Hyperbolic Approximation of Kinetic Equations Using Quadrature-Based Projection Methods

Transcription

1 Master Thesis Hyperbolic Approximation of Kinetic Equations Using Quadrature-Based Projection Methods Julian Köllermeier RWTH Aachen University Supervisor: Prof. Dr. Manuel Torrilhon Center for Computational Engineering Science Mathematics Division RWTH Aachen University

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3 Abstract We derive hyperbolic PDE systems for the solution of the Boltzmann equation. First, we transform the velocity in a highly non-linear way to allow for a physical adaptivity of the method. The unknown distribution function is then approximated by the equilibrium Maxwellian times a series of orthogonal basis functions. The standard continuous projection method for this approach yields a PDE system for the basis coefficients that is in general not hyperbolic. To overcome this problem, we apply quadrature-based projection methods which modify the structure of the system in 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 the emerging system ist hyperbolic and give a proof of hyperbolicity for a Hermite ansatz in one dimension. iii

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5 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 Master of Science in Computational Engineering Science. I would hereby like to thank all the people involved in the work on my thesis and during 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 advices earlier on during the past years as well as for his enduring support and encouragements from the proposal of the topic until the final version of this thesis. The many fruitful discussions and impulses helped a lot in focussing on the important parts of the work and still left the necessary freedom for own ideas and developments. Many thanks also go to my colleagues at MathCCES, especially to my advisor Roman Schaerer who always patiently listened to my questions and deliberations and replied with useful hints and ideas for further investigations. Thanks to Claudia, Marc and Marcus for proofreading my thesis. It must have been a tough job. I do not want to miss my fellow students in the course of Computational Engineering Science (CES) as well as my old and new friends in Aachen and elsewhere, who proved themselves a valuable support during the years of my studies and especially in the past few months. Special thanks I would like to give to my parents and my family, who always animated me to go my own way and gave me a good deal of curiosity to take along that helps me whichever topic I work on. Last but not least, I convey grateful thanks to the Friedrich-Naumann Foundation for Freedom, who greatly helped me during my studies with a fellowship of large value for myself. Apart from the financial support, it opened up new possibilities for me that I had never thought of and gave me the freedom and independence I appreciate so much. v

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7 Contents Abstract Acknowledgment Contents iii v ix 1 Introduction Motivation Aims of the Project Overview Boltzmann Transport Equation Basics of Kinetic Theory Knudsen Number, Applications and Effects of Gas Rarefaction Phase Space and Probability Density Function Macroscopic Quantities of the Flow Field Properties of the Boltzmann Transport Equation Equilibrium Distribution Collision Operator Solution Methods Direct Simulation Monte Carlo Method of Moments Lattice Boltzmann Method Discrete Velocity Method Mathematical Properties Hermite Polynomials Standard Definition Orthonormal Hermite Polynomials Laguerre Polynomials Generalized Laguerre Polynomials Spherical Harmonics Cartesian Spherical Harmonics Jacobi Matrix vii

8 viii Table of contents 3.5 Gaussian Quadrature Gauss-Hermite Quadrature Gauss-Laguerre Quadrature Generalized Gauss-Laguerre Quadrature Non-Classical Quadrature Interpolation Property and Aliasing Motivational Examples One-Dimensional Cases Simple Kinetic Equation Generalized Kinetic Equation c Generalized Kinetic Equation c + c Relation between Quadrature Projection and DVM Multi-Dimensional Cases Simple Kinetic Equation 3D Generalized Kinetic Equation c 2 i 2D Theoretical Concepts Preliminaries Variable Transformation for Physical Adaptivity Derivation of Transformed Boltzmann Equation Expansion Using Basis Functions Compatibility Conditions D Case D Case Coupling of Compatibility Conditions to PDE System Theoretical Concept Generalized Kinetic Equation Discrete Velocity Method Quadrature-based Projection Shifted Boltzmann Equation Discrete Velocity Method Quadrature-Based Projection Hyperbolicity of Shifted BTE and Hermite Ansatz Fully Transformed Boltzmann Equation Discrete Velocity Method Quadrature-Based Projection Hyperbolicity of Transformed BTE and Hermite Ansatz Relation to the Conservation Laws Remark Conclusion Summary Future Work

9 Table of contents ix A Appendix: Compatibility Conditions 3D 79 A.1 Hermite Ansatz A.2 Spherical Harmonics Ansatz References 81

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11 Chapter 1 Introduction 1.1 Motivation Kinetic equations are the basis for many different applications and are widely used in industrial and scientific fields. Especially for rarefied flows they provide an accurate setting for the successful solution of important numerical simulations. In Chapter 2 we will see that there are more or less distinct regions of the flow in which the application of standard fluid dynamic models like the Euler or Navier-Stokes equations is not appropriate for a physical solution. One then has to apply more advanced kinetic models that are motivated directly by kinetic equations. A standard method proposed by Grad in [10] derives equations for the macroscopic flow variables like density, velocity and temperature of the flow by expanding the unknown distribution function of the Boltzmann equation in a Hermite series. The drawback of this rather simple method is that the resulting system of partial differential equations (PDEs) can loose hyperbolicity for certain values of higher moments. The loss of global hyperbolicity is a serious problem, because hyperbolicity is needed for physical solutions and stability of the solution in particular. The admissible region of variables for hyperbolicity 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 that they are hyperbolic in special cases like one-dimensional flows. One of those is based on 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 moment system in one dimension that is globally hyperbolic. They essentially derived the characteristic polynomial of the corresponding matrix analytically and used this information to set certain variables or entries in the matrix to zero so that the new characteristic polynomial has real roots and the system becomes hyperbolic for all values of the variables involved. The approach by Cai et al. gives only limited insight into the underlying theoretical 1

12 2 Introduction foundations of this regularization and it is not really clear how to generalize the procedure to 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 fine discretization in the velocity space with many unknowns. Recent developments by Kauf in [17] show a way to circumvent these problems at the expense of a more difficult PDE involving 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 stable and 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 general framework to derive hyperbolic PDEs for the solution of the Boltzmann equation. With the help of this framework it should be feasible to decide about the hyperbolicity of the emerging system a priori before inserting a special ansatz and performing projections of the 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 particular and analyze the application of those methods with respect to the effects on the structure of the equations as well as on the eigenvalues of the system matrix, which is closely related to the hyperbolicity of the system. The framework should include these quadrature-based projections and give concrete conditions under which the system will be hyperbolic. After the framework has been set up, application to some choices of important classes of functions for the expansion together with related quadrature methods should yield resulting systems that are hyperbolic. 1.3 Overview After this short introduction, Chapter 2 is dedicated to basic properties of the Boltzmann equation and the kinetic approach in contrast to other existing methods. The probability density function f is also introduced and we show how f and the Boltzmann equation are related to macroscopic quantities of the flow field. In Chapter 3 the most important mathematical preliminaries that are used in the following sections are explained. This includes normalized versions of Hermite and Laguerre polynomials as well as spherical harmonics. A large part of the chapter covers the foundations of quadrature methods, especially Gauss-quadrature for the involved polynomials. Some early investigations are summarized in a chapter about motivational examples, see Chapter 4. We here apply quadrature-based projections to simple kinetic equations

13 Introduction 3 and explain the difference to exact projections. 1D as well as multi-dimensional examples show the desirable properties of the emerging systems for the basis coefficients and help for 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 the formulation of the Boltzmann equation under a non-linear transformation of the velocity variable that allows for efficient simulations. Next is the development of the conceptual framework for the derivation of conditions to achieve hyperbolicity. This part covers different kinetic equations and draws an analogy between the discrete velocity method and the quadrature-based projections. Using the mathematical properties and the framework developed before, we also give here a proof for hyperbolicity of the regularized system in the case of the transformed one-dimensional Boltzmann equation with Hermite ansatz and Gauss-Hermite quadrature. We summarize the results in Chapter 6 and discuss future work on the project.

