Faculty of Technology and ScienceMathematics

DISSERTATION

Karlstad University Studies2008:33

Eva Mossberg

Some numerical and analytical methods for equations of wave propagation and kinetic theory

Karlstad University Studies2008:33

Eva Mossberg

Some numerical and analytical methods for equations of wave propagation and kinetic theory

Eva Mossberg. Some numerical and analytical methods for equations of wave propaga-tion and kinetic theory

DISSERTATION

Karlstad University Studies 2008:33ISSN 1403-8099 ISBN 978-91-7063-192-4

© The Author

Distribution:Faculty of Technology and ScienceMathematics651 88 Karlstad054-700 10 00

www.kau.se

Printed at: Universitetstryckeriet, Karlstad 2008

Abstract

This thesis consists of two different parts, related to two different fields in mathemati-

cal physics: wave propagation and kinetic theory of gases. Various mathematical and

computational problems for equations from these areas are treated.

The first part is devoted to high order finite difference methods for the Helmholtz equa-

tion and the wave equation. Compact schemes with high order accuracy are obtained

from an investigation of the function derivatives in the truncation error. From the actual

equation, high order derivatives are transferred to derivatives of lower order, and time

derivatives are transferred to space derivatives. For the Helmholtz equation, a compact

scheme based on this principle is compared to standard schemes and to deferred correc-

tion schemes. Subsequently, the characteristics of the errors for the different methods

are demonstrated and discussed. For the wave equation, a finite difference scheme with

fourth order accuracy in both space and time is constructed and applied to a problem in

discontinuous media.

The second part addresses a set of problems related to kinetic equations. A direct simu-

lation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and

numerical tests are performed. A formal derivation of the method from the Boltzmann

equation with grazing collisions is presented. The linear and linearized Boltzmann col-

lision operators for the hard sphere molecular model are studied using exact reduction of

integral equations to ordinary differential equations. It is demonstrated how the eigen-

values of the operators are obtained from these equations, and numerical values are

computed. A proof of existence of non-zero discrete eigenvalues is presented. The ordi-

nary differential equations are also used to investigate the Chapman-Enskog distribution

function with respect to its asymptotic behavior.

iii

List of Papers

This thesis is based on the following papers, which are referred to in the text by their

Roman numerals.

I E. Mossberg. Higher order finite difference methods for the Helmholtz

equation. Published as part of Lic. thesis IT 2002-001, Dept. of Informa-

tion Technology, Uppsala Univ., Uppsala, Sweden, 2002.

II B. Gustafsson and E. Mossberg. Time compact high order difference

methods for wave propagation. Reprinted from SIAM J. Sci. Comput.,26:259–271, 2004, by permission of the publisher.

III A. V. Bobylev, E. Mossberg, and I. F. Potapenko. A DSMC method for

the Landau-Fokker-Planck equation. Two appendices have been added to

the version published in the book edited by M. S. Ivanov and A. K. Re-

brov: Rarefied Gas Dynamics (Proc. 25th International Symposium onRarefied Gas Dynamics, St. Petersburg, Russia), pages 479–483, 2007.Reprinted by permission of the publisher, the Siberian Branch of the Rus-

sian Academy of Sciences, Novosibirsk.

IV A. V. Bobylev and E. Mossberg. On some properties of linear and lin-

earized Boltzmann collision operators for hard spheres.

v

Results from my research have also been published in the following journal and confer-

ence papers.

M. Thuné, E. Mossberg, P. Olsson, J. Rantakokko, K. Åhlander, and

K. Otto. Object-oriented construction of parallel PDE solvers. In

E. Arge, A. M. Bruaset, and H. P. Langtangen, editors, Modern SoftwareTools for Scientific Computing, pages 203–226. Birkhäuser, Boston, MA,

1997.

E. Mossberg, K. Otto, and M. Thuné. Object-oriented software tools

for the construction of preconditioners. Sci. Programming, 6:285–295,1997.

M. Mossberg and E. Mossberg. Estimation of parameters for signals

in exponential form. In Proc. 22nd IASTED Int. Conf. on Modelling,Identification and Control, Innsbruck, Austria, pages 73–77, 2003.

