definition can be normalized using a normalizationfactor

Nl,m =

√2l + 1

2π(l −m)!(l +m)!

N+l,m if m > 0

1√2N+l,m if m = 0

N−l,−m if m < 0

. (3.40)

According to [15], this version of the solid spherical harmonics can easily be writtenin the cartesian coordinates (x, y, z) = (r sin(θ) cos(φ), r sin(θ) sin(φ), r cos(θ)) using the


following recursion formula

N+0,0 =1,

N−0,0 =0,

N+m,m=− 1

2m

(xN+

m−1,m−1 − yN−m−1,m−1

),

N−m,m=− 1

2m

(yN+

m−1,m−1 + xN−m−1,m−1

),

N±l,m =1

(l +m)(l −m)

((2l − 1)zN±l−1,m − r

2N±l−2,m

).

(3.41)

The cartesian version of the SSH is by definition orthogonal with respect to the scalarproduct defined in (3.35) because we have

< Nl,m, Nl′,m′ >= rl+l′δm,m′δl,l′ . (3.42)

The definition (3.41) now allows for a representation of the solid spherical harmonicsin the cartesian basis. As the corresponding integrals are very easy to solve (in fact, evena simple cartesian quadrature rule of sufficient order of exactness gives exact results),this can be a possibility to speed up computations. On the other hand, we can noweasily identify the basis functions at the different levels l,m of the spherical harmonicswith simple cartesian polynomials and see the differences to a full tensor product witha polynomial basis, for example.

3.4 Jacobi Matrix

In this context, we will briefly mention the so-called Jacobi matrix, which is a tridiagonalmatrix containing the coefficients from the recursion of orthonormal polynomials (e.g.(3.11), note that the Jacobi matrix is not to be mixed up with the so-called Jacobianmatrix that is the first derivative of the numerical or analytical flux calculation). Wewill later see this matrix when we show examples for the computation of eigenvalues ofthe system matrix. It turns out that the eigenvalues of the Jacobi matrix are just thezeros of the (n+ 1)st orthonormal basis function they correspond to (see also [24]).

Our sets of orthonormal polynomials Φn satisfy a recursion rule like

xΦi(x) = aiΦi+1(x) + biΦi−1(x). (3.43)

In case of the orthonormal Hermite polynomials, for example, we have

xΦi(x) =√i+ 1Φi+1(x) +

√iΦi−1(x). (3.44)

Thus

aHi =√i+ 1, bHi =

√i. (3.45)


The Jacobi matrix Jn corresponding to a set of orthonormal polynomials is definedas

Jn :=

0 a0 0 . . . 0

b1 0 a1. . .

...

0 b2. . .

. . . 0...

. . .. . . 0 an−1

0 · · · 0 bn 0

. (3.46)

And the characteristic polynomial is actually some factor γ ∈ R times the (n+ 1)storthonormal polynomial.

χ(J) = det(Jn − λ · In) = γΦn+1(λ). (3.47)

This leads to the fact that the eigenvalues (as roots of the characteristic polynomial)are the zeros of Φn+1, too. Note that the Jacobi matrix for Hermite polynomials issymmetric, due to the coefficients (3.45).

In Section 4.1, we will recognize the Jacobi matrix as the system matrix of the PDEsystem for the basis coefficients after the projection.

3.5 Gaussian Quadrature

Like every quadrature rule, Gaussian quadrature approximates integrals of a certain kindusing evaluations of the integrand at discrete points. In the case of Gaussian quadraturethe integrand consists of a weighted product of a function f and a weighting functionw. Gaussian quadrature of order N ∈ N is performed according to

b∫a

f(x)w(x)dx =N∑i=1

wif(xi), (3.48)

where for i = 1, ..., N the wi are called weights and the xi are the sampling points.For a proper choice of the weights and the corresponding sampling points, the

Gaussian quadrature rule is exact for all polynomials f up to degree 2N − 1.We will further consider the special case, where the weighting function is equivalent

to the weighting function of the Hermite or Laguerre polynomials.It is well known that the sampling points are the roots of the N -th corresponding

orthogonal basis polynomial pn. The weights wi can be calculated according to theformula

wi =aNaN−1

b∫aw(x)pN−1(x)2dx

p′N (xi)pN−1(xi), (3.49)

where aN is the coefficient in front of xN in the respective polynomial pN .It can be shown that the weights are all positive and all the sampling points lie inside

the interval (a, b).


3.5.1 Gauss-Hermite Quadrature

If we choose the weighting function in Equation (3.48) to be w(x) = 1√2πe−x

2/2, and

a = −∞, b = +∞ we are approximating integrals of the kind

+∞∫−∞

f(x)1√2πe−x

2/2dx (3.50)

by a weighted sum of function evaluations.The sampling points xi for the function evaluation are the roots of the N -th Hermite

polynomial, which is given in (3.9) and here denoted as pN .According to the general formula (3.49), the weights for the Gauss-Hermite quadra-

ture can be calculated to be

wi=aNaN−1


p′N (xi)pN−1(xi)

=

√(N − 1)!√N !

1√NpN−1(xi)2

=1

NpN−1(xi)2.

(3.51)

3.5.2 Gauss-Laguerre Quadrature

For another weighting function w(x) = e−x and a = 0, b = +∞ we end up calculatingthe following integrals by the so-called Gauss-Laguerre quadrature

+∞∫0

f(x)e−xdx (3.52)

as a weighted sum of function evaluations.The sampling points xi for the function evaluation are now the roots of the N -th

Laguerre polynomial pN .Similar to the previous version (compare Section 3.5.1), the weights for the Gauss-

Laguerre quadrature are

wi=aNaN−1



=xi

(N + 1)2pN+1(xi)2.

(3.53)

Thus, the Gauss-Laguerre quadrature is essentially performed in the very same wayas the Gauss-Hermite quadrature with the important difference of another weightingfunction as well as different interval bounds.


3.5.3 Generalized Gauss-Laguerre Quadrature

For weighting function w(x) = xαe−x and a = 0, b = +∞ we calculate the followingintegrals

+∞∫0

f(x)e−xxαdx (3.54)

as a weighted sum of function evaluations.The sampling points xi for the function evaluation are now the roots of the N -th

generalized Laguerre polynomial pN .The weights for the generalized Gauss-Laguerre quadrature are

wi=aNaN−1



=xi

(N + 1)2pN+1(xi)2.

(3.55)

3.5.4 Non-Classical Quadrature

There are quadrature formulas for various types of weighting functions in combinationwith the respective domains of integration and the corresponding orthogonal polyno-mials. The so-called classical quadrature formulas include Gauss-Hermite-, Gauss-Laguerre- and Gauss-Legendre quadrature, for example.

However, in special cases one might be confronted with a different weighting function,a so-called non-classical weight, or different domains for the integration for that none ofthe formulas above is applicable. Under certain conditions, it is possible to find corre-sponding weights and quadrature points. As we are interested in a Gaussian-quadrature,we also need a set of orthogonal polynomials with respect to the desired integral. Wewill explain the general procedure of finding the orthogonal polynomials and the weightsin this section.

For our non-classical quadrature, we consider integrals of the type

b∫a

f(x)w(x)dx (3.56)

and want to compute the exact value for polynomials f(x) up to a certain degree usinga quadrature formula like in (3.48)

First, we have to check, whether a Gaussian-quadrature rule with orthogonal poly-nomials exists. According to [19] this requires the following Hankel-matrix to be non-singular

∆N :=

µ0 µ1 . . . µN−1µ1 µ2 . . . µN...

.... . .

...µN−1 µN . . . µ2N−2

, (3.57)


with

µi =

b∫a

xiw(x)dx. (3.58)

Regularity of (3.57) ensures the existence of an orthonormal set of polynomials.

The next step is to construct a basis for the space of polynomials that are orthogonal(or orthonormal) with respect to the given integral (3.48). There are several possibilitiesto do this, of which one of the easiest would be to apply a Gram-Schmidt method toorthonormalize the monomials and end up with a set of orthonormal polynomials pj(x).Other possibilities are the method of moments, the Stieltjes procedure and the Lanczosalgorithm, for more information, see [19].

As we have seen before, compare Section 3.5, the quadrature points xi are now justthe roots of the Nth orthonormal polynomial if we are interested in a formula thatis exact up to degree 2N − 1. As the Gram-Schmidt method is already a numericalprocedure, the calculation of the roots is usually also done numerically and thus veryefficient.

Next is the calculation of the corresponding weights. The weights are determined bythe condition of exactness of the formula up to degree N−1. In principle, one can use thegeneral quadrature formula (3.48) for every monomial xi, for i = 0, . . . , N − 1 and get alinear system of equations by the requirement that the quadrature reproduces the exactresult. Alternatively, the condition is imposed for the orthonormal basis polynomialspj(x). Thus, one has to solve the set of linear equations (see also [21])

p0(x1) . . . p0(xN )p1(x1) . . . p1(xN )

.... . .

...pN−1(x1) . . . pN−1(xN )

w1

w2...wN

=

b∫ap0(x)w(x)dx

b∫ap1(x)w(x)dx

...b∫apN−1(x)w(x)dx

. (3.59)

Note that the right hand side usually includes many zero entries, because the polynomialspj(x) are orthogonal to the constant function p0(x). This may not be the most efficientway to calculate accurate weights, because the solution of the linear system (3.59) canbe unstable, especially for large N . It is also possible to use the general formula (3.49)from above. The reader is referred to [21] for more advanced methods.

We will later again see matrices similar to the one on the left hand side in Equation(3.59). If the polynomials pi form an orthogonal basis and the xi are the correspondingquadrature points, then this matrix will be non-singular, because a set of non-zeroweights wi has to exist.

In the end, we have an orthonormal set of basis polynomials, the roots of theN−th polynomial and the corresponding weights for a Gaussian-quadrature formulato calculate the integrals with respect to non classical weighting functions.


3.5.5 Interpolation Property and Aliasing

With the help of Gaussian quadrature, it is possible to prove an interpolation property ofan expansion of the arbitrary function f in terms of an orthonormal set of polynomials.

We therefore consider two expansions of the original function f up to a certain ordern− 1 ∈ N as follows

f(x) =n−1∑j=0

αjΦj(x)w(x),

f(x) =

n−1∑j=0

αjΦj(x)w(x).

(3.60)

with basis coefficients αi, αi, basis polynomials Φi, the corresponding weighting func-tion w (e.g. w(x) = e−x for Laguerre polynomials) and i = 0, ..., n− 1.

The basis coefficients are either computed by exact integration as

αj =

b∫a

Φj(x)f(x)dx (3.61)

or employing a Gaussian quadrature approximation for the integral using the same setof orthonormal functions Φi

αj =

n∑i=1

wiΦj(xi)1

w(xi)f(xi). (3.62)

Although the second method is somewhat an approximation to the exact integral, itis possible to derive an interesting property for the quadrature based coefficients. Usingthese approximated values, we have

f(xi) = f(xi) for i = 1, ..., n. (3.63)

Which means that the value of the quadrature based approximation f and the valueof the exact function f are the same at the sampling points of the quadrature rule. Thisis not necessarily the case for the coefficients computed with exact integration.

The reason for this is the so-called aliasing error that is introduced by the exactintegration. It is caused by sort of high frequencies in f that spoil the point interpolationproperty.

Equation (3.63) can be proven by the assumption that we have

f(xi) 6= f(xi) for one i = 1, ..., n. (3.64)


Computing αk we get:

αk=n∑i=1

wiΦk(xi)1

w(xi)f(xi)

6=n∑i=1

wiΦk(xi)1

w(xi)f(xi)

=n∑i=1

wiΦk(xi)1

w(xi)

n−1∑j=0

αjΦj(xi)w(xi)

=

n−1∑j=0

αj

n∑i=1

wiΦk(xi)1

w(xi)Φj(xi)w(xi)

=n−1∑j=0

αj

n∑i=1

wiΦk(xi)Φj(xi)

=

n−1∑j=0

αj

∫ ∞−∞

Φk(x)Φj(x)w(x)dx

=n−1∑j=0

αj δk,j

=αk,

(3.65)

where we used the exactness up to degree 2n− 1 of the quadrature formulaFrom αk 6= αk, we then deduce, that our assumption was wrong and we thus have

f(xi) = f(xi) for all i = 1, ..., n, (3.66)

which completes the proof.The approximation f is of course still converging to the right solution f as n→∞.

Chapter 4

Motivational Examples

Prior to the development of an abstract framework as explained in Chapter 5, wewant to show some tests using different kinetic equations and basis functions for theexpansion in order to see how the emerging system for the basis coefficients looks like.The experimental results of this chapter help us to understand the difference betweenexact and quadrature-based projections as we will explain the derivation of the PDEsystem.

We start with some small one-dimensional examples and consider different simplekinetic equations with generalized advection velocities that are closely related to thefully transformed Boltzmann equation, which we will see in the next Chapter 5. Wewill observe that the use of recursion formulas for the basis functions allows an exactcalculation of the system matrix.

Furthermore, we extend the examples to multiple spatial dimensions using a Hermitetensor ansatz. The results are consistent with the one-dimensional case and show thatthe properties of the system are closely related to the choice of the basis functions andthe projection method.