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15 Chapter 2 The Boltzmann Transport Equation Before we turn our attention towards the detailed mathematical discussion and the different models that we want to develop, we should first explain the Boltzmann equation itself and the context it is used in. We therefore describe the kinetic setting, important properties of the Boltzmann equation and the most common solution methods together with their benefits and drawbacks. 2.1 Basics of Kinetic Theory In standard fluid dynamics the fluid is modeled as a continuum meaning that the atoms and molecules or particles as we will call them from now on in general are in constant contact with the other particles. This is obviously valid for a fluid, which usually also has a relatively large density. When it comes to rarefied gases, particles do only interact rarely and are in free flight for the most of the time. This fact can be related to low density, for example for low ambient pressures. At this point, the flow behavior is more and more influenced by binary collisions of the particles. It is therefore required to model the interactions between individual particles in a different way than the standard continuum approach. In the following chapter we want to explain the viewpoint of kinetic theory and briefly explain the most important properties and basic terms related to this point 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 the book by Heinz [12] Knudsen Number, Applications and Effects of Gas Rarefaction Many flow problems can be characterized by small or moderate velocities and ambient conditions. However, the advanced technical capabilities made it possible to reach more extreme values for all parameters involved. In order to further distinguish different flow regimes, the Knudsen number Kn was introduced. It is the quotient of the mean free 5

16 6 Boltzmann Transport Equation path length λ, e.g. the average distance a particle travels between two collisions, and a characteristic length L of the flow problem, e.g. the size of a plane or the diameter of a pipe. The definition of the Knudsen number Kn reads Kn = λ L. (2.1) As a dimensionless flow parameter, the Knudsen number is an important quantity that influences the behavior of the flow. Standard models like the Navier-Stokes equations or the Euler equations are only valid for very small Knudsen numbers, because they rely on the assumption of a continuum in the so-called equilibrium. According to Kauf [17] and Struchtrup [22], the Knudsen number can be used to roughly divide the flow field into different regimes as follows: Kn 0.01: equilibrium or hydrodynamic regime, which is accurately described by the Navier-Stokes equations; 0.01 Kn 0.1: slip flow regime, where the Navier-Stokes equations need additional slip boundary conditions to be still valid; 0.1 Kn 1: transition regime, in which the Navier-Stokes equations are not valid, Boltzmann equation or advanced models are needed; 1 Kn 10: kinetic regime, here the Boltzmann equation is also valid, but a direct 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 and is closely related to the numerical solution approach to solve the flow problem. For the rarefied gases with Kn 0.1, one is interested in efficient and accurate methods to solve the Boltzmann equation, which is the topic of this thesis. In the literature (see e.g. [17], [6]), there are many relevant applications that are covered by the kinetic regime of a rarefied gas. Among those are: Reentry flights of spacecrafts at very high altitude: Gas pressure and density are very 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 the heat flux close to the thermal shield is crucial in this example. Shock waves at very high speed: The velocity jumps from super- to sub-sonic yields sharp gradients over only a very small distance. The shocks also influence the behavior of the flow further downstream. Microscopic channel flows: At very small length scales, the Knudsen number will become large even for ambient conditions. Examples are porous media or ion channels in membranes.

17 Boltzmann Transport Equation 7 When dealing with applications above, it is absolutely necessary to use extended models and methods to correctly predict the various effects of gas rarefaction. The use of standard methods like Navier-Stokes, for example, may lead to wrong predictions of the relevant macroscopic quantities. The method therefore fails to correctly simulate the typical effects for large Knudsen numbers, the so-called kinetic effects (see also [17]) like the Knudsen paradox. It says that the mass flow through a tube is decreasing with the diameter of the tube only until the diameter reaches a certain value of the order of the mean free path λ. From then on, the mass flow increases again. This is not consistent with 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]). It has no moving parts and works only with temperature differences along the wall. Due to that, the gas inside moves from the cold end to the warm end. This allows for a precise and reliable control of the gas flow Phase Space and Probability Density Function When it comes to rarefied gases, one might think about a straightforward method that tracks the way of every single particle. The collisions between particles then couple the evolution of the particles positions. The problem is that this procedure leads to the solution of a vast number of coupled partial differential equations, as even a rarefied gas still consists of too many particles per volume. For each of those particles, one would have to determine the corresponding three-dimensional positions x and velocities c at every 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 distribution function f(x, c, t). The PDF f is related to the number of particles with velocities c at position x and time t. The number of particles with velocities in [c, c + dc] in a certain interval [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 value of f at all times, positions as well as velocities to have complete knowledge about the state of the gas Macroscopic Quantities of the Flow Field Assuming a given PDF f, it is important to recover the macroscopic quantities of the flow field, because we are usually interested in variables like the overall velocity of the gas or the temperature. These and other quantities are all computed with the help of so-called moments of f. The mass density ρ is simply the mass m of one particle times the integral of the PDF f over the velocity space R 3 : ρ := m f dc := mn, (2.2) R 3 with number density n := R 3 fdc.

18 8 Boltzmann Transport Equation The mean velocity v can be computed by means of the momentum density ρv ρv = m c f(t, x, c) dc, (2.3) R 3 or in componentwise/tensor notation ρv i = m R 3 c i f(t, x, c) dc. (2.4) Higher moments can be defined using the peculiar velocities C i, where C i := c i v i. (2.5) With this definition the thermal energy (sometimes also called internal energy) u is given as ρu = m 2 Cii f(t, x, c) dc, (2.6) 2 R 3 where the notation is Cii 2 := C2 1 + C2 2 + C2 3 in three dimensions. For an ideal gas, the temperature T is closely related to the internal energy u by u = 3 k 2 mt, where k is the Boltzmann constant. Writing the temperature in energy units, we define a new variable θ = k m T. Lastly, we give the definition of the pressure tensor p ij and heat flux q i m 2 p ij = m C i C j f(t, x, c)dc, q i = C 2 jjc i f(t, x, c)dc. (2.7) R 3 R 3 Apart from these definitions, there are several thermodynamical laws connecting different 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 the Boltzmann equation for the unknown PDF f. The Boltzmann equation is a partialintegro-differential equation for f with usually seven independent variables t, x, c: t f(t, x, c) + c i f(t, x, c) + G i f(t, x, c) = S(f), (2.8) x i c i here, the first term denotes the change in time, the second term is due to the convective transport with velocity c i and the third term on the left hand side denotes changes in velocity in the presence of external forces G i, such as gravity. The operator S(f) on the right hand side is the so-called collision operator that models collisions of particles with other particles. For most of this thesis, we will neglect external forces and consider only the convective part as the main difficulty.