M. Mossberg and E. Mossberg. On optimal criteria for optimal set point

tracking. In Proc. 2005 American Control Conf., Portland, OR, pages4482–4483, 2005.

M. Mossberg, E. K. Larsson, and E. Mossberg. Fast estimators for large-

scale fading channels from irregularly sampled data. IEEE Trans. onSignal Processing, 54:2803–2808, 2006.

M. Mossberg, E. K. Larsson, and E. Mossberg. Estimation for power

control of large-scale fading channels from irregularly sampled noisy

data. In Proc. 2006 American Control Conf., Minneapolis, MN, pages5745–5750, 2006.

M. Mossberg and E. Mossberg. Identification of local oscillator noise in

sampling mixers. In Proc. 12th IEEE Digital Signal Processing Work-shop, Grand Teton, WY, pages 572–575, 2006.

M. Mossberg, E. K. Larsson, and E. Mossberg. Estimation of large-scale

fading channels from sample covariances. In Proc. 45th IEEE Conf. onDecision and Control, San Diego, CA, pages 2992–2996, 2006.

I. F. Potapenko, A. V. Bobylev, and E. Mossberg. Deterministic and

stochastic methods for nonlinear Landau-Fokker-Planck kinetic equa-

tions with applications to plasma physics. Transport Theory Statist.Phys., Accepted for publication.

vi

Acknowledgments

I am deeply and humbly grateful for all the help and support I have received during

many years from my supervisors Prof. Alexander Bobylev at Karlstad University and

Prof. Bertil Gustafsson at Uppsala University. I also appreciate the contributions from

my co-authors and from my former and present colleagues in Uppsala and Karlstad. I

want to extend a special thank you to my husband Magnus who not only has provided

me with personal support, but also with scientific help as a colleague and co-author.

This work was partially funded by the Swedish Research Council. This support is grate-

fully acknowledged.

vii

That which is static and repetitive is boring.That which is dynamic and random isconfusing. In between lies art.

Often attributed to John Locke

1 Introduction

This thesis includes work on different topics in applied mathematics, with two papers

describing numerical computation methods based on finite difference discretization, and

two papers treating kinetic equations. These rather diverse scientific areas pose different

mathematical and computational challenges, but our ultimate goal when studying them

is the same: to make us understand, describe, and simulate the world we live in.

All matter is formed of particles. Looking around us (or inside us) on a microscopic

level, the complexity in how molecules, atoms and sub-atomic particles are distributed

in relation to each other and how they interact is astonishing. But in everyday life,

there are many macroscopic properties like average velocity, pressure, density, force,

and temperature, that we can model ignoring the microscopic scale, treating matter as a

continuum. In other cases, when this is not possible, we may still be able to model the

particle interactions by equations for distribution functions.

The mathematical modeling of dynamics of motion leads to differential equations. Ex-

cept for a limited number of special cases, it is in general not possible to find the analyt-

ical solution functions of these equations, so in order to describe the functions we need

to study asymptotics and approximate solutions, and perform computer simulations of

deterministic or stochastic type. But in order to perform numerical computations, we

need to replace continuous problems with discrete ones, which makes it unavoidable to

introduce errors.

For discretization solution methods like finite difference or finite elements schemes, we

measure the accuracy of the method in powers of some discretization step size h. A

method with error O(hp) is said to be of order p, where a method with p > 2 usually

is referred to as a higher order method. In Paper I, Paper II, and Paper IV, we use

the notation h = x j− x j−1, k = tn− tn−1, unj = u(x j, tn), and some abbreviations for the

1

standard finite difference approximations of space derivatives:

D+u j =u j+1−u j

h, D−u j =

u j−u j−1h

D0u j =u j+1−u j−1

2h,

D1/2u j =u j+1 +u j

2, D−1/2u j =

u j +u j−12

.

For time derivatives, we use the expressions

Dt+un =

un+1−un

k, Dt

−un =un−un−1

k.