4.1 One-Dimensional Cases

Before we cover kinetic equations in three spatial dimensions, we will first take a lookat the simpler one-dimensional equation and explain the ideas and methods, which wewill essentially also use for the more important case with three spatial dimensions.

4.1.1 Simple Kinetic Equation

In order to get a better understanding, we start with the standard kinetic equation,choose Hermite basis functions for test and ansatz space and apply either an exactprojection or a quadrature-based projection to investigate possible differences.

We consider the following equation

∂

∂tf(t, x, c) + a(c)

∂

∂xf(t, x, c) = 0, (4.1)

31

32 Motivational Examples

which can be seen as the easiest case of a collision-free Boltzmann equation, when wethink of a(c) = c or a general kinetic equation for any a(c).

We want to use an ansatz for the unknown distribution function f . We expand faround the equilibrium distribution (or weighting function) w(c) = 1√

2πe−c

2/2 by a series

of polynomials Φi as follows

f(t, x, c) =1√2πe−c

2/2n∑i=0

αi(t, x)Φi(c), (4.2)

which (using the Einstein sum convention) leads to

w(c) (∂tαi(t, x)Φi(c) + ∂xa(c)αi(t, x)Φi(c)) = 0 (4.3)

⇒ w(c) (Φi(c)∂tαi(t, x) + a(c)Φi(c)∂xαi(t, x)) = 0. (4.4)

Consider now the case a(c) = c. Exemplarily, we here choose normalized Hermitepolynomials for Φ, so we have Φi(c) = Hi(c). Using the recursion formula from Equation(3.11), we can express cHi(c) in terms of Hermite polynomials only

w(c)(Hi(c)∂tαi +

(√i+ 1Hi+1(c) +

√iHi−1(c)

)∂xαi

)= 0. (4.5)

We now project the emerging equation using different projection methods. First,consider the continuous projection

Pj(f) =

+∞∫−∞

f(t, c, x)Hj(c)dc =< f,Hj >w, (4.6)

which means that we basically multiply with the j−th basis function and integrate overthe whole velocity domain.

The second projection method employs a quadrature rule for the calculation andthus reads

Pj(f) =n+1∑k=1

wkf(t, ck, x)1

w (ck)Hj (ck) =< f,Hj >N (4.7)

with quadrature weights wi and sampling points ci according to the specific quadraturerule. The subscript N = n + 1 is the number of sampling points or the order of thequadrature rule. Note that we have to multiply with the inverse of the weighting functionto cancel out the weighting function inside f .

Applying the continuous projection Pj(f) =< f,Hj >, we obtain

< Hi, Hj >w ∂tαi +√i+ 1 < Hi+1, Hj >w ∂xαi +

√i < Hi−1, Hj >w ∂xαi = 0. (4.8)

Now we make use of the orthonormality property of the Hermite polynomials (seeEquation (3.10), i.e. < Hi, Hj >w= δi,j . For technical reasons, we set αj = 0 for j < 0and also for j > n. Thus

δi,j ∂tαi +√i+ 1 δi+1,j ∂xαi +

√i δi−1,j ∂xαi = 0 (4.9)

⇒ ∂tαj +√j∂xαj−1 +

√j + 1∂xαj+1 = 0 (4.10)

Motivational Examples 33

for j = 0, ..., nWe have now derived a system of equations for the unknown basis coefficients αi. It

is possible to write the emerging system in compact matrix form using α = {αi}i=0,...,n

as∂tα+Ac∂xα = 0 (4.11)

with system matrix Ac ∈ Rn+1×n+1

Ac =

0√

1 0 . . . 0√

1 0√

2. . .

...

0√

2. . .

. . . 0...

. . .. . . 0

√n

0 · · · 0√n 0

. (4.12)

Interestingly, the eigenvalues of the matrix are exactly the roots of the (n + 1)-stnormalized Hermite polynomial Hn+1:

σ (Ac) = {λ ∈ R|Hn+1 (λ) = 0}. (4.13)

Going back to the previous Section 3.4, this becomes clear, as the system matrix is justthe Jacobi matrix of the Hermite basis.

If we use the quadrature-based projection method (4.7), the result would not changeat all, because we just replace the standard scalar product < f,Φj >w by the correspond-ing quadrature rule < f,Φj >N and we know from Section 3.5 that the quadrature ruleis exact up to order 2N − 1 = 2(n + 1) − 1 = 2n + 1 of the integrand. As the integralwith the highest order integrand is < Hn+1, Hn >N , the integrand thus has a polynomialdegree of 2n+ 1 and is just integrated exactly.

4.1.2 Generalized Kinetic Equation c2

Now we consider another kinetic equation with a(c) = c2. The reason for that is thatin the transformed Equation (5.4) we will later have quadratic and mixed terms likeξiξj which we want to understand in an easier setting before. We can make use of therecursion relation (see Equation (3.11)) twice, to transform the occurring term a(c)Hi(c)as follows:

c2Hi(c) =√

(i+ 1)(i+ 2)Hi+2(c) + (2i+ 1)Hi(c) +√i(i− 1)Hi−2(c). (4.14)

After that, we again perform the continuous projection (4.6) first and use the or-thonormality property (see Equation (3.10)) of the Hermite polynomials to arrive at

δi,j∂tαi +√

(i+ 1)(i+ 2)δi+2,j∂xαi+(2i+ 1)δi,j∂xαi+√i(i− 1) δi−2,j∂xαi =0

(4.15)

⇒ ∂tαj+√

(j − 1)j ∂xαj−2 +(2j + 1)∂xαj +√

(j + 2)(j + 1)∂xαj+2=0(4.16)


for j = 0, ..., n, which leads to the following system of equations

∂tα+Ac2∂xα = 0 (4.17)

with symmetric system matrix Ac2 ∈ Rn+1×n+1

Ac2 =

1 0√

1 · 2 0 . . . 0

0 3 0√

2 · 3 . . ....

√1 · 2 0 5

. . .. . . 0

0√

2 · 3 . . .. . .

. . .√

(n− 1) · n...

. . .. . .

. . . 2n− 1 0

0 · · · 0√

(n− 1) · n 0 2n+ 1

. (4.18)

In this case, the eigenvalues of the matrix Ac2 are not exactly the zeros of then + 1-st normalized Hermite polynomial Hn+1. The eigenvalues are actually a mixtureof the squared zeros of the n+ 1-st and the n+ 2-nd Hermite polynomial:

λ2 ∈ σ (Ac)⇔ Hn+1 (λ) = 0 ∨Hn+2 (λ) = 0. (4.19)

The relation to the Jacobi matrix is that we can take Jn+1 · Jn+1, delete the lastcolumn and row and then end up with Ac2 . This holds, because we have

Jn+1 · Jn+1 =

1 0√

1 · 2 0 . . . 0

0 3 0√

2 · 3 . . ....

√1 · 2 0 5

. . .. . . 0

0√

2 · 3 . . .. . .

. . .√n · (n+ 1)

.... . .

. . .. . . 2n+ 1 0

0 · · · 0√n · (n+ 1) 0 n+ 1

.

(4.20)

Contrarily, we obtain

Jn · Jn =

1 0√

1 · 2 0 . . . 0

0 3 0√

2 · 3 . . ....

√1 · 2 0 5

. . .. . . 0

0√

2 · 3 . . .. . .

. . .√

(n− 1) · n...

. . .. . .

. . . 2n− 1 0

0 · · · 0√

(n− 1) · n 0 n

. (4.21)

So that the last value (Ac2)n+1,n+1 = 2n+ 1 differs from the last value of Jn · Jn, whichleads to the change of the eigenvalues.


Like before, we now apply the quadrature-based projection (4.7) and will notice verysoon that there is a difference in the emerging system of equations. The integral withthe integrand of highest degree is now

√(n+ 2)(n+ 1) < Hn+2, Hn >N , which is not

integrated exactly, due to n+2+n = 2n+2 = 2(n+1) = 2N > 2N−1. Exact integrationwould lead to

√(n+ 2)(n+ 1) < Hn+2, Hn >w= 0 due to orthogonality, but we get a

value of√

(n+ 2)(n+ 1) < Hn+2, Hn >N= −(n + 1) = −N using quadrature. Thischanges the last entry of the matrix to n.

The new system matrix Ac2 then is

Ac2 =

1 0√

1 · 2 0 . . . 0

0 3 0√

2 · 3 . . ....

√1 · 2 0 5

. . .. . . 0

0√

2 · 3 . . .. . .

. . .√

(n− 1) · n...

. . .. . .

. . . 2n− 1 0

0 · · · 0√

(n− 1) · n 0 n

. (4.22)

The small difference in the last entry of the matrix leads to a change in the eigen-values. Now the eigenvalues of Ac2 are purely the squared roots of the n+ 1-st Hermitepolynomial:

σ(Ac2) = {λ2 ∈ R|Hn+1 (λ) = 0}. (4.23)

The reason is that now the system matrix is Ac2 = Jn · Jn, including the smaller entry(Ac2)n+1,n+1 = n.

4.1.3 Generalized Kinetic Equation c+ c2

As we have seen in the previous case, the quadrature method is able to change theeigenvalues of the system to be simply the squared roots of a Hermite polynomial. Forour shifted and scaled equation later on, we will also need another type of equation,which is very similar to equation (4.1) using a(c) = c+ c2.

With this setting, it is now possible to redo the same calculation from above andcalculate the eigenvalues of the corresponding system. After insertion of the ansatz (4.2),we can perform a projection.

For the continuous projection (4.6) we get eigenvalues that do not correspond to theroots of a Hermite polynomial at all.

If we use the quadrature-based projection method (4.7), the eigenvalues are againchanged and in fact related to the roots of the (n + 1)-st Hermite polynomial in thefollowing way:

σ(Ac+c2

)= {λ+ λ2 ∈ R|Hn+1 (λ) = 0}. (4.24)

It therefore seems apparent that the quadrature projection always changes the eigen-values corresponding to the equation using a(c) to be the roots λ of the n+1-st Hermitepolynomial inserted into a (λ). We also did tests with different basis functions, for


example Legendre polynomials or Laguerre polynomials and the eigenvalues alwaysbehaved in the respective way.

4.1.4 Relation between Quadrature Projection and Discrete VelocityMethod

In the previous Section 4.1, we have seen that simple quadrature-based projections sufficeto change the system slightly to systematically obtain specific eigenvalues of the systemmatrix. This is closely related to the procedure during a discrete velocity method (DVM),which we have described in Section 2.3.4. Motivated by this examples, we will now takea closer look at the relation between the two methods and see that they are essentiallythe same for the simple kinetic equations we used in the previous sections.

Regarding the PDE (4.1) and a specific ansatz with arbitrary basis function Φi, sothat w(c) = 1, the distribution function reads

f(t, x, c) =n∑i=0

αi(t, x)Φi(c). (4.25)

If we insert the ansatz (4.25) into Equation (4.1), we can evaluate the equation atdiscrete velocities cj for j = 0, . . . , n to obtain n different equations

n∑i=0

Φi(cj)∂tαi(t, x) + a(cj)Φi(cj)∂xαi(t, x) = 0. (4.26)

This system of equations can be written in matrix vector form using the followingdefinitions

α = (α0(t, x), . . . , αn(t, x))T , (4.27)

B =

Φ0(c0) . . . Φn(c0)...

. . ....

Φ0(cn) . . . Φn(cn)

, (4.28)

A = diag(a(c0), . . . , a(cn)). (4.29)

The system can then be written in the following very compact form

B∂tα+AB∂xα = 0. (4.30)

Assuming B is regular, we can multiply by its inverse B−1 from the left to end up withthe system

∂tα+B−1AB∂xα = 0. (4.31)

We can now directly see that the eigenvalues of the system are the diagonal entries ofA, as the multiplication with the regular matrices B and B−1 does not change theeigenvalues of A. The eigenvalues of the system are therefore point evaluations of the


function a(c) at the discrete velocities.

Now we proceed differently using the following quadrature formula (with respect toweighting function w(c) = 1)) for the projection of Equation (4.1) with the projectionoperator Pj(f)

Pj(f) =n∑k=0

wkf(t, x, ck)Φj(ck) =< f,Φj >N . (4.32)

This leads to the PDE system (j = 0, . . . , n)

n∑i=0

n∑k=0

wkΦi(ck)Φj(ck)∂tαi(t, x) + wka(ck)Φi(ck)Φj(ck)∂xαi(t, x) = 0. (4.33)

Similar to the system before, we write this system in matrix vector form using thefollowing additional definitions

C =

w0Φ0(c0) . . . wnΦn(c0)...

. . ....

w0Φ0(cn) . . . wnΦn(cn)

, (4.34)

W = diag(w0, . . . , wn). (4.35)

Thus, the system readsCB∂tα+CAB∂xα = 0 (4.36)

or using C = BWBWB∂tα+BWAB∂xα = 0. (4.37)

Assuming that B is regular and the weights are non zero (which is usual for quadraturerules), we again obtain the same system as before

∂tα+B−1AB∂xα = 0, (4.38)

which means that the eigenvalues of the system are still the diagonal entries of the matrixA, i.e. point evaluations of the function a(c) at the distinct quadrature points.