19 Boltzmann Transport Equation Equilibrium Distribution The right hand side operator S(f) in Equation (2.8) forces the process towards its equilibrium state and is zero, if equilibrium is achieved. At equilibrium, the density function f has the form of a local Maxwellian that is in d dimensions defined as follows f M (t, x, c) = ρ(t, x) m ( 1 d exp (c i v i (t, x)) 2 2πθ(t, x) 2θ(t, x) ). (2.9) As density, mean velocity and temperature may vary with t and x, a local Maxwellian is assigned to each point in time and space. Proper definitions of the collision operator S(f) have to ensure that S(f M ) = 0. In the non-equilibrium case, the density function f differs from a Maxwellian. When we use a special ansatz for the form of the density function f later on, it is nevertheless important to have the Maxwellian in the solution space in order to give the right solution in the equilibrium case. This will later justify some particular choices for a basis of the ansatz space Collision Operator The form of the collision operator has a huge impact on the behavior of the distribution function 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 and was derived using the so-called Stosszahlansatz S (Boltz) (f) = 1 ρ R 5 ( gb f(c) f(c1 ) f(c)f(c 1 )) db dɛ dc 1, (2.10) where the notation f indicates the PDF for the post-collision velocity. In principle it is possible to use this approach for numerical simulations, but its high dimensionality makes already the evaluation of S (Boltz) at discrete points in t and x very costly. For more 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 derive a linearized collision operator from the Boltzmann operator that is known as the BGK model [2]: S (BGK) (f) = 1 τ BGK (f M f). (2.11) This ansatz basically represents a relaxation towards the equilibrium distribution f M (see (2.9)) with relaxation time τ BGK. The BGK model is often chosen because of its simplicity. Note that the Boltzmann equation (2.8) together with the BGK model (2.11) is not a linear equation, because the Maxwellian f M still includes exponentials of the macroscopic variables ρ, v i and θ, which are in turn integrals of the distribution function according to Section

20 10 Boltzmann Transport Equation There is also another approach, which assumes small velocity changes due to collisions and results in a Fokker-Planck operator for the collisions S (FP) (f) = ( ) ( ) 1 (c i v i )f + 2 2es f, (2.12) c i τ F P c k c k 3τ F P with relaxation time τ FP and sensible energy e s = 3 2 mt in three spatial dimensions. The derivation of this operator is shown in detail in [6]. An application using the Fokker-Planck model is described in [13]. 2.3 Solution Methods Numerical methods for the solution of the Boltzmann equation have been under development since the late 1960s. During the past fifty years, different methods have been successfully applied to various problems. We will now give a short overview about the most important classes of methods Direct Simulation Monte Carlo It was Bird, who proposed the first method to actually solve rarefied gas flows using the so-called Direct Simulation Monte Carlo method (DSMC) [3]. His method was later improved 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 is actually not modeled by a PDE like the Boltzmann equation, but the gas is described by a system of particles. Each of the particles has a position and a velocity at every time. Note that in a real computation the number of numerical particles is substantially smaller than the real number of particles, which is of the order of So usually some hundreds 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 initial values. (2) in each time step, the particles first have a free-flight phase, where they are moved according to their assigned velocities for the time interval t. (3) the free-flight phase is followed by a collision phase, where particles undergo collisions that are modeled by a collision probability. Binary collisions then change the velocities of the involved particles. The particles thus move and collide in every time step. The calculation is by definition unsteady and steady results are obtained by asymptotic limits in time. Macroscopic quantities can later be derived from the particles velocities by averaging over small cells of the flow field. k

21 Boltzmann Transport Equation 11 The benefit of this type of method is clearly its simplicity. It is straightforward to implement, once the collision probabilities are modeled. On the other hand, the number of particles needs to be sufficiently high to enable accurate solutions. This was especially a problem during the development of the method. Furthermore, the number of required particles increases with the density of the gas, making the method less suitable for problems with moderate Knudsen numbers Method of Moments A relatively new method is the so-called method of moments (MoM), in which equations for the macroscopic moments are directly derived from the Boltzmann equation. A good summary of the method and the problem of the closure below is given by Levermore in [18]. The general procedure can be demonstrated for a simple, one-dimensional equation like (2.13) f(t, x, c) + c f(t, x, c) = 0. (2.13) t x Now the integral operator I n ( ) (2.14) is applied to the PDE, where I n multiplies the equation by c n and integrates over the velocity space I n ( ) := c n dc. (2.14) After the application of I n ( ), we can identify the so-called moments (2.15) in the equation M n (t, x) := f(t, x, c)c n dc. (2.15) The lower moments have a direct physical meaning. For example, we have M 0 = ρ, M 1 = v. The one-dimensional Boltzmann equation now transforms to t M i(t, x) + x M i+1(t, x) = 0, i = 0,..., n. (2.16) By this simple trick, it is possible to eliminate the velocity dependence. We will essentially use the same procedure for our quadrature-based projection methods later on. Note that the convective term leads to the appearance of the higher moment M n+1. Now, one has n equations for n + 1 variables, as the last equation also contains M n+1. The difficult part is now to find a so-called closure that is an additional relation to close the system of equations. The easiest way would be to simply set M n+1 = 0. Unfortunately, this simple approach does not yield satisfactory results. It can lead to negative values for the density,

22 12 Boltzmann Transport Equation for example. There are more advanced methods to close the system that relate the highest moment to the lower moments. One of these approaches is the maximum entropy method. The value of the highest moment is chosen to maximize the mathematical entropy, leading to physical solutions. A disadvantage is that the relation can no longer be written down in closed form, but is the solution of an optimization procedure in every step of a numerical method. Moment methods are also often used in kinetic models for radiative transfer, where the three-dimensional velocity of the particles is written in terms of direction and energy of the particles, see [9] for an example Lattice Boltzmann Method The Lattice Boltzmann method (LBM) is actually a modification of another particle method, the Lattice Gas Automata method (more information can be found in [7]) in which 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 takes place, 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 to reduce statistical noise. Most of the time LBMs uses a BGK model for the collisions, where a collision step is followed by a free-flight step just like in case of the DSMC method. The important difference is that the velocity space is discretized and only allows for particle velocities along the lattice to the next lattice grid point. One possible velocity space discretization in two dimensions could be c V h = {(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 discrete values 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]) Discrete Velocity Method Another method that is in fact closely related to our quadrature-based projection methods is the Discrete Velocity Method (DVM). A good description of the method can be found in [6]. The DVM discretizes the Boltzmann equation in distinct points of the velocity space. The discretization points c n are called discrete velocities. In the one-dimensional case, this means we end up with a system of equations for each of the discrete velocities t f(t, x, c n) + c n x f(t, x, c n) = 0. (2.18) Note that this special discretization of the velocity space leads to the unknowns f n (t, x) := f(t, x, c n ) that do only depend on t and x. The discretization of the right

23 Boltzmann Transport Equation 13 hand side collision operator is more delicate, as the collision invariants and conservation properties impose some restrictions for a meaningful discretization. There is a relation between DVM and the MoM. In the case of MoM, the equation is multiplied by the function c n and then integrated over the velocity space. DVM evaluates the equation at certain velocities c n which is equivalent to a multiplication with the dirac function δ(c c n ) followed by integration over the velocity space. Thus, the DVM can be seen as another projection method, where the test functions (which are used 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 distribution function. With the point evaluations f k = f(t, x, c k ) this ansatz has the form f(t, x, c) = n f k δ (c c k ). (2.19) k=1 Note that the regularity of this ansatz is very weak. The expansion in delta functions is not differentiable and there is no meaningful interpretation of f for intermediate values c c k as well as for the derivative of f at any point c.