When we study mathematical models like kinetic equations, involving distribution func-

tions and multidimensional integral operators, conventional numerical discretization

methods give prohibitively large problems. Therefore, we apply particle simulation

methods based on a large, but finite, number of samples, representing “all” the particles.

For time-dependent equations, this is usually done by a splitting of the process in a pure

transport phase and an interaction phase, as in Paper III.

2 Higher order methods for wave propagation problems

2.1 Background

Wave propagation equations appear in many applications, such as acoustics [19], geo-

physics [13], and electromagnetics [10]. A general model for wave propagation is the

equation

1

a(x)∂ 2φ∂ t2

= ∇ · (b(x)∇φ)+H(x, t). (1)

By introduction of the new variables

ϕ(x, t) =∂φ∂ t

+H1(x, t), ψ(x, t) = b(x)∇φ +H2(x, t) ,

we can rewrite this as the equivalent first order system

ϕt = a(x)∇ ·ψ +F(x, t) ,ψt = b(x)∇ϕ +G(x, t) .

(2)

As an example of a wave phenomenon, we consider sound propagation in a fluid [6].

For a fluid with negligible viscosity and heat-conductivity, the flow is described by the

2

Euler equations of conservation of mass and momentum, an equation for the entropy

and an equation of state:

∂ρ∂ t

+∂

∂x j

(ρu j)

= 0 ,

∂ui

∂ t+u j

∂ui

∂x j+ρ−1

∂ p∂xi

= 0 ,

∂S∂ t

+u ·∇S = 0 ,

p = p(ρ,S)

relating the velocity field u(x, t), the density ρ(x), the pressure p(x, t), and the entropy

S(x, t). Then we obtain

∂ p∂ t

+u ·∇p = c2∂ρ∂ t

+ c2u ·∇ρ ,

with

c2(x) =∂ p∂ρ

∣∣∣∣S> 0 for all x.

For a one-dimensional flow, we get

pt +upx + c2ρux = 0 ,

ρ(ut +uux)+ px = 0 .

If we linearize these last two equations around the state where u = 0, ρ(x) = R(x), andp is constant, we obtain the system

pt + c2(x)R(x)ux = 0 ,

ut +R−1(x)px = 0 ,

or the scalar acoustic wave equation of second order

1

c2ptt =−Ruxt =−Rutx = R

(R−1px

)x .

If we assume that (1) has a time-periodic solution, so that we can write φ and F as

Fourier seriesφ(x, t) = ∑

ωuω(x)eiωt ,

F(x, t) = ∑ω

Fω(x)eiωt ,

3

the problem of solving the wave equation is reduced to the problem of solving the

Helmholtz equation

a(x)∇ · (b(x)∇uω(x))+ω2uω(x) =−Fω(x) .

For the acoustic wave equation

1

c2ptt = ρ

(1

ρpx

)x

we get the corresponding Helmholtz equation

ρ(1

ρu′ω(x)

)′+κ2uω(x) = 0 ,

with κ = ω/c(x), or in homogeneous media

u′′(x)+κ2u(x) = 0 . (3)

When computing non-decaying solutions to the wave equation in an unbounded do-

main, we need to impose absorbing boundary conditions of Engquist-Majda type [14]

to create a finite computational area. These well-posed boundary conditions minimize

the artificial reflections from the boundary. For (1) with a = b = 1 in x � 0, t � 0, the

first absorbing boundary condition of this family at the line x = L is

∂φ∂ t

+∂φ∂x

∣∣∣∣x=L

= 0.

In one dimension, this outflow condition at x = L is exact, giving φ = φ(x− t). Un-

fortunately, we are only aware of stable discretizations of second order accuracy, but

not of higher order. If, instead, we work with the Helmholtz equation (3), this is not a

problem. To suppress all reflections at x = L and allow only solutions like u = σeiωx,

yielding φ = σeiω(x−t), we use

u′(L)− iκu(L) = 0 .