Our observation did not rely on the specific choice of basis functions. The onlyrequirement is that the matrix B is invertible, including the fact that the quadraturepoints or discrete velocities respectively have to be pairwise distinct.

The observation can be made, because the quadrature method is basically evaluatingthe equation at distinct points and combining the resulting equations in a linear way toend up with a more complicated system than the discrete velocity method. But as wehave seen, the systems and their properties are essentially the same.

If we now choose a standard projection method that computes exact integrals over thevelocity space, this linear combination property is not given anymore. The appearanceof the function a(c) inside the integral leads to the use of recursion formula, as we have


done it in the previous Chapter 3. But the result cannot be generalized to arbitraryfunctions a(c), as recursion formulas are usually only available for a limited class offunctions, such as polynomials. Even for polynomial a(c), recursion formulas lead todifferent values than for the quadrature case, as we have seen before (see Section 4.1.2),changing the behavior of the system as well as the eigenvalues.

This result is a first glimpse at the generalization of the projection procedure,here still for a very simple equation. We will later build up on the formulations anddevelop an abstract framework to understand the quadrature-based projections also inthe transformed Boltzmann equation in Chapter 5.

4.2 Multi-Dimensional Cases

In more than one spatial dimension, there are different possible ways to propose anansatz for the distribution function f . The most simple one is the consistent extensionof the 1D example from the previous Section 4.1. Now we describe a Hermite tensoransatz in three dimensions by taking a basis that is a Hermite polynomial in everyvelocity direction.

4.2.1 Simple Kinetic Equation 3D

The first 3D kinetic equation we consider is again rather simple

∂

∂tf(t,x, c) + a(c)

∂

∂xf(t,x, c) = 0 (4.39)

Where we first only cover the case a(c) = c and use the consistent extension of our

one-dimensional ansatz from before (see Equation 4.2) with weight w(c) = 1√2πe−c

T c/2

f(t,x, c) =1√

2π3 e−cT c/2

n∑i,j,k=0

αi,j,k(t,x)Φi,j,k(c). (4.40)

But the basis function is now a Hermite polynomial in every direction:

Φi,j,k(c) = Hi(cx) Hj(cy) Hk(cz). (4.41)

The projections are either done by analytical computation of the integrals

Pl,m,n(f) =

+∞∫−∞

+∞∫−∞

+∞∫−∞

f(t,x, c)Φl,m,n(c)dc =< f,Φl,m,n > (4.42)

or by a Hermite-Gauss quadrature formula of order N as follows

Pl,m,n(f) =N+1∑

k1,k2,k3=1

wk1wk2wk3f(t, ck1 , ck2 , ck3 ,x)

w (ck1 , ck2 , ck3)Φl,m,n(ck1 , ck2 , ck3) =< f,Φl,m,n >N .

(4.43)


The derivation of the system of equations can be carried out just like in the 1Dcase (see Section 4.1.1). The only difference in this 3D case is that we have more termsemerging from the expansion in each of the three velocity components. We will omit thederivation here and go directly to the results of the different projection methods.

Using the Hermite tensor ansatz from above (see Equation (4.40)), we cannot expectrotational symmetry of the problem, as we have used an expansion in cartesian coordi-nates along each single axis that is not invariant under rotational transformation. Wecan therefore only expect symmetry with respect to one of the coordinate axes for bothprojection methods.

The projection of Equation (4.39) with the operator (4.42) or (4.43) leads to a coupledsystem of PDEs in three spatial variables. The general form of the system is

A∂tα+B∂xα+C∂yα+D∂zα = 0, (4.44)

where we usually have A = In for orthonormal basis functions.

In order to check hyperbolicity of the system and get some knowledge about thebehavior of the system, we take a look at the generalized system matrix for unit vectorβ = (β1, β2, β3) of length one, so that ‖ β ‖= 1. The system matrix can be written as

Asys = A−1 (Bβ1 +Cβ2 +Dβ3+) . (4.45)

This is also a consistent extension of the one-dimensional case, because the systemmatrix is evaluated for every direction β, where the unit vector β can be seen as avector pointing in the direction of interest. The system is said to be hyperbolic, if Asys

has real eigenvalues for all directions β.

For numerical calculations, it is possible to plot the eigenvalues of the system matrixdepending on the direction β in a 3D plot for different n.

Figures 4.1a and 4.2a show the full plot for Hermite functions up to degree n = 1 andn = 2 respectively. The eigenvalues lie on circles that cross through points (cx, cy, cz) attheir largest distance from the origin, where the coordinates cx, cy, cz are combinationsof zeros of the (n+ 1)-st Hermite polynomial Hn+1(c).

We can see the same behavior for the case n = 2, but the high number of degrees offreedom (16) leads to a very dense plot. Thus we cut the plots at cz = 0 showing valuesinside the x − y−plane in Figures 4.1b and 4.2b. There we can in principle observeanalogous results for the 2D case. Circles extending up to the coordinates of roots ofthe Hermite polynomials are combined in the plane.

For the sake of completeness, we give the roots of the involved polynomials

H1(c) = 0⇔ c ∈ {−1, 1}, (4.46)

H2(c) = 0⇔ c ∈ {−√

3, 0,√

3}. (4.47)

As we have seen, the results are in perfect agreement to the one-dimensional case fromthe previous Section 4.1. But in addition we now have eigenvalues of Asys dependingon the direction of the flow. Due to the Hermite tensor ansatz in cartesian coordinates,


(a) full plot (b) plot cut at cz = 0

Figure 4.1: 3D spherical plot of eigenvalues for n = 1

the system’s behavior is no longer isotropic. We therefore have certain directions of theflow in which information is propagated faster than in other directions. This is usuallynot desired, because the discrete model looses its symmetry on the one hand and is notlonger Galilei invariant on the other hand. An ansatz as simple as the Hermite tensorin 3D can obviously not overcome these problems.

Note that the results of the projection are the same for both the continuous (4.42)and quadrature projection (4.42), because the degree of exactness of the quadratureformula ensures exact integration of all the involved terms. This will not be the case fora(c)i = c2i in the following section.

4.2.2 Generalized Kinetic Equation c2i 2D

After assuring comparable behavior of the 3D case regarding the standard simple kineticequation (4.39), we now turn our attention to the multi-dimensional version of the secondtest case from the previous section. Replacing the convective velocities cx, cy, cz by theirsquares or simply setting a(c)i = c2i , we obtain the equation

∂

∂tf(t,x, c) +

3∑i=1

c2i∂

∂xif(t,x, c) = 0. (4.48)

Using the exact same ansatz and projection methods (see Equations (4.40) and (4.42)or (4.43)), we can derive a new system of equations for the unknown basis coefficients αof the form (4.44).

The eigenvalues of the system matrix Asys do again depend on the directional vectorvariable β, but can be plotted for various values of β and n.


(a) full plot (b) plot cut at cz = 0

Figure 4.2: 3D spherical plot of eigenvalues for n = 2

Figures 4.3a, 4.4a and 4.5a show the resulting plots for the continuous projectionaccording to the 2D case of Equation (4.42). These plots can be obtained by the settingβ3 = 0 in the definition of the system matrix Asys, for example.

We see (n + 1)2 circles going through the marked points at their largest distancefrom the origin. These points’ coordinates are combinations of all squared nonzero rootsof the (n+ 1)-st and the (n+ 2)-nd Hermite basis polynomial. This corresponds to the1D case, where eigenvalues were also related to either the (n+ 1)-st and the (n+ 2)-ndHermite polynomial.

The results for the quadrature-based projection method as defined in the general3D setting in Equation (4.43) are very different from the continuous projection. Theeigenvalues for a quadrature projection can be seen in Figures 4.3b, 4.4b and 4.5b.

For n = 1 (see Figure 4.3b), there is only one circle, because all four eigenvaluesdegenerate to one with algebraic multiplicity 4. This is in agreement with the 1D case,because the corresponding squared zero of the (n+1)-st Hermite polynomial is also only1, which is in fact the cx and cy coordinate of the circle’s point with the largest distancefrom the origin.

Proceeding to the case n = 2, the relevant squared zeros of the third Hermitepolynomial are 0 and 3, which can be combined to the coordinates of the circles’ pointswith the largest extension from the origin, again. Note that this time, there is also acircle with radius zero, represented by the origin itself.

For n = 3, we see the same behavior regarding the squared zeros of the 4-th Hermitepolynomial, which are 3±

√6. Combinations of those values mark the specific points in


-1 1 2 3 4

-1

1

2

3

4

(a) continuous projection

-1 1 2 3 4

-1

1

2

3

4

(b) quadrature-based projection

Figure 4.3: 2D polar plot of eigenvalues for n = 1

-2 2 4 6

-2

2

4

6


-2 2 4 6

-2

2

4

6




-4 -2 2 4 6 8 10

-4

-2

2

4

6

8

10


-4 -2 2 4 6 8 10

-4

-2

2

4

6

8

10



Figure 4.5b.Summarizing, we see a very similar result compared to the 1D case. The eigenvalues

are combinations of roots of Hermite polynomials of degree n + 1 and n + 2 if we usecontinuous (exact) projections. On the other hand, we purely have a relation to thesquared zeros of the (n+ 1)-st Hermite polynomial for the quadrature-based projectionmethod.

The results show a different behavior of the quadrature-based projection methodscompared with the exact projections. The emerging system of PDEs is always hyper-bolic, because there is no variable transformation involved, the PDEs are linear in theunknowns αi. As we will see in the next chapters, the hyperbolicity of the system dependson the variables themselves, if we perform a transformation of the velocity variable.

44

Chapter 5

Theoretical Concepts

This chapter is divided into two sections. The first section deals with a transformationof the velocity variable in the Boltzmann equation to allow for physical adaptivity ofour discretizations. The result is a transformed Boltzmann equation that is coupledto the macroscopic flow variables by so called compatibility conditions. In the secondsection we develop the abstract framework to understand and prove hyperbolicity of theemerging PDE system. After the derivation of hyperbolicity conditions, we show thatthese conditions hold in the case of a Hermite ansatz in one dimension.

5.1 Preliminaries

The construction of our conceptual framework relies on the fact that the solution ofthe Boltzmann Transport Equation still poses difficult challenges for modern methods,because the discretization of the velocity space adds to the complexity of the scheme.The velocity space is in general unbounded and especially in real applications high-dimensional due to lack of symmetry. This leads to major drawbacks when it comesto standard discrete velocity methods, see also Section 2.3.4, because the number ofdiscrete velocities needed to accurately resolve the physical domain makes the schemecomputationally very expensive. Other methods like DSMC, compare Section 2.3.1, areonly applicable for reasonable high Knudsen numbers.

Here we want to follow the philosophy of moment methods, where a small numberof variables (the so-called moments) describes the behavior of the flow with sufficientaccuracy. We are in principle deriving sets of closed partial differential equations whichmodel rarefied gas flows. The set of equations includes the balance laws for mass,momentum and energy and the closure is done by a coupling of the heat flux to the restof the equations.

We closely follow the approach proposed by Kauf in [17], but perform the derivationsin a general d−dimensional setting and take a closer look at the emerging equations inorder to examine stability properties of the PDE system. The central ideas of our workare summarized in the following sections.

45

46 Theoretical Concepts

First, we show how a transformation of variables combined with a flexible expansionallows for an efficient solution of the Boltzmann equation. We will also cover the problemof a closure of the emerging system by so-called compatibility conditions. Later, we wantto establish the theoretical framework that enables us to apply certain quadrature-basedprojection methods and achieve hyperbolic equations.

5.1.1 Variable Transformation for Physical Adaptivity

One of the reasons why the Boltzmann equation is so difficult to solve is that a dis-cretization requires a very large mesh, because the mean velocity of the flow can inprinciple attain very large values in magnitude and even for moderate mean velocitieswe can observe large microscopic velocity values. It is nevertheless possible to rescalethe equation, such that the mean velocity is zero and the microscopic velocity is alwaysof the order O(1). One then only has to consider a relatively small domain for thediscretization of the velocity space and can still capture most of the relevant effects.

Likewise to [17], the velocity is transformed in a highly non-linear way to allow forintrinsic physical adaptivity of the scheme. As a probability distribution, the densityfunction f usually extends over a large domain in velocity space, but in some cases itcan also exhibit a very small region of non-negligible values. The position and size ofthis region are all connected to the mean and the variance of the distribution function,which correspond to the velocity v and the temperature θ. A reasonable transformationnow shifts the microscopic velocity c by its macroscopic counterpart v and scales by thevariance

√θ, such that the transformed density function is centered around a transformed

mean of zero and has variance of one.

The corresponding Galilei-invariant variable transformation reads

c(t,x) =⇒ c− v(t,x)√θ(x, t)

=: ξ(t,x, c). (5.1)

Now ξ can be seen as a shifted and scaled (or dimensionless) velocity variable. Thedistribution function f(ξ) is usually close to zero for already moderate values of ξ andcentered around ξ = 0 because of the velocity shift in the definition of ξ.