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25 Chapter 3 Mathematical Properties This chapter is about the necessary mathematical elements needed for the formulation and the expansion of the transformed Boltzmann equation as well as the solution using quadrature-based projection methods. We first introduce the different types of basis functions along with their respective properties. These functions will later be used as ansatz or test functions in the various settings and will determine the structure of the emerging PDE system. We derive normalized versions of the respective functions and show important recursion results that will be used in Chapters 4 and 5. The last part of this chapter is about quadrature methods and Gaussian quadrature in particular. Together with the different quadrature methods, we also give an introduction to 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 orthonormal set of Hermite functions that we will later use for the expansion of the unknown distribution function of the Boltzmann equation. The interested reader is referred to [1] for a more detailed summary of properties and formulas concerning Hermite polynomials 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-called probabilists Hermite polynomials H n that are defined in the following way for n 0 H n (ξ) = ( 1) n e ξ2 /2 dn dξ n e ξ2 /2. (3.1) 15

26 16 Mathematical Properties The other type of definition, leading to the so-called physicists Hermite polynomials He n, is dn He n (ξ) = ( 1) n e ξ2 dξ n e ξ2. (3.2) Note the missing factor in the exponent that leads to the conversion formula H n (ξ) = 2 n/2 Hen (ξ/ ) 2. (3.3) In the context of the Boltzmann equation, the probabilists Hermite polynomials are usually considered, because they are closely related to the Maxwellian that is the equilibrium distribution of the Boltzmann equation. This is the reason to use them as a set of basis functions for the expansion of the distribution function in one spatial dimension. The first Hermite polynomials can be easily calculated as H 0 (ξ)=1, H 1 (ξ)=ξ, H 2 (ξ)=ξ 2 1, H 3 (ξ)=ξ 3 3ξ, (3.4) H 4 (ξ)=ξ 4 6ξ The corresponding polynomials of higher degree can be derived using the following recursion formula that holds because of the definition (3.1) H n+1 (ξ) = ξ H n (ξ) n H n 1 (ξ). (3.5) Furthermore, the derivative of a Hermite polynomial can be expressed in terms of the lower order polynomial according to H n(ξ) = n H n 1 (ξ). (3.6) It is easy to show that the Hermite polynomials are an orthogonal basis of the corresponding 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 2π e ξ2 /2. According to that, we can compute for m N < H n, H m > w = n!δ nm, (3.8) which shows that the Hermite polynomials are in fact orthogonal but not normalized.

27 Mathematical Properties Orthonormal Hermite Polynomials As we have seen in Equation (3.8), the standard version of the Hermite polynomials are not normalized with respect to the weighted scalar product defined above. We therefore define a normalized Hermite polynomial as follows H n (ξ) := 1 n! Hn (ξ). (3.9) Due to their definition, we directly conclude the orthonormality of these polynomials (compare Equation (3.8)) < H n, H m > w = δ nm. (3.10) Using the definition (3.9) and the properties from Section 3.1.1, we can derive recurrence relations and the derivative of the normalized Hermite polynomial: ξh n (ξ) = n + 1H n+1 (ξ) + nh n 1 (ξ) (3.11) and H n(ξ) = n H n 1 (ξ). (3.12) Another useful property is d dξ (H n(ξ)w(ξ))=w(ξ) ( H n(ξ) ξh n (ξ) ) = w(ξ) n + 1H n+1 (ξ). (3.13) Combining (3.11) and (3.13), we obtain the relation ξ d dξ (H n(ξ)w(ξ))=ξw(ξ) ( H n(ξ) ξh n (ξ) ) = w(ξ) n + 1 ( n + 2H n+2 (ξ) + n + 1H n (ξ) ). (3.14) These relations will play a major role when we use a Hermite ansatz for our distribution function in the Boltzmann equation (see Section ). 3.2 Laguerre Polynomials The Hermite polynomials are defined such that they fulfill an orthogonality relation when integrated over the whole ξ R. But in special cases it is necessary to have similar polynomials but with different weights and integration intervals. One different approach are the so-called Laguerre polynomials, which are defined as d n L n (ξ) = eξ n! dξ n e ξ ξ n. (3.15)

28 18 Mathematical Properties The first Laguerre polynomials are L 0 (ξ)=1, L 1 (ξ)= ξ + 1, L 2 (ξ)= 1 ( ξ 2 4ξ + 2 ), 2 L 3 (ξ)= 1 ( ξ 3 + 9ξ 2 18ξ + 6 ), 6 L 4 (ξ)= 1 ( ξ 4 16ξ ξ 2 96ξ + 24 ). 24 (3.16) The polynomials also follow a recursion rule for the computation of polynomials of higher degree (using the definition L 1 (ξ) := 0) (n + 1)L n+1 (ξ) = (2n + 1 ξ)l n (ξ) nl n 1 (ξ). (3.17) The recursion formula for the derivative looks rather different from the one for Hermite polynomials (compare Equation (3.11)). Derivatives can be calculated using L n(ξ) = L n 1(ξ) L n 1 (ξ) (3.18) or from ξl n(ξ) = nl n (ξ) nl n 1 (ξ). (3.19) The Laguerre polynomials are already orthonormal as they are defined in (3.15) with respect to the scalar product < φ, ψ > wl = + 0 φ(ξ)ψ(ξ)w(ξ)dξ, (3.20) with weighting function w L (ξ) = e ξ when integrated over the positive domain ξ R + Consequently, we have for m N < L n, L m > w = δ nm. (3.21) The Laguerre polynomials therefore form a set of orthonormal basis functions. It is possible to use them for the expansion of the unknown distribution function of the three-dimensional Boltzmann equation in the radial velocity direction Generalized Laguerre Polynomials The Laguerre polynomials introduced in (3.15) are orthogonal with respect to the weighted standard scalar product (3.20). But for a transformation of variables, additional terms appear in the integrals due to the Jacobian of the transformation rule. For spherical

29 Mathematical Properties 19 velocity coordinates basically the term r 2 sin(θ) has to be considered for a proper definition of spherical coordinates (r, θ, φ). In the radial velocity direction, we then have to compute integrals of the following form 0 f(r)w(r)r 2 dr. (3.22) We therefore use polynomials that are orthogonal in the proper sense. This can be achieved by so-called generalized Laguerre polynomials L α n of degree n that are defined as follows L α n(ξ) = ξ α e ξ d n n! dξ n e ξ ξ n+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 L α 0 (ξ)=1, L α 1 (ξ)= ξ + α + 1, L α 2 (ξ)= ξ2 2 (α + 2)(α + 1) (α + 2)ξ +. 2 (3.24) Similar to the standard Laguerre polynomials, there exist some recursion rules and a formula for derivatives. One important formula is the following shift of the parameter α: L α n(ξ) = L α+1 n (ξ) α+1 L n 1 (ξ). (3.25) The formula remains valid for n = 0, if we define L α n(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)w Lα (ξ)dξ, (3.26) with weighting function w Lα (ξ) = e ξ ξ α, because we have + 0 L α n(ξ) L α m(ξ)w Lα (ξ)dξ = Γ(n + α + 1), (3.27) n! for Gamma-function Γ. Consequently, we can write down a normalized version L α n! ξ α e ξ d n n(ξ) = Γ(n + α + 1) n! dξ n e ξ ξ n+α, (3.28)

30 20 Mathematical Properties that leads to < L α n, L α m > wlα = δ nm. (3.29) Corresponding to (3.25), we can also derive a recursion formula for the normalized functions using the definition (3.28) as follows L α n(ξ) = n + α + 1L α+1 n (ξ) nl α+1 (ξ). (3.30) This formula can be used for the analytical computation of integrals emerging from a special 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 onedimensional part of a more complex application. We aim at a full three-dimensional setting 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 functions for 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 coordinates that can be evaluated for every point on the unit sphere and return a single value. We only consider real SH, but there are also complex-valued SH functions. We will now introduce a normalized version of the spherical harmonics, such that the upcoming integrals are easy to calculate and the system matrix will become symmetric. First, we need the so-called normalized associated Legendre polynomials L m l of degree l N and order m N, which can be calculated from the associated Legendre polynomials Pl m using the following formula L m (2l + 1)(l m)! l := Pl m (ξ) (3.31) 2(l + m)! and P m l := ( 1)m 2 l l! ( 1 x 2 ) m 2 d l+m n 1 dx l+m ( x 2 1 ) l. (3.32) Together with the setting L m l := 0 for l > m, the normalized associated Legendre polynomials satisfy a set of recursion relations that are very useful for computations with the spherical harmonics later. Now the real SH function of degree l and order m ( l m l) is defined as follows Y 0 l (θ, φ) :=Y 0,l(θ, φ) := 1 2π L0 l (cos(θ)) (m = 0), Y m l (θ, φ):=y m 1 (θ, φ) := Lm π l (cos(θ)) cos(mφ) (m > 0), 1,l Yl m (θ, φ):=y m 2,l (θ, φ):= 1 L m π l (cos(θ)) sin( m φ)(m < 0). (3.33)