We are interested in accurate and efficient finite difference solution methods for the

time-dependent wave equation (1) or (2), and the time-independent Helmholtz equa-

tion (3), with the extended goal to give additional insight in how to treat other real-

world wave propagation problems. Numerical discretization methods for the wave equa-

tion have been studied extensively for decades, but real-world applications, where the

wavelength is very small compared to the computational area, give unacceptably large

algebraic problems for low order methods if the wave oscillations are to be properly

4

resolved. This means that we prefer higher order methods [21, 39], but we still need to

pay attention to computer memory issues so that we limit the number of values we store

during the computations. If we need to solve systems of linear equations, the coefficient

matrices should have a narrow bandwidth. In other words, we are looking for methods

with a compact computational stencil.

The standard approach to construction of a high order finite difference scheme is to start

with a lower order approximation of a derivative, look at the leading error term in the

approximation and eliminate this term by subtracting a finite difference approximation.

This process can easily be automated, a construction algorithm is given in [16], to give

finite difference schemes of arbitrary order. But for increased order of accuracy, the

stencils gets wider and wider. The first two papers in this thesis give alternate principles

suitable for wave propagation problems, resulting in more compact schemes.

The first idea, to cancel error terms by repeated use of the equation, was originally used

by Numerov in the 1920s for ODEs, see the original papers [26, 27] and the classical

textbook [12] as well as more recent work [5, 11, 28, 29, 37, 43]. The second idea is due

to Fox and has been frequently used by Pereyra for many years, see e.g. [17, 33]. The

approach here is to use a lower order, more cheaply computed, solution for correction of

error terms. This method is referred to as the deferred correction method, and examples

of work in this area published after Paper I include [22, 23]. By application of the

Numerov idea for the wave equation written as a first order system, we can construct

a compact scheme with high accuracy in both space and time. After the publication of

Paper II, this approach has been used in several papers including [41, 42, 44].

2.2 Summary of Paper I

Higher order finite difference methods for the Helmholtz equation

For time-period solutions to the wave equation, we need only to solve the time-indepen-

dent Helmholtz equation. In Paper I, the one-dimensional model problem

u′′(x)+κ2u(x) = 0, 0 < x < 1,

u′(0)+ iκu(0) = 2iκ,

u′(1)− iκu(1) = 0.

(4)

is studied, and high order finite difference schemes are constructed with different tech-

niques and compared. For a given wave number, and a given resolution tolerance, we

can prescribe the number of discretization points required for each scheme to resolve

the wave properly. Below, the order schemes are presented as examples, but Paper I

5

describes the principles for obtaining the corresponding schemes of arbitrary even order

of accuracy.

A standard construction of a sixth order scheme for Eq. (4) gives the scheme

[(I− 1

12h2D+D−+

1

90h4(D+D−)2

)h2D+D−+h2κ2

]u j = 0,

involving six neighbors u j±1,2,3 of u j for every inner point x j, j = 1, . . . ,n, and intro-

ducing four extra ghost points u−2,−1,n+2,n+3 at the boundaries. This means that in

the one-dimensional case, a hepta-diagonal linear system must be solved, compared to

a tri-diagonal system for a second order method. This is more or less a question of

computational power, but more problematic is the fact that the wide stencil creates dif-

ficulties if we want to partition the computational domain for a domain decomposition

method.

An analysis of the error from the computation shows that the error in the computation

is dominated by the phase shift (and not by the amplitude error), so the number of

necessary grid points is in principle determined by the phase error. For wave number κand tolerance ε , this number is n � 0.3κ7/6ε−1/6.

By repeated usage of the differential equation, it is possible to construct an elegant and

cost-efficient sixth order compact scheme of Numerov type

[(I +

1

12h2κ2 +

1

240h4κ4

)h2D+D−+h2κ2

]u j = 0,

giving only a tri-diagonal system to solve for n � 0.2κ7/6ε−1/6 grid points. Also in thiscase, the computational error is dominated by the phase shift. The total error from a

computation is shown in Figure 1.

For the model problem (4), this is an excellent approach, but the usefulness for real-

world applications is somewhat limited. The construction is complicated by mixed

derivatives in the error terms, so the idea does not painlessly lend itself to multi-dimen-

sional computations. A two-dimensional fourth order scheme of this type, is constructed

and used in [28].