Note that the transformation is more complicated than it looks like, because thevariables involved are in fact moments of the distribution function, see (2.3) and (2.6).

The transformation to ξ enables a better approximation quality, as we can see froman 1D example in Figures 5.1a and 5.1b.

Using the transformed variable ξ, the distribution function f is centered around zeroand decays rapidly for already moderate values of ξ. Therefore, an accurate resolutiononly requires a limited number of point values for ξ inside the domain [−3, 3], for example.It is important to mention that the general form of the distribution function stays likethat during the calculation, while the velocity v and temperature θ change.

Theoretical Concepts 47

-10 -5 0 5 10c

0.2

0.4

0.6

0.8

1.0fHcL

f2

f1

(a) untransformed Gaussians, depending on c

-4 -2 0 2 4Ξ

0.2

0.4

0.6

0.8

1.0

fHΞL

f2

f1

(b) transformed Gaussians, depending on ξ

Figure 5.1: Gaussians f1 with ρ1 = 1, v1 = 3, θ1 = 0.3 and f2 with ρ2 = 0.8, v2 = −4,θ2 = 5

5.1.2 Derivation of Transformed Boltzmann Equation

We will now see how the Boltzmann equation

∂

∂tf(t,x, c) + ci

∂

∂xif(t,x, c) = S(f) = 0 (5.2)

is changed when applying the transformation from above. As a first example, we considerthe 1D version of (5.2) and perform the transformation as specified in (5.1.


According to Kauf [17] the derivatives change because of the chain rule and theBoltzmann equation turns into a slightly more complicated equation as follows (usingthe convective time derivative Dt := ∂t + v∂x)

Dtf +√θξ∂xf + ∂ξf

(− 1√

θ

(Dtv +

√θξ∂xv

)− 1

2θξ(Dtθ +

√θξ∂xθ

))= 0. (5.3)

The full multi-dimensional case is essentially a consistent extension of the simple 1Dcase from above. The d-dimensional Boltzmann equation (5.2) is transformed using thechange of variables from above and results in

Dtf +√θξj∂xjf + ∂ξjf

(− 1√

θ

(Dtvj +

√θξi∂xivj

)− 1

2θξj

(Dtθ +

√θξi∂xiθ

))= 0.

(5.4)Note the difference between the original version (5.2) and the new transformed

versions (5.3) or (5.4) as additional terms depending on v and θ appear due to thetransformation of the velocity variable ξ. If we want to utilize the advantageous adaptiveformulation from above, we have to deal with these additional terms. Note that thedensity ρ did not yet enter the equations, this happens only after the insertion of theansatz specified in the next subsection 5.1.3 as the ansatz also depends on ρ.

A simpler transformation takes only the velocity shift into account and does not usethe temperature scaling. The transformation thus reads

c(t,x) =⇒ c− v(t,x) =: ξ(t,x, c). (5.5)

This leads to a simplified version of (5.4) without any temperature appearance. Theequation can be obtained by analogous calculations from above or by setting θ ≡ 1 inEquation (5.4). The result is the shifted Boltzmann equation

Dtf + ξj∂xjf + ∂ξjf (−Dtvj − ξi∂xivj) = 0. (5.6)

When it comes to the compatibility conditions (see Section 5.1.4), Equation (5.6) onlyneeds the conditions for the velocity, because the density ρ and the temperature θ donot enter the equation. This equation is thus easier to analyze, but still exhibits thegeneral problems we want to overcome.

5.1.3 Expansion Using Basis Functions

Additional to the transformation of the velocity variable, we propose a similar form ofansatz that Kauf used in [17]. The expansion of the unknown distribution function f isdone around its equilibrium distribution, which is a Maxwellian (compare (2.9)). Thismakes sense, as we usually have perturbations from the Maxwellian distribution if we arein non-equilibrium. The specific form of the perturbations is modeled by the coefficientsof a set of arbitrary basis functions Φ. The expansion in d dimensions thus reads

f(t,x, ξ) =ρ

√2πθ

dexp

(−ξ2/2

) n∑i=0

αi(t,x)Φi(ξ) (5.7)


or with another notation

f(t,x, ξ) =ρ√θdw(ξ)

n∑i=0

αi(t,x)Φi(ξ) =ρ√θdf(t,x, ξ). (5.8)

This type of expansion uses the information about the overall shape of the distribu-tion function, which is close to a Maxwellian most of the time. Deviations from thatshape should be efficiently expressed by the basis functions Φi.

The type of basis function is not yet specified here. The Φi can in principle beany function, e.g. polynomials, spherical harmonics, wavelets, splines. This gives theframework some kind of flexibility and variation to choose from later.

Using the notation from Equation (5.8), we can insert the ansatz into the d-dimensionalBoltzmann equation (5.4) and obtain an equation for the basis expansion f . Thederivatives of f with respect to some variable g transform to partial derivatives of f(or f) according to

df

dg=∂f

∂ρ

∂ρ

∂g+∂f

∂θ

∂θ

∂g+∂f

∂g(5.9)

with partial derivatives

∂f

∂ρ=

1

ρf,

∂f

∂θ=− d

2θf.

(5.10)

Inserting this for the expressions Dtf and ∂xjf , we obtain

(1

ρDtρ−

d

2θDtθ

)f +Dtf+

√θξj

((1

ρ∂xjρ−

d

2θ∂xjθ

)f + ∂xj f

)+

∂ξj f

(− 1√

θ

(Dtvj +

√θξi∂xivj

)− 1

2θξj

(Dtθ +

√θξi∂xiθ

))=0.

(5.11)

Where all the derivatives can be seen as partial derivatives, so that the prefactor ρ√θd

has been factored out and we multiplied with its inverse to make the equation simpler.

5.1.4 Compatibility Conditions

Since the macroscopic variables ρ, v and θ are already included in the ansatz (5.7), weneed additional relations in order to end up with a closed system of equations later. Wetherefore insert the expansion (5.7) into the definition of the macroscopic variables fromSection 2.1.3. We only consider dimensionless masses of the particles, i.e. m = 1.


5.1.4.1 1D Case

To begin with the 1D case, we have to compute integrals over the one-dimensionalvelocity space. For the transformation of the integrals, we use dξ = dc√

θ. For the density,

this yields

ρ=

+∞∫−∞

f(t, x, c)dc =ρ√θ

+∞∫−∞

f(t, x, ξ)√θdξ = ρ

+∞∫−∞

f(t, x, ξ)dξ

⇒ 1= < f(ξ), 1 > .

The definition of the macroscopic velocity results in

ρv=

+∞∫−∞

cf(t, x, c)dc =ρ√θ

+∞∫−∞

(√θξ + v

)f(t, x, ξ)

√θdξ

⇒ ρv=√θ

+∞∫−∞

ξf(t, x, ξ)dξ + vρ

⇒ 0 = < f(ξ), ξ > .

Third, we use the energy to derive the last condition

ρθ=

+∞∫−∞

(c− v)2 f(t, x, c)dc =ρ√θ

+∞∫−∞

√θ2ξ2f(t, x, ξ)

√θdξ

⇒ 1 = < f(ξ), ξ2 > .

Summarized, our three compatibility conditions for one-dimensional problems are

1= < f(ξ), 1 >,

0= < f(ξ), ξ >,

1= < f(ξ), ξ2 > .

(5.12)

It is important to point out that these conditions are independent from the particularchoice of the basis functions Φi. With the help of specific basis functions, the conditions(5.12) translate into conditions for the basis coefficients αi(t, x).

5.1.4.2 3D Case

Similar as for the one-dimensional case, we need to derive compatibility conditions for thebasis coefficients. The only difference is here that we need to perform three-dimensionalintegrals over the whole velocity space, as specified in the definition of the macroscopic


quantities. For the transformation of the integrals, we now use dξi = dci√θ. We start with

the density

ρ=

∫∫∫f(t,x, c)dc =

ρ√θ3

∫∫∫f(t,x, ξ)

√θ3dξ =

∫∫∫f(t,x, ξ)dξ

⇒ 1= < f(ξ), 1 > .

For the velocity components, we proceed analogously (i = 0, 1, 2)

ρvi=

∫∫∫cif(t,x, c)dc =

ρ√θ3

∫∫∫ (√θξi + vi

)f(t,x, ξ)

√θ3dξ

⇒ ρvi=√θ

∫∫∫ξif(t,x, ξ)dξ + viρ

⇒ 0 = < f(ξ), ξi > .

The definition of the energy gives the third condition (index i uses sum convention)

3ρθ=

∫∫∫(ci − vi)2 f(t,x, c)dc =

ρ√θ3

∫∫∫ √θ2ξ2i f(t,x, ξ)

√θ3dξ

⇒ 3 = < f(ξ), ξ2i > .

We see that the five compatibility conditions are in fact a consistent extension of theone-dimensional conditions (5.12):

1= < f(ξ), 1 >,

0= < f(ξ), ξi > , i = 0, 1, 2,

3= < f(ξ), ξ2i > .

(5.13)

For a specific choice of basis functions, these conditions will impose conditions on thebasis coefficients of the expansion ansatz.

5.1.4.3 Coupling of Compatibility Conditions to PDE System

After the derivation of the compatibility conditions (5.13), we have a set of PDEs anda set of algebraic equations for the variables αi as well as ρ, v and θ. The total setof equations is closed with respect to the variables involved, but one still has to thinkabout how to incorporate the algebraic equations into the PDE system.

There are several different approaches, of which Kauf [17] already mentioned aprojection method that numerically tries to project the ansatz space to the admissiblespace by trying to minimize the distance. For the optimization problem, one could applythe various methods described in [8], for example. Another method proposed by Kaufis to construct a new basis by linear combination of old basis functions such that thenew basis satisfies the compatibility condition. This all leads to the reduction of the


dimension of the ansatz space, so that the PDE system can be used solely with the newgiven basis.

In contrast to our procedure, Kauf replaces the total derivatives of the macroscopicvariables ρ, v and θ by the conservation laws and then adds the conservation laws tothe set of equations, together with a coupling through the heat flux.

Instead, we aim to proceed in a different way: We use the algebraic compatibilityconditions to eliminate a set of basis coefficients of the expansion ansatz and then addthe macroscopic variables in their place. This leaves us with a reduced set of basiscoefficients αi and a closed system for in general n variables and n PDEs.

In the general case, we can incorporate the compatibility conditions into the PDEsystem by the following procedure that is related to the treatment of constrained ODEs,so-called differential-algebraic equations (DAEs, compare [11]):

Consider a specific ansatz f(t,x, c) =∑n

i=0 αi(t,x)Φi(c) for the unknown distri-bution function and a given quadrature formula for the calculation of the integralsin the compatibility conditions (5.13). Assuming a total number of l conditions, thecompatibility conditions are simply an underdetermined system of equations accordingto

Qα = g, with Q ∈ Rl×n+1, g ∈ Rl. (5.14)

Now we perform a decomposition of the matrix Q, for example a singular valuedecomposition, such that the matrix Q can be written as

Q = SMT , with S ∈ Rl×l, M ∈ Rl×n+1, T ∈ Rn+1×n+1, (5.15)

for invertible square matrices S and T . The matrix M has the form

M =

σ0 0 . . . 0. . .

.... . .

...σl 0 . . . 0

=:(Q,0

)(5.16)

with the singular values of Q on the diagonal in case of a singular value decomposition.We introduce a set of transformed variables β by the definition

β := Tα. (5.17)

Now it is possible to rewrite the discretized compatibility conditions to

Qα = g⇒ SMTα = g⇒ SMβ = g⇒ Mβ = S−1g

⇒ M

(β1

β2

)= S−1g

⇒ β1 = Q−1S−1g.

(5.18)


for β1 ∈ Rl and β2 ∈ Rn−l, so that we can directly solve for l variables of β, namelythe first l variables β1.

As a decomposition like in (5.15) is always possible, we can always eliminate thenecessary number of variables from our unknown vectorα (or β respectively) and therebyclose the system of PDEs by inserting

α = T−1(β1

β2

)=((T−1

)1,(T−1

)2

)( β1

β2

)=(T−1

)1Q−1S−1g +

(T−1

)2β2

(5.19)for a proper splitting of the columns of T−1 =

((T−1

)1,(T−1

)2

).

With the help of the compatibility conditions, the set of n+ 1 basis coefficients α isthus reduced to a set of n− l variables β2 that leads to a closed PDE system with thesame number of equations and unknowns.

The concrete compatibility conditions for 1D Hermite ansatz functions are used inSection 5.2.3.3 and an example for compatibility conditions in a three-dimensional settingwith either spherical harmonics and Laguerre polynomials or Hermite ansatz functionscan be found in the Appendix A.

5.2 Theoretical Concept

Using the ansatz in Equation (5.7) and the Boltzmann equation as written in Equa-tion (5.3) together with the coupling compatibility conditions, we can in principle decideon a specific basis for the ansatz space and then project the equations using a secondset of test functions. As we will see later, the choice of the projection method will havea major impact on the structure of the equations as well as on their stability properties.The result after the projection is a set of equations for the reduced basis coefficients αiand ρ, v, θ.