31 Mathematical Properties 21 Figure 3.1: SH degree l = 0: Y 0 0 At first sight it is important that Yl 0 does not depend on φ and all the SH functions are polynomials in trigonometric functions of θ and φ. Furthermore, we have 2l +1 functions for each degree l. The first few SHs are: Y0 0 (θ, φ) = 1 2 π, Y1 1 (θ, φ)= π sin(θ) sin(φ), Y1 0 (θ, φ) = π cos(θ), Y1 1 (θ, φ) = 1 3 sin(θ) cos(φ), 2 π Y2 2 (θ, φ)= π sin2 (θ) sin(2φ), Y2 1 (θ, φ)= 1 15 sin(2θ) sin(φ), 4 π Y2 0 (θ, φ) = 1 5 (3 cos(2θ) + 1), 8 π Y2 1 (θ, φ) = 1 15 sin(2θ) cos(φ), 4 π Y 2 2 (θ, φ) = π sin2 (θ) cos(2φ). (3.34) In order to get an impression of how a spherical harmonics looks like, it is possible to plot Yl m in the following way: we scale each vector that points from the origin to the unit sphere by the absolute value of Yl m for this particular choice of θ and φ corresponding to the direction in which the vector points. The results are shown in Figures 3.1 to 3.3e.

32 22 Mathematical Properties (a) Y1 1 (b) Y10 (c) Y11 Figure 3.2: SH degree l = 1 (a) Y2 1 (b) Y20 (d) Y2 2 (c) Y21 (e) Y22 Figure 3.3: SH degree l = 2

33 Mathematical Properties 23 By construction, the SH functions are orthonormal with respect to the scalar product < f, g >= meaning that we simply have 2π π 0 0 f(θ, φ)g(θ, φ) sin(θ)dθdφ, (3.35) < Yl m, Yl m >= δ m,m δ l,l. (3.36) The set of spherical harmonics is therefore an orthonormal basis for all functions defined on the unit sphere. It is possible to use them as part of the ansatz functions in a full three-dimensional setting Cartesian Spherical Harmonics As we have seen in the previous section, the spherical harmonics are part of a basis of all polynomial functions in x, y, z in the three-dimensional space. But they are naturally formulated in the spherical coordinates r, θ, φ together with a radial part that depends solely on the radius r. Consistently, one would have to calculate the emerging integrals using spherical integration, too. Depending on the context, this can be inefficient or inflexible. In [15] a cartesian version of the solid spherical harmonics is defined. The solid spherical harmonics Nl m (r, θ, φ) are related to the spherical harmonics Yl m (θ, φ) by an additional factor r l : N l,m (r) = r l Yl m (θ, φ). (3.37) The additional factor r l enables the representation of the solid spherical harmonics in cartesian coordinates x, y, z. The cartesian solid spherical harmonics are first defined as follows { r l Ñ + l,m = (1+ m )! P m l (cos(θ) cos( m φ)) if m 0 Ñ + l,m if m < 0, (3.38) Ñ l,m = r l (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 solid spherical harmonics r l Yl m (θ, φ). The definition can be normalized using a normalization factor 2l + 1 Ñ + l,m if m > 0 N l,m = (l m)!(l + m)! 1 2π 2 Ñ + l,m if m = 0. (3.40) Ñ l, m if m < 0 According to [15], this version of the solid spherical harmonics can easily be written in the cartesian coordinates (x, y, z) = (r sin(θ) cos(φ), r sin(θ) sin(φ), r cos(θ)) using the

34 24 Mathematical Properties following recursion formula N + 0,0 =1, N 0,0 =0, N + m,m= 1 2m N m,m= 1 2m N ± l,m = 1 (l + m)(l m) ( ) xn m 1,m 1 + yn m 1,m 1, ( ) yn m 1,m xn m 1,m 1, ( (2l 1)zN ± l 1,m r2 N ± l 2,m ). (3.41) The cartesian version of the SSH is by definition orthogonal with respect to the scalar product defined in (3.35) because we have < N l,m, N l,m >= rl+l δ m,m δ l,l. (3.42) The definition (3.41) now allows for a representation of the solid spherical harmonics in the cartesian basis. As the corresponding integrals are very easy to solve (in fact, even a 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 now easily identify the basis functions at the different levels l, m of the spherical harmonics with simple cartesian polynomials and see the differences to a full tensor product with a polynomial basis, for example. 3.4 Jacobi Matrix In this context, we will briefly mention the so-called Jacobi matrix, which is a tridiagonal matrix 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 Jacobian matrix that is the first derivative of the numerical or analytical flux calculation). We will later see this matrix when we show examples for the computation of eigenvalues of the system matrix. It turns out that the eigenvalues of the Jacobi matrix are just the zeros of the (n + 1)st orthonormal basis function they correspond to (see also [24]). Our sets of orthonormal polynomials Φ n satisfy a recursion rule like xφ i (x) = a i Φ i+1 (x) + b i Φ i 1 (x). (3.43) In case of the orthonormal Hermite polynomials, for example, we have xφ i (x) = i + 1Φ i+1 (x) + iφ i 1 (x). (3.44) Thus a H i = i + 1, b H i = i. (3.45)

35 Mathematical Properties 25 as The Jacobi matrix J n corresponding to a set of orthonormal polynomials is defined 0 a b 1 0 a.. 1. J n :=. 0 b (3.46) an b n 0 And the characteristic polynomial is actually some factor γ R times the (n + 1)st orthonormal polynomial. χ(j) = det(j n λ I n ) = γφ n+1 (λ). (3.47) This leads to the fact that the eigenvalues (as roots of the characteristic polynomial) are the zeros of Φ n+1, too. Note that the Jacobi matrix for Hermite polynomials is symmetric, due to the coefficients (3.45). In Section 4.1, we will recognize the Jacobi matrix as the system matrix of the PDE system for the basis coefficients after the projection. 3.5 Gaussian Quadrature Like every quadrature rule, Gaussian quadrature approximates integrals of a certain kind using evaluations of the integrand at discrete points. In the case of Gaussian quadrature the integrand consists of a weighted product of a function f and a weighting function w. Gaussian quadrature of order N N is performed according to b a f(x)w(x)dx = N w i f(x i ), (3.48) where for i = 1,..., N the w i are called weights and the x i 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 p n. The weights w i can be calculated according to the formula w i = a N a N 1 b a i=1 w(x)p N 1 (x) 2 dx p N (x i)p N 1 (x i ), (3.49) where a N is the coefficient in front of x N in the respective polynomial p N. It can be shown that the weights are all positive and all the sampling points lie inside the interval (a, b).

36 26 Mathematical Properties Gauss-Hermite Quadrature If we choose the weighting function in Equation (3.48) to be w(x) = 1 2π e x2 /2, and a =, b = + we are approximating integrals of the kind + 1 f(x) e x2 /2 dx (3.50) 2π by a weighted sum of function evaluations. The sampling points x i for the function evaluation are the roots of the N-th Hermite polynomial, which is given in (3.9) and here denoted as p N. According to the general formula (3.49), the weights for the Gauss-Hermite quadrature can be calculated to be b w i = a w(x)p N 1 (x) 2 dx N a a N 1 p N (x i)p N 1 (x i ) (N 1)! 1 = N! NpN 1 (x i ) 2 (3.51) 1 = Np N 1 (x i ) Gauss-Laguerre Quadrature For another weighting function w(x) = e x and a = 0, b = + we end up calculating the following integrals by the so-called Gauss-Laguerre quadrature + 0 f(x)e x dx (3.52) as a weighted sum of function evaluations. The sampling points x i for the function evaluation are now the roots of the N-th Laguerre polynomial p N. Similar to the previous version (compare Section 3.5.1), the weights for the Gauss- Laguerre quadrature are w i = a N a N 1 b a w(x)p N 1 (x) 2 dx p N (x i)p N 1 (x i ) x i = (N + 1) 2 p N+1 (x i ) 2. (3.53) Thus, the Gauss-Laguerre quadrature is essentially performed in the very same way as the Gauss-Hermite quadrature with the important difference of another weighting function as well as different interval bounds.