If our primary concern is to avoid systems of large bandwidth, we may apply the method

of deferred correction, where we repeatedly use lower order derivative approximations

to form a sequence of algebraic systems to be solved. By this, the solution is iteratively

improved from, say, second order accurate to fourth order accurate, to sixth order accu-

rate and so on. The resulting sixth order scheme obtained in this fashion for the interior

6

Figure 1: The errors obtained using the sixth order Numerov scheme.

points u j, j = 1, . . . ,n, in our problem reads

[h2D+D−+h2κ2

]u<2>

j = 0,[h2D+D−+h2κ2

]u<4>

j =1

12h4(D+D−)2u<2>

j ,

[h2D+D−+h2κ2

]u<6>

j =[1

12h4(D+D−)2− 1

90h6(D+D−)3

]u<4>

j .

Note that the coefficient matrix of the linear systems remain the same throughout the

sequence, enabling us to take advantage of an existing factorization in the solution pro-

cess. The right hand side expressions contains wide stencils, but applied to known

values, and the additional ghost points values needed can be found by extrapolation.

Unfortunately, the errors will be larger with this approach than with the Numerov-type

scheme, as seen in Figure 2. Both a phase shift and an amplitude error will contribute

and the necessary number of grid points will increase to n � 0.2κ9/6ε−1/6. The advan-tage of this method is its generality; it can be applied to other equations, as in Paper II,

and it can easily by extended to multi-dimensional problems, as in [18].

7

Figure 2: The errors obtained using the sixth order deferred correction scheme.

2.3 Summary of Paper II

Time compact high order difference methods for wave propagation

For the time-dependent wave equation, we write the equation as a first order system[pu

]t=[

0 a(x)b(x) 0

][pu

]x+[

FG

], t � 0, (5)

with positive a and b, or, for stability analysis purposes,

pt =√

a(√

bu)x,

ut =√

b(√

ap)x,(6)

using the homogeneous case and the scaled variables p = p/√

a, u = u/√

b. The con-tinuous problem fulfills the conservation property

ddt

(||p(t)||2 + ||u(t)||2) = 0,

8

so we prefer numerical schemes with similar qualities.

For the finite difference approximation, we use a staggered Yee-type grid, illustrated in

Figure 3, and

pn+1/2j+1/2 = p(( j +1/2)h,(n+1/2)k),

unj = u( jh,nk).

x

t

x j x j+1

tn

tn+1

unj

pn+1/2j+1/2

Figure 3: The unknowns are staggered in both space and time.

The standard second order difference scheme for (5) has truncation errors k2/24pttt +ah2/24uxxx +O(k4)+O(h4) for the p-equation, and k2/24uttt +bh2/24pxxx +O(k4)+O(h4) for the u-equation. From the equations themselves, we find that

pttt = a(b(aux)x)x +a(bFx)x +aGxt +Ftt ,

uttt = b(a(bpx)x)x +b(aGx)x +bFxt +Gtt ,

and by subtraction of finite difference approximations for these terms in the scheme, we

get a scheme that is fourth-order accurate in both space and time:

pn+1/2j+1/2 = pn−1/2

j+1/2 + ka j+1/2D+unj

+k24

a j+1/2(k2D+b jD−a j+1/2D+−h2D2

+D−)unj

+ kFnj+1/2 +

k3

24

((a(bFx)x)n

j+1/2 +(aGxt)nj+1/2 +(Ftt)n

j+1/2

),

un+1j = un

j + kb jD−pn+1/2j+1/2

+k24

b j(k2D−a j+1/2D+b jD−−h2D+D2−)pn+1/2

j+1/2

+ kGn+1/2j +

k3

24

((b(aGx)x)

n+1/2j +(bFxt)

n+1/2j +(Gtt)

n+1/2j

).

9

As seen in Figure 4, this scheme is more compact in time than a traditional fourth order

scheme. This explicit scheme is stable for

x

t

x j x j+1

tn

tn+1

unj

pn+1/2j+1/2

Figure 4: Compact fourth order scheme.

kh√

c1c2 <

√12

13+β,

with

c1 = maxj

1

2

(√a j+1/2b j +

√a j+1/2b j+1

),

c2 = maxj

1

2

(√a j+1/2b j +

√a j−1/2b j

),

β = maxj

(max

{√a j+3/2/a j+1/2 ,

√a j−1/2/a j+1/2

}).