Now we turn our attention towards the projection method and the systematic inves-tigation of the emerging PDEs.

Our aim is to work on a framework that enables the solution of the BoltzmannTransport Equation for real applications. We want to be able to choose from differenttypes of basis functions for the ansatz and test space and specify a projection methodof choice to derive the PDE system for the coefficients with incorporated compatibilityconditions. The question of hyperbolicity of the system should be answered a-prioriby the particular choices we have made. It is crucial to end up with a hyperbolicsystem, because a loss of hyperbolicity would lead to a totally nonphysical behavior ofthe solution.

Here is where the quadrature-based projection methods come into play. Unlikestandard projection methods employing exact integration over the velocity space, thesequadrature-based methods can modify the equations in the exactly right way to resultin global hyperbolicity even for the transformed equation, as we will see. But it is stillnot known, which specific quadrature method together with the specific basis functionsleads to the desired hyperbolicity.


We therefore want to develop a theory that gives us necessary and sufficient con-ditions for hyperbolicity. The framework depends largely on the considered PDE andwe will start with a simple so-called generalized kinetic equation. The similarity ofthis equation with the transformed Boltzmann equation (5.3) allows us to conductcomparable results in a still rather straightforward setting.

For the case of the standard Boltzmann equation with BGK collision operator (with-out velocity shift or variable transformation), Shan and He [20] have already shown theequivalence of a LBM (which is very similar to DVM) and an expansion using Hermitefunctions, when evaluating the collision operator at the nodes of Hermite quadrature. Wego much further by considering the transformed equation (5.4) and observe a comparablestructure of the emerging system of equations with respect to the DVM.

As discrete velocity methods are hyperbolic by construction, compare Section 2.3.4,we will start with the DVM and then draw an analogy using quadrature-based projectionmethods.

5.2.1 Generalized Kinetic Equation

First, we develop the theory for the so-called generalized kinetic equation in d dimensions,which reads

∂

∂tf(t,x, c) + ai(c)

∂

∂xif(t,x, c) = 0 (5.20)

for d real-valued functions ai(c), the so-called advection velocities.

5.2.1.1 Discrete Velocity Method

The following Lemma (5.2.1) shows that the DVM indeed leads to a hyperbolic systemof equations of a very simple form.

Lemma 5.2.1 Let f : R+ × Rd × Rd → R, (t,x, c) 7→ f(t,x, c) be a probability densityfunction and

∂

∂tf(t,x, ck) + ai(ck)

∂

∂xif(t,x, ck) = 0 (5.21)

a discrete velocity model of Equation (5.20) with discrete velocities ck ∈ Rd, k = 1, . . . , n.Then, the system of point evaluations (5.21) in the ck is hyperbolic with generalized

eigenvalues λk =∑d

i=1 βiai(ck) for k = 1, . . . , n and ||β||2 = 1

Proof Point evaluations of Equation (5.20) in ck, k = 1, . . . , n lead to the system (5.21).We define

f := (f(t,x, c1), . . . , f(t,x, cn))T ∈ Rn (5.22)

andAi := diag (ai(c1), . . . , ai(cn)) ∈ Rn×n. (5.23)

The system (5.21) can then be rewritten as

∂

∂tf +Ai

∂

∂xif = 0. (5.24)


In order to check hyperbolicity of (5.24), we look at the generalized system matrixAsys := βiAi (||β||2 = 1):

Asys = βiAi = diag (βiai(c1), . . . , βiai(cn)) ∈ Rn×n. (5.25)

The eigenvalues λ : det (In − λAsys) = 0 of the (diagonal) system matrix Asys are justthe diagonal entries of the matrix:

λk = βiai(ck), k = 1, . . . , n (5.26)

�

5.2.1.2 Quadrature-based Projection

The major difference between exact and quadrature-based projection methods is thatquadrature rules use point evaluations of the integrand. Therefore, the resulting systemhas to be related to the discrete velocity method somehow. The relation is stated by thefollowing lemma:

Lemma 5.2.2 Let f : R+ × Rd × Rd → R, (t,x, c) 7→ f(t,x, c) be a probability densityfunction.

Furthermore, let ck ∈ Rd, k = 1, . . . , n be discrete quadrature points with correspond-ing quadrature weights wk ∈ R. Let Φi : Rd → R and Φi : Rd → R be the ansatz and testfunctions, respectively (i = 1, . . . , n). The expansion of f using summation conventionreads

f(t,x, c) = αj(t,x)Φj(c). (5.27)

Then, the projected system of PDEs using quadrature-based projection is hyperbolicwith real eigenvalues λk = βiai(ck), k = 1, . . . , n if the matrices B = (Φj(ci))i,j=1,...,n

and B =(

Φj(ci))i,j=1,...,n

are invertible.

Proof After insertion of the ansatz (5.27) into Equation (5.20) and using a discrete,quadrature-based projection with the quadrature formula specified by the cl and wl andthe test functions Φk, the system reads

wlΦj(cl)Φk(cl)∂

∂tαj(t,x) + ai(cl)wlΦj(cl)Φk(cl)

∂

∂xiαj(t,x) = 0 (5.28)

for k = 1, . . . , n. Now we define

α := (α1, . . . , αn)T ∈ Rn, (5.29)

Ai := diag (ai(c1), . . . , ai(cn)) ∈ Rn×n, (5.30)

B = (Φj(ci))i,j=1,...,n ∈ Rn×n, B =(

Φj(ci))i,j=1,...,n

∈ Rn×n, (5.31)

W = diag (w1, . . . , wn) ∈ Rn×n. (5.32)


The system can then be written in the following form

BTWB

∂

∂tα+ B

TWAiB

∂

∂xiα = 0. (5.33)

B and BT

are invertible by assumption and W is also invertible as a diagonal matrixof non-zero quadrature weights (quadrature weights are usually positive). Thus we canmultiply by their inverse matrices and obtain

∂

∂tα+B−1AiB

∂

∂xiα = 0. (5.34)

The generalized system matrix is now (||β||2 = 1)

Asys = βiB−1AiB (5.35)

and has real eigenvalues

λk = βiai(ck), k = 1, . . . , n. (5.36)

because the multiplication with B−1 from the left and B from the right is a similaritytransformation, not changing the eigenvalues of the matrix βiAi.

�

Remark With the help of

fi(t,x) = f(t,x, ci) = αj(t,x)Φj(ci) (5.37)

we can relate the evaluations fi(t,x) of the DVM and the coefficients αj(t,x) of thequadrature-based projection method by the simple equation

f = Bα (5.38)

There is thus an one-to-one relation between the two sets of variables if the matrixB is invertible and the discrete velocities are equal to the quadrature points ck. Theevaluations of the distribution function f at the discrete velocities ck is only a linearcombination of the coefficients of the quadrature formulation.

If we indeed only multiply (5.33) by(BTW)−1

and use the relation (5.38), we

obtain Equation (5.24). This shows that the quadrature method is in fact equivalent toa pointwise evaluation of the equation.

Remark It is important to note that the eigenvalues do not depend on the specificchoice of basis functions for the ansatz and test space. As long as the matrices Band B are invertible, the eigenvalues only depend on the evaluations of the advectionfunctions ai(c) at the quadrature points and the respective direction β.


5.2.2 Shifted Boltzmann Equation

The transformed Boltzmann equation (5.4) or (5.11) respectively, is actually much harderto analyze, because of the coupling with the macroscopic variables ρ, v and θ. Wetherefore first consider the following shifted Boltzmann equation (5.39)(compare with(5.6))

Dtf + ξj∂xjf + ∂ξjf (−Dtvj − ξi∂xivj) = 0 (5.39)

and perform basically the same procedure as before (compare Section 5.2.1), but withthe notable difference of the additional v-dependence and some algebraic constraints.An extension to the full transformed Equation (5.4) is then straightforward and onlyrequires some technical arguments and definitions as can be seen in Section 5.2.3.

As we want to draw an analogy between discrete velocity methods and the quadrature-based projection methods, we proceed similar to the previous Section 5.2.1 and firstderive the complete system of equations (including the algebraic coupling with thecompatibility conditions) for the DVM followed by the quadrature-based methods ina second step.


For a set of discrete velocities ξk ∈ Rd, k = 1, . . . , n, we consider pointwise evaluations ofEquation (5.39). This leads to the following relation for the unknowns fk (k = 1, . . . , n)(see also Equation (5.22)):

Dtfk +

d∑i=1

(ξk)i∂xifk + ∂ξif

∣∣∣∣ξk

−Dtvi +

d∑j=1

(ξk)j∂xjvi

= 0. (5.40)

Again, we define

f := (f1, . . . , fn)T ∈ Rn (5.41)

and

Ai := diag ((ξ1)i, . . . , (ξn)i) ∈ Rn×n, i = 1, . . . , d. (5.42)

The evaluation of the derivatives is for now summarized by the matrix Df :

Df =

∂ξ1f

∣∣∣∣ξ1

· · · ∂ξdf

∣∣∣∣ξ1

.... . .

...

∂ξ1f

∣∣∣∣ξn

· · · ∂ξdf

∣∣∣∣ξn

∈ Rn×d. (5.43)

The unknowns v and f are combined into one vector u as follows

u := (v1, . . . , vd, f1, . . . , fn) ∈ Rn+d. (5.44)


This allows to write system (5.40) in compact notation as

(−Df , In)D

Dtu+

d∑i=1

Ai (−Df , In)∂

∂xiu = 0. (5.45)

The system (5.45) states n equations for n+ d unknowns u. As additional relations, wehave the compatibility conditions for the velocity:

0 =< f, ξi >, i = 1, . . . , d. (5.46)

In total, we then obtain n + d equations for n + d unknown variables. The systemtherefore seems to make sense as a system of equation for the unknowns.

However, it is more than unclear how to evaluate the derivative terms in the definitionof Df (5.43). There are only point values of the distribution function f available andthere is no natural way to define a derivative or a point evaluation of this derivative atthe discrete velocities. In general, one could suggest simple finite differences on the pointvalues or do a reconstruction of the point values and then compute the derivative of thereconstruction to evaluate this at the discrete velocities. We get back to this problemlater.

We will now make an attempt to incorporate the compatibility conditions (5.46)into the PDE system (5.45) by directly eliminating d unknowns from the system. Thismethod can also be seen as an exact subspace projection, provided that such a solutionexists.

The integral version of Equation (5.46) reads

0 =

∫Rdξkf(t,x, ξ)dξ, k = 1, . . . , d. (5.47)

Using the DVM ansatz for f (compare Equation (2.19)), we can evaluate the integralin Equation (5.47) to get d linear equations coupling the unknowns fk by the followingformula (or any other quadrature rule like the trapezoidal rule, for example)

0 =n∑i=1

(ξk)ifk, k = 1, . . . , d. (5.48)

According to Section 5.1.4.3, we write the emerging system as Qf = 0, i.e. g = 0,with Q ∈ Rd×n. Splitting up the variables as shown in Section 5.1.4.3, we have β1 = 0,

because of β1 = Q−1S−1g and g = 0. We then insert our new variables into the system

(compare to (5.19)):f =

(T−1

)2β2. (5.49)

Introducing the new unknown vector u

u :=

(vβ2

)∈ Rn (5.50)


we are able to rewrite the system (5.45) with only n variables as follows

(−Df ,

(T−1

)2

) DDtu+

d∑i=1

Ai

(−Df ,

(T−1

)2

) ∂

∂xiu = 0. (5.51)

As we can see, system (5.51) includes now only n variables and consists of n equations.Analogously to Lemma (5.2.2), hyperbolicity of the system is guaranteed, if the matrixin front of D

Dt u is invertible.

We do not want to talk about the calculation of Df in great detail for the moment,one could simply use a finite difference as an approximation of the derivatives, but theeffect on the system is still to be investigated. The calculation of the corresponding termwill be more straightforward in the quadrature-based case, which we will cover now.

5.2.2.2 Quadrature-Based Projection

Again, we consider the shifted Equation (5.39) in d dimensions and want to make use ofa quadrature-based projection to arrive at a similar system like (5.51) in the end.

First, we use the ansatz

f(t,x, c) =n∑i=1

αj(t,x)Φj(c) (5.52)

with some basis coefficients αj and ansatz functions Φj . We then need a quadratureformula to compute the following integrals of an arbitrary function g∫

Rdg(ξ)dξ ≈

N∑k=1

wkg(ξk) (5.53)

with weights wk and quadrature points ξk, for k = 1, . . . , N . We will later only considerthe case n = N , so that there is an one-to-one relation between the basis coefficientsand the quadrature rule. Note that the weighting function that we used before is nowhidden in the definition of the Φj or g respectively.

After insertion of (5.52) into Equation (5.39) and multiplication with some testfunction Φm, we can apply the quadrature-based projection according to Equation (5.53)and obtain the following system of equations

N∑k=1

n∑l=1

(wkΦl(ξk)Φm(ξk)Dtαk +

d∑i=1

((ξk)iwkΦl(ξk)Φm(ξk)∂xiαk

))+

N∑k=1

n∑l=1

d∑i=1

wkαl∂ξiΦl(ξ)

∣∣∣∣ξk

Φm(ξk)

−Dtvi +

d∑j=1

(ξk)j∂xjvi

= 0

(5.54)

for m = 1, . . . , n.