37 Mathematical Properties Generalized Gauss-Laguerre Quadrature For weighting function w(x) = x α e x and a = 0, b = + we calculate the following integrals + f(x)e x x α dx (3.54) 0 as a weighted sum of function evaluations. The sampling points x i for the function evaluation are now the roots of the N-th generalized Laguerre polynomial p N. The weights for the generalized Gauss-Laguerre quadrature are w i = a N a N Non-Classical Quadrature b a w(x)p N 1 (x) 2 dx p N (x i)p N 1 (x i ) x i = (N + 1) 2 p N+1 (x i ) 2. (3.55) There are quadrature formulas for various types of weighting functions in combination with the respective domains of integration and the corresponding orthogonal polynomials. 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 of the formulas above is applicable. Under certain conditions, it is possible to find corresponding 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. We will explain the general procedure of finding the orthogonal polynomials and the weights in 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 using a quadrature formula like in (3.48) First, we have to check, whether a Gaussian-quadrature rule with orthogonal polynomials exists. According to [19] this requires the following Hankel-matrix to be nonsingular µ 0 µ 1... µ N 1 µ 1 µ 2... µ N N :=......, (3.57) µ N 1 µ N... µ 2N 2

38 28 Mathematical Properties with µ i = b a x i w(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 possibilities to do this, of which one of the easiest would be to apply a Gram-Schmidt method to orthonormalize the monomials and end up with a set of orthonormal polynomials p j (x). Other possibilities are the method of moments, the Stieltjes procedure and the Lanczos algorithm, for more information, see [19]. As we have seen before, compare Section 3.5, the quadrature points x i are now just the roots of the Nth orthonormal polynomial if we are interested in a formula that is exact up to degree 2N 1. As the Gram-Schmidt method is already a numerical procedure, the calculation of the roots is usually also done numerically and thus very efficient. Next is the calculation of the corresponding weights. The weights are determined by the condition of exactness of the formula up to degree N 1. In principle, one can use the general quadrature formula (3.48) for every monomial x i, for i = 0,..., N 1 and get a linear system of equations by the requirement that the quadrature reproduces the exact result. Alternatively, the condition is imposed for the orthonormal basis polynomials p j (x). Thus, one has to solve the set of linear equations (see also [21]) p 0 (x 1 )... p 0 (x N ) p 1 (x 1 )... p 1 (x N )..... p N 1 (x 1 )... p N 1 (x N ) w 1 w 2. w N = b a b a b a p 0 (x)w(x)dx p 1 (x)w(x)dx. p N 1 (x)w(x)dx. (3.59) Note that the right hand side usually includes many zero entries, because the polynomials p j (x) are orthogonal to the constant function p 0 (x). This may not be the most efficient way to calculate accurate weights, because the solution of the linear system (3.59) can be 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 p i form an orthogonal basis and the x i are the corresponding quadrature points, then this matrix will be non-singular, because a set of non-zero weights w i has to exist. In the end, we have an orthonormal set of basis polynomials, the roots of the N th polynomial and the corresponding weights for a Gaussian-quadrature formula to calculate the integrals with respect to non classical weighting functions.

39 Mathematical Properties Interpolation Property and Aliasing With the help of Gaussian quadrature, it is possible to prove an interpolation property of an 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 order n 1 N as follows f(x) = f(x) = n 1 j=0 n 1 j=0 α j Φ j (x)w(x), α j Φ j (x)w(x). (3.60) with basis coefficients α i, α i, basis polynomials Φ i, the corresponding weighting function 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 α j = b a Φ j (x)f(x)dx (3.61) or employing a Gaussian quadrature approximation for the integral using the same set of orthonormal functions Φ i α j = n 1 w i Φ j (x i ) w(x i ) f(x i). (3.62) i=1 Although the second method is somewhat an approximation to the exact integral, it is possible to derive an interesting property for the quadrature based coefficients. Using these approximated values, we have f(x i ) = f(x i ) for i = 1,..., n. (3.63) Which means that the value of the quadrature based approximation f and the value of the exact function f are the same at the sampling points of the quadrature rule. This is 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 exact integration. It is caused by sort of high frequencies in f that spoil the point interpolation property. Equation (3.63) can be proven by the assumption that we have f(x i ) f(x i ) for one i = 1,..., n. (3.64)

40 30 Mathematical Properties Computing α k we get: α k = = n 1 w i Φ k (x i ) w(x i ) f(x i) n 1 w i Φ k (x i ) w(x i ) f(x i ) i=1 i=1 n 1 w i Φ k (x i ) w(x i ) i=1 n 1 = j=0 n 1 = j=0 n 1 = j=0 n 1 = j=0 = α k, α j α j n 1 j=0 α j Φ j (x i )w(x i ) n 1 w i Φ k (x i ) w(x i ) Φ j(x i )w(x i ) i=1 n w i Φ k (x i )Φ j (x i ) i=1 α j Φ k (x)φ j (x)w(x)dx α j δ k,j (3.65) where we used the exactness up to degree 2n 1 of the quadrature formula From α k α k, we then deduce, that our assumption was wrong and we thus have f(x i ) = f(x i ) 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.

41 Chapter 4 Motivational Examples Prior to the development of an abstract framework as explained in Chapter 5, we want to show some tests using different kinetic equations and basis functions for the expansion 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 between exact and quadrature-based projections as we will explain the derivation of the PDE system. We start with some small one-dimensional examples and consider different simple kinetic equations with generalized advection velocities that are closely related to the fully transformed Boltzmann equation, which we will see in the next Chapter 5. We will observe that the use of recursion formulas for the basis functions allows an exact calculation of the system matrix. Furthermore, we extend the examples to multiple spatial dimensions using a Hermite tensor ansatz. The results are consistent with the one-dimensional case and show that the properties of the system are closely related to the choice of the basis functions and the projection method. 4.1 One-Dimensional Cases Before we cover kinetic equations in three spatial dimensions, we will first take a look at the simpler one-dimensional equation and explain the ideas and methods, which we will essentially also use for the more important case with three spatial dimensions 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 exact projection or a quadrature-based projection to investigate possible differences. We consider the following equation f(t, x, c) + a(c) f(t, x, c) = 0, (4.1) t x 31

42 32 Motivational Examples which can be seen as the easiest case of a collision-free Boltzmann equation, when we think 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 f around the equilibrium distribution (or weighting function) w(c) = 1 2π e c2 /2 by a series of polynomials Φ i as follows f(t, x, c) = 1 2π e c2 /2 n α i (t, x)φ i (c), (4.2) i=0 which (using the Einstein sum convention) leads to w(c) ( t α i (t, x)φ i (c) + x a(c)α i (t, x)φ i (c)) = 0 (4.3) w(c) (Φ i (c) t α i (t, x) + a(c)φ i (c) x α i (t, x)) = 0. (4.4) Consider now the case a(c) = c. Exemplarily, we here choose normalized Hermite polynomials for Φ, so we have Φ i (c) = H i (c). Using the recursion formula from Equation (3.11), we can express ch i (c) in terms of Hermite polynomials only ( ( i w(c) H i (c) t α i + + 1Hi+1 (c) + ) ) ih i 1 (c) x α i = 0. (4.5) We now project the emerging equation using different projection methods. First, consider the continuous projection P j (f) = + f(t, c, x)h j (c)dc =< f, H j > w, (4.6) which means that we basically multiply with the j th basis function and integrate over the whole velocity domain. The second projection method employs a quadrature rule for the calculation and thus reads n+1 1 P j (f) = w k f(t, c k, x) w (c k ) H j (c k ) =< f, H j > N (4.7) k=1 with quadrature weights w i and sampling points c i according to the specific quadrature rule. The subscript N = n + 1 is the number of sampling points or the order of the quadrature rule. Note that we have to multiply with the inverse of the weighting function to cancel out the weighting function inside f. Applying the continuous projection P j (f) =< f, H j >, we obtain < H i, H j > w t α i + i + 1 < H i+1, H j > w x α i + i < H i 1, H j > w x α i = 0. (4.8) Now we make use of the orthonormality property of the Hermite polynomials (see Equation (3.10), i.e. < H i, H j > w = δ i,j. For technical reasons, we set α j = 0 for j < 0 and also for j > n. Thus δ i,j t α i + i + 1 δ i+1,j x α i + i δ i 1,j x α i = 0 (4.9) t α j + j x α j 1 + j + 1 x α j+1 = 0 (4.10)

43 Motivational Examples 33 for j = 0,..., n We have now derived a system of equations for the unknown basis coefficients α i. It is possible to write the emerging system in compact matrix form using α = {α i } i=0,...,n as t α + A c x α = 0 (4.11) with system matrix A c R n+1 n+1 A c = n 0 0 n 0. (4.12) Interestingly, the eigenvalues of the matrix are exactly the roots of the (n + 1)-st normalized Hermite polynomial H n+1 : σ (A c ) = {λ R H n+1 (λ) = 0}. (4.13) Going back to the previous Section 3.4, this becomes clear, as the system matrix is just the Jacobi matrix of the Hermite basis. If we use the quadrature-based projection method (4.7), the result would not change at all, because we just replace the standard scalar product < f, Φ j > w by the corresponding quadrature rule < f, Φ j > N and we know from Section 3.5 that the quadrature rule is exact up to order 2N 1 = 2(n + 1) 1 = 2n + 1 of the integrand. As the integral with the highest order integrand is < H n+1, H n > N, the integrand thus has a polynomial degree of 2n + 1 and is just integrated exactly Generalized Kinetic Equation c 2 Now we consider another kinetic equation with a(c) = c 2. The reason for that is that in the transformed Equation (5.4) we will later have quadratic and mixed terms like ξ i ξ j which we want to understand in an easier setting before. We can make use of the recursion relation (see Equation (3.11)) twice, to transform the occurring term a(c)h i (c) as follows: c 2 H i (c) = (i + 1)(i + 2)H i+2 (c) + (2i + 1)H i (c) + i(i 1)H i 2 (c). (4.14) After that, we again perform the continuous projection (4.6) first and use the orthonormality property (see Equation (3.10)) of the Hermite polynomials to arrive at δ i,j t α i + (i + 1)(i + 2)δ i+2,j x α i +(2i + 1)δ i,j x α i + i(i 1) δ i 2,j x α i =0 (4.15) t α j + (j 1)j x α j 2 +(2j + 1) x α j + (j + 2)(j + 1) x α j+2 =0 (4.16)

44 34 Motivational Examples for j = 0,..., n, which leads to the following system of equations t α + A c 2 x α = 0 (4.17) with symmetric system matrix A c 2 R n+1 n+1 A c 2 = (n 1) n n (n 1) n 0 2n + 1. (4.18) In this case, the eigenvalues of the matrix A c 2 are not exactly the zeros of the n + 1-st normalized Hermite polynomial H n+1. The eigenvalues are actually a mixture of the squared zeros of the n + 1-st and the n + 2-nd Hermite polynomial: λ 2 σ (A c ) H n+1 (λ) = 0 H n+2 (λ) = 0. (4.19) The relation to the Jacobi matrix is that we can take J n+1 J n+1, delete the last column and row and then end up with A c 2. This holds, because we have J n+1 J n+1 = n (n + 1) n n (n + 1) 0 n + 1 (4.20) Contrarily, we obtain J n J n = (4.21) (n 1) n n (n 1) n 0 n So that the last value (A c 2) n+1,n+1 = 2n + 1 differs from the last value of J n J n, which leads to the change of the eigenvalues.

45 Motivational Examples 35 Like before, we now apply the quadrature-based projection (4.7) and will notice very soon that there is a difference in the emerging system of equations. The integral with the integrand of highest degree is now (n + 2)(n + 1) < H n+2, Hn > N, which is not integrated exactly, due to n+2+n = 2n+2 = 2(n+1) = 2N > 2N 1. Exact integration would lead to (n + 2)(n + 1) < H n+2, Hn > w = 0 due to orthogonality, but we get a value of (n + 2)(n + 1) < H n+2, Hn > N = (n + 1) = N using quadrature. This changes the last entry of the matrix to n. The new system matrix Ãc 2 then is à c 2 = (4.22) (n 1) n n (n 1) n 0 n The small difference in the last entry of the matrix leads to a change in the eigenvalues. Now the eigenvalues of Ãc2 are purely the squared roots of the n + 1-st Hermite polynomial: σ(ãc 2) = {λ2 R H n+1 (λ) = 0}. (4.23) The reason is that now the system matrix is Ãc 2 = J n J n, including the smaller entry (Ãc 2) n+1,n+1 = n Generalized Kinetic Equation c + c 2 As we have seen in the previous case, the quadrature method is able to change the eigenvalues of the system to be simply the squared roots of a Hermite polynomial. For our 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 + c 2. With this setting, it is now possible to redo the same calculation from above and calculate 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 the roots of a Hermite polynomial at all. If we use the quadrature-based projection method (4.7), the eigenvalues are again changed and in fact related to the roots of the (n + 1)-st Hermite polynomial in the following way: ) σ (Ãc+c 2 = {λ + λ 2 R H n+1 (λ) = 0}. (4.24) It therefore seems apparent that the quadrature projection always changes the eigenvalues corresponding to the equation using a(c) to be the roots λ of the n + 1-st Hermite polynomial inserted into a (λ). We also did tests with different basis functions, for

46 36 Motivational Examples example Legendre polynomials or Laguerre polynomials and the eigenvalues always behaved in the respective way Relation between Quadrature Projection and Discrete Velocity Method In the previous Section 4.1, we have seen that simple quadrature-based projections suffice to change the system slightly to systematically obtain specific eigenvalues of the system matrix. This is closely related to the procedure during a discrete velocity method (DVM), which we have described in Section Motivated by this examples, we will now take a closer look at the relation between the two methods and see that they are essentially the 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 Φ i, so that w(c) = 1, the distribution function reads f(t, x, c) = n α i (t, x)φ i (c). (4.25) i=0 If we insert the ansatz (4.25) into Equation (4.1), we can evaluate the equation at discrete velocities c j for j = 0,..., n to obtain n different equations n Φ i (c j ) t α i (t, x) + a(c j )Φ i (c j ) x α i (t, x) = 0. (4.26) i=0 This system of equations can be written in matrix vector form using the following definitions α = (α 0 (t, x),..., α n (t, x)) T, (4.27) Φ 0 (c 0 )... Φ n (c 0 ) B =....., (4.28) Φ 0 (c n )... Φ n (c n ) A = diag(a(c 0 ),..., a(c n )). (4.29) The system can then be written in the following very compact form B t α + AB x α = 0. (4.30) Assuming B is regular, we can multiply by its inverse B 1 from the left to end up with the system t α + B 1 AB x α = 0. (4.31) We can now directly see that the eigenvalues of the system are the diagonal entries of A, as the multiplication with the regular matrices B and B 1 does not change the eigenvalues of A. The eigenvalues of the system are therefore point evaluations of the