In order to write this system in matrix-vector form, we need to make the followingdefinitions analogous to the non-shifted case

α := (α1, . . . , αn)T ∈ Rn, (5.55)

Ai := diag ((ξ1)i, . . . , (ξn)i) ∈ RN×N , i = 1, . . . , d, (5.56)

B = (Φj(ξi))i=1,...,N, j=1,...,n ∈ RN×n, B =(

Φj(ξi))i=1,...,N, j=1,...,n

∈ RN×n, (5.57)

W = diag (w1, . . . , wN ) ∈ RN×N . (5.58)

The evaluation of the derivatives with respect to ξi goes into the definition of the matrixDiB:

DiB =

∂ξiΦ1(ξ)

∣∣∣∣ξ1

· · · ∂ξiΦn(ξ)

∣∣∣∣ξ1

.... . .

...

∂ξiΦ1(ξ)

∣∣∣∣ξN

· · · ∂ξiΦn(ξ)

∣∣∣∣ξN

∈ RN×n, i = 1, . . . , d (5.59)

With the help of the definitions above, we can write Equation (5.54) in matrix-vectorform

BTWB

D

Dtα+

d∑i=1

(BTWAiB

∂

∂xiα

)−

d∑i=1

BTWDiBα

D

Dtvi +

d∑j=1

(BTWAjDiBα

∂

∂xjvi

) = 0

(5.60)

Now we sum up all the unknowns into one vector u according to

u := (v1, . . . , vd, α1, . . . , αn) ∈ Rd+n (5.61)

and define for simplicity

DB := (D1Bα, . . . ,DdBα) ∈ RN×d. (5.62)

This allows us to obtain the full system of n equations for the n+ d unknowns u asfollows

BTW (−DB,B)

D

Dtu+

d∑i=1

BTWAi (−DB,B)

∂

∂xiu = 0. (5.63)

Provided that the matrices BT

andW are invertible (this is only possible for N = n),we can reduce the system (5.63) to the simpler version

(−DB,B)D

Dtu+

d∑i=1

Ai (−DB,B)∂

∂xiu = 0. (5.64)


This does indeed look very similar to the DVM case (5.45).

System (5.64) gives us n equations for n + d unknowns u. To close the system,we employ the velocity compatibility conditions (5.46) like before and get additional drelations.

In contrast to the DVM case, we now have a system in which we can in principleeasily evaluate each expression and directly compute solutions via numerical methodsfor constrained PDEs, because the derivatives inside the matrix DB can be calculated,once a specific basis for the ansatz space is chosen.

Continuing in the fashion of the DVM approach before, we directly want do build inthe constraints into the system in order to reduce the dimension of the unknown vector uand have a quadratic system to analyze. Using the quadrature method proposed before,we can directly evaluate the integrals in (5.46) and get discrete relations between thebasis coefficients in α

0 =N∑l=1

n∑i=1

wlΦi(ξl)(ξk)lαi, k = 1, . . . , d. (5.65)

Similar to the DVM case, we write the emerging system as Qα = 0, i.e. g = 0,with Q ∈ Rd×n. Splitting up the variables as shown in Section 5.1.4.3, we have β1 = 0,

because of β1 = Q−1S−1g and g = 0. We then insert our new variables into the system

(compare to (5.19)):

α =(T−1

)2β2 (5.66)

We now only need to insert this into the PDE system (5.64) and obtain a closed set

of n equations for n variables u =

(ug2

). We therefore define

DB :=(D1B

(T−1

)2g2, . . . ,DdB

(T−1

)2g2)∈ RN×n−d (5.67)

and

B := B(T−1

)2∈ RN×d. (5.68)

We end up with the system

(−DB, B

) D

Dtu+

d∑i=1

Ai

(−DB, B

) ∂

∂xiu = 0. (5.69)

Equation (5.69) is now a system of n equations for a total number of n unknown

variables. Hyperbolicity is obtained, if the matrix(−DB, B

)in front of the time

derivative is regular and thus invertible, because the system matrix then behaves justlike in the simpler cases (see Section 5.2.1).

After this lengthy calculations, we can now summarize the conditions that have tobe fulfilled to arrive at a hyperbolic PDE system using the quadrature-based projectionmethod. The conditions are:


(1) Regularity of matrix B,

(2) Regularity of matrix W or equivalently wi 6= 0, ∀i = 1, . . . , N,

(3) Regularity of matrix(−DB, B

)including rank

(DB

)= d and rank

(B)

= n− d.(5.70)

The conditions 5.70 show that it is sufficient to analyze the involved matrices forregularity to prove hyperbolicity. The choice of the ansatz functions together with thecompatibility conditions are the necessary ingredients for the derivation.

A verification of these conditions 5.70 in the case of Hermite functions for the ansatzspace in one dimension is made in the following Section 5.2.2.3.

5.2.2.3 Hyperbolicity of Shifted BTE and Hermite Ansatz

The previous Section 5.2.2.2 allows us to construct hyperbolic PDE systems for thesolution of the shifted Boltzmann equation. We want to use this rather abstract settingand show an example in which we can apply the framework to actually obtain theresulting hyperbolic system.

With the concrete conditions (5.70) at hand, we now want to check, whether thespecific choice of Hermite functions for the ansatz and test space satisfies the conditionsso that we arrive at a hyperbolic system of equations.

To shorten notation, we consider the one-dimensional case d = 1 (one dimensional xand v). We use the following expansion of the unknown distribution function:

f(t, x, ξ) =1√2πe−ξ

2/2n∑i=0

αi(t, x)ΦHi (ξ) (5.71)

Where ΦHi are the normalized Hermite polynomials from Section 3.1.2. The expansion

can be interpreted by taking

Φi(ξ) :=1√2πe−ξ

2/2ΦHi (ξ) (5.72)

in the setting of the previous calculations in Section 5.2.3.To be consistent with this choice of basis functions, we use Gauss-Hermite quadrature

with N = n, where the quadrature points ξk are the zeros of ΦHN+1 and the wk are the

associated weights.Now we have to check the conditions (5.70) one after another:

The matrix B can be written as

B =1√2πdiag

(e−ξ

20/2, . . . , e−ξ

2n/2) ΦH

0 (ξ0) · · · ΦHn (ξ0)

.... . .

...ΦH0 (ξn) · · · ΦH

n (ξn)

=: EBH (5.73)


The matrix E is regular as a diagonal matrix with non-zero diagonal entries and thematrix B is actually the same matrix as in (3.59) for the case of Hermite polynomials,where this matrix is used to compute the quadrature weights. As the quadratureweights wk correspond to the Gauss-Hermite quadrature, this matrix B has to beinvertible.

For later use, we write B columnwise with bj = (Bi,j)i=0,...,n

B = (b0, . . . , bn) (5.74)

The weighting matrix

W = diag (w1, . . . , wN ) ∈ RN×N (5.75)

is also invertible, because the diagonal entries are the quadrature weights and they areguaranteed to be positive for Gauss-Hermite quadrature.

The last and a bit more difficult part to prove is the regularity of the matrix(−DB, B

). Here we first have to calculate the derivatives of the ansatz functions

Φi(ξ)d

dξΦi(ξ) = −

√i+ 1Φi+1(ξ) (5.76)

where we used (3.13) and then the definition of the basis function (5.72).The matrix D1B can then be explicitly derived

D1B = −

√

1Φ1(ξ0) · · ·√n+ 1Φn+1(ξ0)

.... . .

...√1Φ1(ξn) · · ·

√n+ 1Φn+1(ξn)

(5.77)

or in terms of (5.74)

D1B = − (b1, . . . , bn+1) · diag(√

1, . . . ,√n+ 1

)(5.78)

Note that the last column of D1B or bn+1 is in fact zero, because the ξi are just theroots of Φn+1.

Now we have to figure out, which variable αi we can solve for in order to reducethe number of variables of the system by one. In the case of Hermite polynomials withthe respective additional weighting function, the compatibility condition for the velocityreduces to a very simple relation for the second basis coefficient α1. The condition simplydemands

α1 = 0 (5.79)

In the framework of Section 5.1.4.3, we can denote for this special case of Hermitebasis functions:

Q = (0, 1, 0, . . . , 0) . (5.80)


This leads to the following possible decomposition of Q:

Q = SMT (5.81)

withS = 1, M = (1, 0, 0, . . . , 0) , (5.82)

and

T =

0 1 0 . . . 01 0 0 . . . 0

0 0 1. . .

......

.... . .

. . . 00 0 . . . 0 1

(5.83)

T is thus nothing more than a permutation of the first two elements of the correspondingvector. Together with the relation β1 = 0 (see (5.66) for example), we have

α =(T−1

)2β2 = (β1, 0, β2, ..., βn) (5.84)

or respectivelyβ = (0, α0, α2, ..., αn) . (5.85)

As T−1 = T , it is

(T−1

)2

=

1 0 . . . 00 0 . . . 0

0 1. . .

......

. . .. . . 0

0 . . . 0 1

∈ Rn+1×n (5.86)

meaning that the application of(T−1

)2

to a matrix is equal to the effect of deleting thesecond column of that matrix.

Thus, the other matrices B and D1B can be written without their respective secondcolumn as

B = B(T−1

)2

= (b0, b2, . . . , bn) (5.87)

and (remember that bn+1 = 0)

D1B = − (b1, b3, . . . , bn+1) · diag(√

1,√

3, . . . ,√n+ 1

)(5.88)

The combined matrix DB is here actually a vector and reads

DB = D1Bβ2 = −(√

1b1α0 +√

3b3α2 + · · ·+√nbnαn−1

)(5.89)

Together, this yields the matrix(−DB, B

)as follows:(

−DB, B)

=(√

1b1α0 +√

3b3α2 + · · ·+√nbnαn−1, b0, b2, . . . , bn

)(5.90)


Regularity of the matrix from Equation (5.90) demands that this matrix has columnrank n, which is valid if α0 6= 0. Otherwise, the first column is a linear combinationof the last n − 1 columns. The condition α0 6= 0 is only technical in this calculation,because in a full shifted and scaled equation, there would be additional compatibilityconditions requiring to set α0 = 1 for example in the Hermite case.

We can thus conclude that the system of PDEs is hyperbolic for non-vanishing α0.

5.2.3 Fully Transformed Boltzmann Equation

The previous Section 5.2.2 shows that we will achieve a hyperbolic system if we useHermite polynomials for the expansion of the distribution function and perform thecorresponding Gauss-Hermite quadrature for the projections at least in the case of ashifted Boltzmann equation.

Now, we will turn our attention to the fully transformed Boltzmann equation with ascaled ansatz as specified in Equation (5.8). We therefore consider the Equation (5.11)and again want to derive conditions for the hyperbolicity of the emerging system.

The PDE we analyze is(1

ρDtρ−

d

2θDtθ

)f +Dtf+

√θξj

((1

ρ∂xjρ−

d

2θ∂xjθ

)f + ∂xj f

)+

∂ξj f

(− 1√

θ

(Dtvj +

√θξi∂xivj

)− 1

2θξj

(Dtθ +

√θξi∂xiθ

))=0

(5.91)

We will first use an DVM approach and derive the corresponding system of equationsand later pass over to the quadrature-based projections by some simple redefinitions. Wewill further write f instead of f to ease notation.


This section is somewhat analogous to Section 5.2.2.1 with the difference that we nowconsider the fully transformed Boltzmann equation (5.91) from above in d dimensions.

For a set of discrete velocities ξk ∈ Rd, k = 1, . . . , n, we consider pointwise evaluationsof Equation (5.91). This leads to the following relation for the unknowns ρ, vj , θ, fk(k = 1, . . . , n):(

1

ρfk,−

1√θ∂ξjf

∣∣∣∣ξk

,− d

2θfk −

1

2θ∂ξjf

∣∣∣∣ξk

(ξk)j , In

)Dtu+ (5.92)

d∑i=1

√θ

(1

ρ(ξk)i fk,−

1√θ∂ξjf

∣∣∣∣ξk

(ξk)i ,−d

2θfk (ξk)i −

1

2θ∂ξjf

∣∣∣∣ξk

(ξk)j (ξk)i , (ξk)i

)∂ξiu=0.

(5.93)


For the unknown variables, we define the vector u as

u := (ρ, v1, . . . , vd, θ, f1, . . . , fn)T ∈ Rn+d+2. (5.94)

Again, we define

Ai := diag ((ξ1)i, . . . , (ξn)i) ∈ Rn×n, i = 1, . . . , d (5.95)

and for the derivatives

DF =

∂ξ1f

∣∣∣∣ξ1

· · · ∂ξdf

∣∣∣∣ξ1

.... . .

...

∂ξ1f

∣∣∣∣ξn

· · · ∂ξdf

∣∣∣∣ξn

∈ Rn×d. (5.96)

In addition we haveDjF := j-th column of DF (5.97)

and a diagonal matrix storing some scaling variables

Λ =

1√θ

ρθId

ρ2θ3/2

ρ√θIn

∈ Rn+d+2×n+d+2. (5.98)

This allows us to write the system emerging from (5.91) in compact notation as

AΛD

Dtu+

d∑i=1

AiA√θΛ

∂

∂xiu = 0 (5.99)

with the matrixA = (f ,−DF ,−d · f −Aj (DjF ) , In) . (5.100)

The system (5.99) states n equations for n+d+2 unknowns u. As additional relations,we have the compatibility conditions for the density, velocity and temperature:

1= < f(ξ), 1 >,

0= < f(ξ), ξi >, i = 0, 1, d,

d= < f(ξ), ξ2i > .

(5.101)

In total, we then obtain n+ d+ 2 equations for n+ d+ 2 unknown variables.As for the shifted case, it is again more than unclear, how to evaluate the derivatives

in the definition of DF , because there is no continuous or differentiable reconstructionof the distribution function available for the DVM. This problem does not arise in thequadrature-based projections, because we have an expansion of f that is differentiabledue to its basis representation.


5.2.3.2 Quadrature-Based Projection

Instead of deriving the whole system from the governing PDE (5.91) using the expansion(5.7), we proceed by redefining the matrices in the system (5.99).

We use the ansatz

f(t,x, c) =n∑i=1

αj(t,x)Φj(c) (5.102)

with some basis coefficients αj and ansatz functions Φj . For the projections, we multiply

by test function Φj and integrate over the whole velocity space. We then need aquadrature formula to compute the following integrals of an arbitrary function g∫

Rdg(ξ)dξ =

n∑k=1

wkg(ξk) (5.103)

with weights wk and quadrature points ξk, for k = 1, . . . , n.We first need to introduce some notation again:

α = (α1, . . . , αn)T ∈ Rn (5.104)

B = (Φj(ξi))i=1,...,n, j=1,...,n ∈ Rn×n, B =(

Φj(ξi))i=1,...,n, j=1,...,n

∈ Rn×n (5.105)

W = diag (w1, . . . , wN ) ∈ Rn×n (5.106)

The evaluation of the derivatives goes into the definition of the matrix DB:

DiB =

∂ξiΦ1(ξ)

∣∣∣∣ξ1

· · · ∂ξiΦn(ξ)

∣∣∣∣ξ1

.... . .

...

∂ξiΦ1(ξ)

∣∣∣∣ξn

· · · ∂ξiΦn(ξ)

∣∣∣∣ξn

∈ Rn×n, i = 1, . . . , d (5.107)

With the previous definitions, the DVM and the quadrature-based method are linkedby the following relations that are easy to verify:

f =Bα

DF =DB = (D1Bα, . . . ,DdBα)

DiF=DiBα

(5.108)

Plugging the Equations (5.108) into (5.100), we obtain the new matrix for thequadrature-based projections

AQ = (Bα,−DB,−dBα−Aj (DjB) ,B) (5.109)

and the whole system reads

BTWAQΛ

D

Dtu+

d∑i=1

BTWAiAQ

√θΛ

∂

∂xiu = 0 (5.110)


for the unknowns

u := (ρ, v1, . . . , vd, θ, α1, . . . , αn)T ∈ Rn+d+2 (5.111)

Note that the multiplication with BTW is due to the quadrature rule, which is

employed after the multiplication with Φj .

We can see that the system is quite similar to the one obtained for the shiftedBoltzmann equation in (5.69).

Next is the addition of the compatibility conditions into the system. As before, weaim at an unconstrained system of size n for n unknown variables rather than a systemof n equations for n+ d+ 2 variables closed by d+ 2 algebraic relations.

Likewise to the shifted case, we evaluate the integrals in the compatibility conditions(5.46) and obtain d+ 2 equations for the basis coefficients in α.

Writing the emerging system as Qα = g (compare (5.14)) with Q ∈ Rd+2×n, wecan decompose the matrix Q by means of singular values decomposition as described in

Section 5.1.4.3. Splitting up the variables as shown above, we have β1 = Q−1S−1g. We

then insert our new variables into the system (compare to (5.19)):

α =(T−1

)1β1 +

(T−1

)2β2 (5.112)

As β1 is constant, we now only need to insert this into the PDE system (5.64) andobtain a closed set of n equations for n variables uT =

(ρ,vT θ,β2

T). We define

DBβ :=(D1B

((T−1

)1β1 +

(T−1

)2β2

), . . . ,DdB

((T−1

)1β1 +

(T−1

)2β2

)),

(5.113)

B := B(T−1

)2

(5.114)

and

Bβ := B((T−1

)1β1 +

(T−1

)2β2

). (5.115)

Next is the introduction of the new unknown vector u

u :=(ρ,vT , θ,βT2

)T ∈ Rn. (5.116)

The matrix AQ essentially transforms to

A =(Bβ,−DBβ,−d · Bβ −AjDjBβ, B

). (5.117)

Furthermore, we have

Λ =

1√θ

ρθId

ρ2θ3/2

ρ√θIn−d−2

. (5.118)


Using all of these definitions, the system is

BTWAΛ

D

Dtu+

d∑i=1

BTWAiA

√θΛ

∂

∂xiu = 0. (5.119)

Equation (5.119) is now a system of n equations for a total number of n unknown

variables. Hyperbolicity is obtained, if the matrix BTWAΛ in front of the time

derivative is regular and thus invertible. The behavior of the system is then characterizedby linear combinations of the eigenvalues of matrix Ai.

The conditions to achieve a hyperbolic system of PDEs in the case of quadrature-based projections for the fully transformed Boltzmann equation are thus:

(1) Regularity of matrix B

(2) Regularity of matrix W or equivalently wi 6= 0, ∀i = 1, . . . , N

(3) Regularity of matrix Λ(see (5.98))

(4) Regularity of matrix A including rankA = n

(5.120)

Again, the result is very similar to the previous case of the shifted Boltzmannequation 5.2.2. Checking the regularity of the matrices in the conditions 5.120, wecan prove hyperbolicity of the emerging PDE system. We only need to choose a specificbasis for the ansatz space and use the compatibility conditions to reduce the number ofunknowns in the described way.

In one dimension, a proof of hyperbolicity is done in the following Section 5.2.3.3 forthe case of Hermite ansatz functions.

5.2.3.3 Hyperbolicity of Fully Transformed BTE and Hermite Ansatz

In Section 5.2.2.3, we have shown that the reduced PDE system is in fact hyperbolicfor the specific choice of Hermite polynomials as ansatz and test functions together withsuitable Hermite-Gauss quadrature as the projection method for the shifted Boltzmannequation. We now want to do the same for the one-dimensional (in x and v) fullytransformed Boltzmann equation using the conditions and notation from the previoussections.

In the following, we thus check the conditions (5.120).We use basically the same ansatz as before (but for f):

f(t, x, ξ) =1√2πe−ξ

2/2n∑i=0

αi(t, x)ΦHi (ξ) (5.121)

where ΦHi are the normalized Hermite polynomials from Section 3.1.2. The expansion

can be interpreted by taking (5.8) and using

Φi(ξ) :=1√2πe−ξ

2/2ΦHi (ξ). (5.122)


From the discussions above, we can already guarantee the regularity of B and W aswe are using Gauss-Hermite quadrature.

The matrix Λ is regular, if and only if ρ 6= 0 ∧ θ 6= 0. These are obvious conditions,because the ansatz (5.8) would not make sense otherwise.

The last thing to check is now the regularity of the matrix A, which we will do inthe following deliberations.

We proceed analogously to Section 5.2.2.3 up to the solution of the compatibilityconditions. In one spatial dimension, we have three constrains to fulfill from which weget three explicit values for α0, α1 and α2:

1= < f(ξ), 1 > ⇒ α0 = 1

0= < f(ξ), ξ > ⇒ α1 = 0

1= < f(ξ), ξ2 >⇒ α2 = 0

(5.123)

According to Section 5.1.4.3, we use (exact) Gauss-Hermite quadrature to evaluatethe integrals in 5.123 and obtain a linear system of equations Qα = g. By applicationof the quadrature rule, we identify the following terms:

Q =

1 0 0 0 . . . 00 1 0 0 . . . 0

1 0√

2 0 . . . 0

and g = (1, 0, 1) . (5.124)

This leads to the following possible decomposition of Q:

Q = SMT (5.125)

with

S =

1 0 00 1 01 0 1

, (5.126)

M =

1 0 0 0 . . . 00 1 0 0 . . . 0

0 0√

2 0 . . . 0

, so that Q =

1 0 00 1 0

0 0√

2

(5.127)

andT = In (5.128)

which leads toα = T−1β = β. (5.129)

It is then possible to explicitly compute

β1 =

β0β1β2

= Q−1S−1g =

100

. (5.130)


On the other hand, this is equivalent to setting (α0,α1,α2) = (1, 0, 0).

As T = In, the application of(T−1

)2

to a matrix is equal to the effect of deleting

the first three columns of that matrix and a multiplication with(T−1

)1

is equal to anextraction of the first three columns.

The matrices in (5.117) are then obtained by

B = B(T−1

)2

= (b3, . . . , bn) , (5.131)

Bβ = B((T−1

)1β1 +

(T−1

)2β2

)= b0 · 1 +

n∑i=3

biαi. (5.132)

For the terms with involved derivatives, we can use the recursion relation of theHermite polynomials to get (compare with (5.76))

d

dξΦi(ξ) = −

√i+ 1Φi+1(ξ). (5.133)

The matrix D1B can then be explicitly derived as

D1B = −

√

1Φ1(ξ0) · · ·√n+ 1Φn+1(ξ0)

.... . .

...√1Φ1(ξn) · · ·

√n+ 1Φn+1(ξn)

(5.134)

or in terms of (5.74)

D1B = − (b1, . . . , bn+1) · diag(√

1, . . . ,√n+ 1

). (5.135)

This leads to the matrix D1Bβ (note that bn+1 = 0)

D1Bβ=D1B((T−1

)1β1 +

(T−1

)2β2

)=−√

1b1 −n∑i=4

√i+ 1bi+1αi

=−√

1b1 −n−1∑i=4

√i+ 1bi+1αi.

(5.136)

The term with AjDjBβ also needs some special attention. It can be derived using


the recursions from Equations (3.13) and (3.14):

(A1D1B)k=

n∑i=0

(ξ ∂ξΦi(ξ))

∣∣∣∣ξk

αi

=n−1∑i=0

(ξ ∂ξΦi(ξ))

∣∣∣∣ξk

αi + (ξ ∂ξΦn(ξ))

∣∣∣∣ξk

αn

=−n−1∑i=0

√i+ 1

(√i+ 2Φi+2(ξ) +

√i+ 1Φi(ξ)

) ∣∣∣∣ξk

αi −(ξ√n+ 1Φn+1(ξ)

) ∣∣∣∣ξk

αn

=−n−1∑i=0

√i+ 1

(√i+ 2Φi+2(ξ) +

√i+ 1Φi(ξ)

) ∣∣∣∣ξk

αi.

(5.137)With the help of

zi :=√i+ 1

(√i+ 2bi+2(ξ) +

√i+ 1bi(ξ)

) ∣∣∣∣ξk

(5.138)

we can write the vector A1D1B as a sum of the ziαi:

A1D1B = −n−1∑i=0

ziαi. (5.139)

In the expression AjDjBβ, the first three αi are replaced by 1, 0, 0, which then leadsto the equation for the missing vector:

A1D1Bβ=−√

1(b2√

2 + b0√

1)−n−1∑i=3

ziαi

=− b0 −√

2b2 −n−1∑i=3

ziαi.

(5.140)

This expression is a linear combination of the columns of B. It does not contain bn+2,as the sum goes only up to n − 1 and the column bn+1 is zero anyway, because thequadrature points are just zeros of the n+ 1-st Hermite Polynomial.

The only thing left is now to check the columns of the matrix

A =(Bβ,−DBβ,−d · Bβ −AjDjBβ, B

)(5.141)

for linear independence.As has been discussed above, every column of A is a linear combination of some bi.

We can therefore prove linear independence by checking where which column bi appears.In total, we must not have more than n columns bi involved and no column of A has tobe linearly dependent of another column in A.


(1) B has columns b3, . . . , bn

(2) DBβ is a combination of the columns of B minus an additional b1 and thus

linearly independent of B

(3) Bβ is a combination of columns of B plus b0 and thus linearly independent of theothers

(4) A1D1Bβ is a combination of columns of B minus b0 and√

2b2 and thus linearlyindependent of the others

Matrix A therefore has full rank and is regular. The PDE system then is hyperbolicfor all values of the αi. We see that the quadrature-based projection methods in factlead to a globally hyperbolic system for the unknown basis coefficients and macroscopicvariables.