47 Motivational Examples 37 function a(c) at the discrete velocities. Now we proceed differently using the following quadrature formula (with respect to weighting function w(c) = 1)) for the projection of Equation (4.1) with the projection operator P j (f) n P j (f) = w k f(t, x, c k )Φ j (c k ) =< f, Φ j > N. (4.32) k=0 This leads to the PDE system (j = 0,..., n) n i=0 k=0 n w k Φ i (c k )Φ j (c k ) t α i (t, x) + w k a(c k )Φ i (c k )Φ j (c k ) x α i (t, x) = 0. (4.33) Similar to the system before, we write this system in matrix vector form using the following additional definitions w 0 Φ 0 (c 0 )... w n Φ n (c 0 ) C =....., (4.34) w 0 Φ 0 (c n )... w n Φ n (c n ) Thus, the system reads or using C = BW W = diag(w 0,..., w n ). (4.35) CB t α + CAB x α = 0 (4.36) BW B t α + BW AB x α = 0. (4.37) Assuming that B is regular and the weights are non zero (which is usual for quadrature rules), we again obtain the same system as before t α + B 1 AB x α = 0, (4.38) which means that the eigenvalues of the system are still the diagonal entries of the matrix A, 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 only requirement is that the matrix B is invertible, including the fact that the quadrature points or discrete velocities respectively have to be pairwise distinct. The observation can be made, because the quadrature method is basically evaluating the equation at distinct points and combining the resulting equations in a linear way to end up with a more complicated system than the discrete velocity method. But as we have seen, the systems and their properties are essentially the same. If we now choose a standard projection method that computes exact integrals over the velocity space, this linear combination property is not given anymore. The appearance of the function a(c) inside the integral leads to the use of recursion formula, as we have

48 38 Motivational Examples done it in the previous Chapter 3. But the result cannot be generalized to arbitrary functions a(c), as recursion formulas are usually only available for a limited class of functions, such as polynomials. Even for polynomial a(c), recursion formulas lead to different 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 and develop an abstract framework to understand the quadrature-based projections also in the transformed Boltzmann equation in Chapter Multi-Dimensional Cases In more than one spatial dimension, there are different possible ways to propose an ansatz for the distribution function f. The most simple one is the consistent extension of the 1D example from the previous Section 4.1. Now we describe a Hermite tensor ansatz in three dimensions by taking a basis that is a Hermite polynomial in every velocity direction Simple Kinetic Equation 3D The first 3D kinetic equation we consider is again rather simple f(t, x, c) + a(c) f(t, x, c) = 0 (4.39) t x 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 2π e ct c/2 f(t, x, c) = 1 2π 3 e ct c/2 n i,j,k=0 α i,j,k (t, x)φ i,j,k (c). (4.40) But the basis function is now a Hermite polynomial in every direction: Φ i,j,k (c) = H i (c x ) H j (c y ) H k (c z ). (4.41) The projections are either done by analytical computation of the integrals P l,m,n (f) = f(t, x, c)φ l,m,n (c)dc =< f, Φ l,m,n > (4.42) or by a Hermite-Gauss quadrature formula of order N as follows P l,m,n (f) = N+1 k 1,k 2,k 3 =1 f(t, c k1, c k2, c k3, x) w k1 w k2 w k3 Φ l,m,n (c k1, c k2, c k3 ) =< f, Φ l,m,n > N. w (c k1, c k2, c k3 ) (4.43)

49 Motivational Examples 39 The derivation of the system of equations can be carried out just like in the 1D case (see Section 4.1.1). The only difference in this 3D case is that we have more terms emerging from the expansion in each of the three velocity components. We will omit the derivation 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 expect rotational symmetry of the problem, as we have used an expansion in cartesian coordinates along each single axis that is not invariant under rotational transformation. We can therefore only expect symmetry with respect to one of the coordinate axes for both projection methods. The projection of Equation (4.39) with the operator (4.42) or (4.43) leads to a coupled system of PDEs in three spatial variables. The general form of the system is A t α + B x α + C y α + D z α = 0, (4.44) where we usually have A = I n for orthonormal basis functions. In order to check hyperbolicity of the system and get some knowledge about the behavior of the system, we take a look at the generalized system matrix for unit vector β = (β 1, β 2, β 3 ) of length one, so that β = 1. The system matrix can be written as A sys = A 1 (Bβ 1 + Cβ 2 + Dβ 3 +). (4.45) This is also a consistent extension of the one-dimensional case, because the system matrix is evaluated for every direction β, where the unit vector β can be seen as a vector pointing in the direction of interest. The system is said to be hyperbolic, if A sys has real eigenvalues for all directions β. For numerical calculations, it is possible to plot the eigenvalues of the system matrix depending 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 and n = 2 respectively. The eigenvalues lie on circles that cross through points (c x, c y, c z ) at their largest distance from the origin, where the coordinates c x, c y, c z are combinations of zeros of the (n + 1)-st Hermite polynomial H n+1 (c). We can see the same behavior for the case n = 2, but the high number of degrees of freedom (16) leads to a very dense plot. Thus we cut the plots at c z = 0 showing values inside the x y plane in Figures 4.1b and 4.2b. There we can in principle observe analogous results for the 2D case. Circles extending up to the coordinates of roots of the Hermite polynomials are combined in the plane. For the sake of completeness, we give the roots of the involved polynomials H 1 (c) = 0 c { 1, 1}, (4.46) H 2 (c) = 0 c { 3, 0, 3}. (4.47) As we have seen, the results are in perfect agreement to the one-dimensional case from the previous Section 4.1. But in addition we now have eigenvalues of A sys depending on the direction of the flow. Due to the Hermite tensor ansatz in cartesian coordinates,

50 40 Motivational Examples (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 the flow in which information is propagated faster than in other directions. This is usually not desired, because the discrete model looses its symmetry on the one hand and is not longer Galilei invariant on the other hand. An ansatz as simple as the Hermite tensor in 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 quadrature formula ensures exact integration of all the involved terms. This will not be the case for a(c)i = c2i in the following section Generalized Kinetic Equation c2i 2D After assuring comparable behavior of the 3D case regarding the standard simple kinetic equation (4.39), we now turn our attention to the multi-dimensional version of the second test case from the previous section. Replacing the convective velocities cx, cy, cz by their squares or simply setting a(c)i = c2i, we obtain the equation 3 X f (t, x, c) + c2i f (t, x, c) = 0. t xi (4.48) i=1 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 α of the form (4.44). The eigenvalues of the system matrix Asys do again depend on the directional vector variable β, but can be plotted for various values of β and n.

51 Motivational Examples (a) full plot 41 (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 projection according to the 2D case of Equation (4.42). These plots can be obtained by the setting β3 = 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 distance from the origin. These points coordinates are combinations of all squared nonzero roots of the (n + 1)-st and the (n + 2)-nd Hermite basis polynomial. This corresponds to the 1D case, where eigenvalues were also related to either the (n + 1)-st and the (n + 2)-nd Hermite polynomial. The results for the quadrature-based projection method as defined in the general 3D setting in Equation (4.43) are very different from the continuous projection. The eigenvalues 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 eigenvalues degenerate 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 only 1, which is in fact the cx and cy coordinate of the circle s point with the largest distance from the origin. Proceeding to the case n = 2, the relevant squared zeros of the third Hermite polynomial are 0 and 3, which can be combined to the coordinates of the circles points with the largest extension from the origin, again. Note that this time, there is also a circle with radius zero, represented by the origin itself. For n = 3, we see the same behavior regarding the squared zeros of the 4-th Hermite polynomial, which are 3 ± 6. Combinations of those values mark the specific points in