5.2.4 Relation to the Conservation Laws

The compatibility conditions serve as a coupling of the macroscopic variables ρ,v, θ tothe basis coefficients αi and the previous section showed how to insert these conditionsdirectly into the PDE system to end up with a closed system of PDEs. It is still to bementioned how the PDE system is related to the classical conservative equations andwhich closure we get. Using the framework from above, it is possible to compute allthe involved matrices for Hermite basis and test functions, for example. We will nowshow, that the first three equations of the PDE system are in fact the conservation lawsincluding a special closure for the heat flux:

The one-dimensional fully transformed system (compare with (5.119)

BTWAΛ

D

Dtu+ B

TWAA

√θΛ

∂

∂xu = 0 (5.142)

can be written as

T ΛD

Dtu+X

√θΛ

∂

∂xu = 0 (5.143)

using the definitions

T := BTWA (5.144)

andX := B

TWAA. (5.145)

The computation of the matrices T and X can either be done numerically or ana-lytically by using the exactness of the quadrature formula. The matrices evaluate to

T =

1 0 00 1 0 0

0 0√

2∗ ∗ ∗...

...... In−3

∗ ∗ ∗

(5.146)


and

X =

0 1 0 0 0 . . . 01 0 2 0 0 . . . 0√3α3

√2 3√

3α3

√3 0 . . . 0

∗ ∗ ∗...

...... Jn,3

∗ ∗ ∗

(5.147)

where the entries denoted by ∗ only enter the last n− 3 equations and contain the basiscoefficients. Jn,3 is the Jacobi matrix Jn without the first three columns and rows, see(3.46) or (4.12) for the Jacobi matrix of Hermite polynomials.

Note that the vector of unknown variables u is

u = (ρ, v, θ, α3, . . . , αn) ∈ R. (5.148)

Writing out the first three equations of the PDE system (5.143), we directly obtain:

1ρDtρ + ∂xv = 01√θDtv +

√θρ ∂xρ+ 1√

θ∂xθ = 0

1√2√θDtθ +

√3√θ∂xα3 +

√3α3

√θρ ∂xρ+ 3

√3

2 α31√θ∂xθ +

√2∂xv = 0

(5.149)

With the help of the convective time derivative Dt := ∂t + v∂x and some modifica-tions, we can transform this set of equations (5.149) into the conservative form of theconservation laws that read

∂tρ + ∂x (ρv) = 0∂t (ρv) + ∂x

(ρv2 + ρθ

)= 0

∂t(ρv2 + ρθ

)+ ∂x

((ρv2 + ρθ

)· v + 2ρθv + q

)= 0

(5.150)

where the closure for the heat flux q can be derived by a comparison of the correspondingterms. It is

q = θ3/2ρ√

6α3. (5.151)

We therefore see a coupling of the conservation laws to the basis coefficients throughthe coefficient α3. Instead of the full system (5.142), we can thus replace the first threeequations by the conservation laws (5.150) and use the closure as specified in (5.151).

5.2.5 Remark

In the end of this chapter, we want to add a remark about the formulation with theconvective time derivative Dt := ∂t + vi∂xi . If we use this shifted formulation, wewill also derive the corresponding shifted eigenvalues. The real eigenvalues that arethe characteristic speeds of the system are obtained by addition of vi in the respectivedirection. As vi is real, this has no effect on the hyperbolicity and is thus not a problem


for our framework. Of course, one has to keep the addition in mind when it comes toreal calculations.

Taking into account the eigenvalues λAik = (ξk)i of the matrix Ai, we get thefollowing eigenvalues for the complete system:

λk =d∑i=1

βi

(λAik

√θ + v

)=

d∑i=1

βi

((ξk)i

√θ + v

), k = 1, . . . , n (5.152)

which is a generalization of the results found by Cai in [5].

76

Chapter 6

Conclusion

6.1 Summary

After the motivation, the thematic introduction and the explanation of the mathematicalingredients in the first chapters, we have successfully developed a theoretical frameworkfor the derivation of hyperbolic PDE systems for the solution of kinetic equations suchas the Boltzmann equation in a very efficient transformed version.

The conceptual framework described in Chapter 5 not only demonstrates the generalprocedure for an analysis of the emerging PDE system, it also yields practical conditionsthat guarantee hyperbolicity of the system. The key to the results obtained in thischapter was the application of quadrature-based projection methods. With the helpof an analogy to discrete velocity methods, it was possible to generalize hyperbolicityconditions for simple kinetic equations to more advanced cases like the transformedBoltzmann equation.

The conditions for hyperbolicity could be verified according to Sections 5.2.2.3 and5.2.3.3 for different one-dimensional applications of Hermite ansatz functions togetherwith Gauss-Hermite quadrature for the projections. The proofs for these examplescontain useful techniques for other types of basis functions paired with the correspondingquadrature formulas.

The one-dimensional Hermite example together with the compatibility conditions infact yielded the well-known conservation laws with a coupling to the basis coefficientsvia the heat flux, which states the closure of the system.

Together with the examples from Chapter 4 we gained more concrete insight intothe application of quadrature-based projections, as the results could be explained fromanother point of view looking directly at the system matrix as well as with the proofsin Chapter 5. The application of quadrature-based projection methods therefore can beseen as a successful way to derive hyperbolic PDE systems for the efficient solution ofthe Boltzmann equation.

77

78 Conclusion

6.2 Future Work

The work on this topic will be continued by the following work on a PhD thesis. Theconceptual work of this master thesis opens many different areas for further research.

One important task would be the extension of the proof from Hermite ansatz func-tions to more arbitrary functions and quadrature formulas. The techniques of the proofcould in principle be used for the generalization to orthogonal basis functions togetherwith their respective Gauss quadrature formulas.

It is also an open question, if one can deal with the problem of the evaluation of thederivatives in the DVM setting so that the PDE system for the point evaluations willbecome hyperbolic. There is possibly again a relation to the quadrature points that wecan use to obtain meaningful spectral methods for the derivative term.

A more practical problem is the consideration of proper quadrature rules for thespherical harmonics expansion in three dimensions. Lebedev quadrature or sphericalt-designs could potentially be useful for this, but it is not clear if they can be used interms of the developed framework.

The theoretic results should also be supplemented by some numerical examplesinvestigating the approximation quality of the quadrature-based methods. For the con-sideration of efficiency, sparse grids could be beneficial in three-dimensional applicationsas well.

In the end, the goal would be a computational tool that lets us choose an equation,a specific ansatz and projection method and guarantees hyperbolicity of the equationsso that the system of equations for the unknowns can be successfully solved by somededicated numerical method in a second step.

Appendix A

Compatibility Conditions 3D

In order to allow for numerical calculations, we want to give the compatibility conditionsfor Hermite and Spherical Harmonics (SH) basis functions. In a fully transformed BTE,these conditions can be used to eliminate the correct number of variables so that thePDE system is closed.

A.1 Hermite Ansatz

In three spatial dimensions, the compatibility conditions for a Hermite ansatz are a bitmore complicated compared to the one-dimensional case (see Equations (5.2.2.3) and(5.2.3.3)) as some of the coefficients depend on each other as we will see.

Using the ansatz

f(t,x, ξ) =ρ

√2πθ

3 e−ξT ξ/2

n∑i,j,k=0

αi,j,k(t,x)Φi,j,k(ξ) (A.1)

with products of normalized Hermite polynomials Φi,j,k (see Equation (4.41), we canevaluate the conditions from Section 5.1.4. Note that we have

f(t,x, ξ) =1√

2π3 e−ξT ξ/2

n∑i,j,k=0

αi,j,k(t,x)Φi,j,k(ξ). (A.2)

The compatibility conditions thus yield 5 additional equations for the basis coeffi-cients αi:

1= < f(ξ), 1 > ⇒ α0,0,0 = 1,

0= < f(ξ), ξi >⇒ α1,0,0 = α0,1,0 = α0,0,1 = 0,

3= < f(ξ), ξ2i >⇒ α2,0,0 + α0,2,0 + α0,0,2 = 0.

(A.3)

Here the last equation leads to a coupling of the three involved coefficients. Of course,the coupling is only linear so that this can still be used in the conceptual framework ofSection 5.2.3.

79

80 Compatibility Conditions 3D

A.2 Spherical Harmonics Ansatz

If we use a Spherical Harmonics ansatz, the unknown distribution function is expandedby

f(t,x, ξ) = f(t,x, r, θ, φ) =ρ√2πθ

3 e−r2/2

n∑k=0

M∑l=0

l∑m=−l

αkl,m(t,x)Ψkl,m(r, θ, φ) (A.4)

with basis function Ψkl,m as a product of a Laguerre function and a SH

Ψkl,m(r, θ, φ) = L

l+ 12

k

(r2

2

)rl · Yl,m(θ, φ). (A.5)

Note that the Laguerre function times rl specifies the radial part and the SH accountsfor the angular part of the basis function.

The functions Ψkl,m form an orthonormal basis of polynomials in R3, as has been

discussed in Sections 3.3 and 3.3.1.The compatibility conditions now again lead to 5 additional equations for the basis

coefficients αkl,m, if we use the transformation to spherical velocity coordinates

(ξ1, ξ2, ξ3) = (r sin(θ) cos(φ), r sin(θ) sin(φ), r cos(θ)), (A.6)

1= < f(ξ), 1 > ⇒ α00,0 =

1

(2π)3/4,

0= < f(ξ), ξi >⇒ α01,1 = α0

1,−1 = α01,0 = 0,

3= < f(ξ), ξ2i >⇒ α10,0 =

√6

(2π)3/4.

(A.7)

Using corresponding quadrature formulas, one can now proceed in the same fashionas in Sections 5.2.2.3 and 5.2.3.3 and start to show hyperbolicity of the correspondingequations with the help of the framework from Chapter 5.

References

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions withFormulas, Graphs, and Mathematical Tables. New York, 1965.

[2] P. L. Bhatnagar, E. P. Gross, and M. Krook. A model for collision processes ingases. Physical Review, 94:511–525, 1954.

[3] G. A. Bird. Direct simulation and the Boltzmann equation. Physical Fluids,13:2676–2687, 1970.

[4] G. A. Bird. Monte Carlo simulation in an engineering context. Rarefied GasDynamics, 1:239–255, 1981.

[5] Z. Cai, Y. Fan, and R. Li. Globally hyperbolic regularization of Grad’s momentsystem in one dimensional space. Communications in Mathematical Sciences,11(2):547–571, 2012.

[6] C. Cercignani. The Boltzmann Equation and its Application. Springer, 1988.

[7] S. Chen and G. D. Doolen. Lattice Boltzmann method for fluid flows. AnnualReview of Fluid Mechanics, 30:329–364, 1998.

[8] P. Deuflhard. Newton Methods for Nonlinear Problems. Affine Invariance andAdaptive Algorithms. Springer, 2004.

[9] M. Frank, C.D. Hauck, and E. Olbrant. Perturbed, entropy-based closure forradiative transfer. Kinet. Relat. Models, 6:557–587, 2013.

[10] H. Grad. On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2(4):331–407, 1949.

[11] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II, Stiff andDifferential-Algebraic Problems. Springer Verlag, 2002.

[12] S. Heinz. Statistical Mechanics of Turbulent Flows. Springer, 2003.

[13] S. Heinz, P. Jenny, and M. Torrilhon. A solution algorithm for the fuiddynamics equations based on a stochastic model for molecular motion. Journalof Computational Physics, 229:1077–1098, 2010.

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82

[14] F. P. Incropera and D. P. DeWitt. Fundamentals of heat and mass transfer, 2001.

[15] J. M. Perez Jorda and W. Yang. A concise redefinition of the solid sphericalharmonics and its use in fast multipole methods. Journal of Chemical Physics,104:8003–8006, 1996.

[16] T. Kataoka, M. Tsutahara, K. Ogawa, Y. Yamamoto, M. Shoji, and Y. Sakai.Knudsen pump and its possibility of application to satellite control. Theoreticaland Applied Mechanics, 53:155–162, 2004.

[17] P. Kauf. Multi-Scale Approximation Models for the Boltzmann Equation. PhDthesis, ETH Zurich, 2011.

[18] C. D. Levermore. Moment closure hierarchies for kinetic theories. Journal ofStatistical Physics, 83:1021–1065, 1996.

[19] G. V. Milovanovic and A. S. Cvetkovic. Special classes of orthogonal polynomialsand corresponding quadratures of Gaussian type. Mathematica Balkanica, 26:169–184, 2012.

[20] X. Shan and X. He. Discretization of the velocity space in the solution of theBoltzmann equation. Physical Review Letters, 80:65–68, 1998.

[21] A. H. Stroud and D. Secrest. Gaussian Quadrature Formulas. Englewood Cliffs,1966.

[22] H. Struchtrup. Macroscopic Transport Equations for Rarefied Gas Flows. Springer,2005.

[23] M. Torrilhon. Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions. Comm. Comput. Phys., 7(4):639–673, 2010.

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83

Eigenstandigkeitserklarung

Hiermit versichere ich, dass ich diese Masterarbeit selbstandig verfasst und keine anderenals die angegebenen Quellen und Hilfsmittel benutzt habe. Die Stellen meiner Arbeit,die dem Wortlaut oder dem Sinn nach anderen Werken entnommen sind, habe ich injedem Fall unter Angabe der Quelle als Entlehnung kenntlich gemacht.

Aachen, im September 2013

Julian Kollermeier

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