Using the discrete norm

‖ f n‖2h = ∑∣∣ f n

j∣∣2 h

we can show that

∥∥∥pn+1/2∥∥∥2

h+∥∥un+1

∥∥2h � K(

∥∥∥p−1/2∥∥∥2

h+∥∥u0∥∥2h),

where K is independent of t, so the solution stays bounded also for very long time

integration. As long as the coefficients a(x) and b(x) are bounded, this is true also if

they are discontinuous, like when an acoustic wave travels from water to sediment.

10

3 Methods related to the Boltzmann equation and theLandau equation

3.1 Background

A central mathematical object in kinetic theory of rarefied gases is the classical Boltz-

mann equation (∂∂ t

+v ·∇x

)f (x,v, t) = Q( f , f ) (7)

for the one-particle distribution function f (x,v, t), proportional to the probability that a

particle exists in position x ∈ R3 with velocity v ∈ R3 at time t ∈ R+. The right hand

side collision integral reads

Q( f , f ) =∫R3×S2

g(|u|,μ)[

f (x,v′, t) f (x,w′, t)− f (x,v, t) f (x,w, t)]

dwdωωω, (8)

where u = v−w, v′ = (v+w+ |u|ωωω)/2, v′ = (v+w−|u|ωωω)/2. Here, g(|u|,μ) =|u|σ(|u|,θ), where σ(|u|,θ) is the differential cross-section at the scattering angle θ =arccosμ ∈ [0, π], with μ = u ·ωωω/|u|.There is a huge amount of literature about the Boltzmann equation, see for example [2,8,

38] and references therein. Here, we just try to outline a background for some particular

problems treated in Paper III and Paper IV.

In the general case of gas flow in three space dimensions, the function f (x,v, t) dependson seven variables, and the collision operator is a five-fold integral. It is therefore not

surprising that the absolute majority of numerical methods for Eqs. (7)–(8) are based

on stochastic simulation techniques, like the Direct Simulation Monte-Carlo (DSMC)

method [1, 2], though there are interesting results based on a deterministic approach

(see [30] for an overview). Regardless of whether the methods are of deterministic or

stochastic nature, they are usually based on a splitting principle. For each time step with

step-size k, two successive problems are solved. First, consider the transport step

ft +v ·∇x f = 0, 0 � t � k,

f∣∣t=0

= f (x,v,0),

and then the collision step

ft = Q( f , f ), 0 � t � k,

f |t=0 = f (x,v,k).

11

The first step is relatively simple, in fact trivial for DSMC methods, and therefore the

problem is in principle reduced to solution of the spatially homogeneous Boltzmann

equation

∂ f∂ t

=∫R3×S2

g(|u|,μ)[

f (v′, t) f (w′, t)− f (v, t) f (w, t)]

dwdωωω. (9)

This is why solution methods for the simplified equation (9) are important also for the

general, spatially inhomogeneous, case.

A special asymptotic case of the Boltzmann equation is the Landau-Fokker-Planck

equation [24]. Its spatially homogeneous version can be used as a model for Coulomb

collisions. A detailed formal derivation of how this equation can be obtained as a graz-

ing collision limit of the Boltzmann equation is given in Paper III. Of course, this is just

a modification of the original Landau derivation [24].

A recent review of some deterministic numerical methods for solving this equation can

be found in [34], including completely conservative finite difference schemes, begin-

ning in the 1970s with [3, 35, 46]. Another important contribution is the spectral meth-

ods based on Fourier series, see [15, 31]. Two original Monte-Carlo schemes were

proposed by Takizuka and Abe [40] and by Nanbu [25]. A detailed comparison of these

methods can be found in the recent paper [45]. Both of these simulation algorithms

were proposed without relation to the Landau-Fokker-Planck equation, but as a way

to treat Coulomb collisions. A formal theoretical framework of collision simulations

for long-range forces was proposed in [4], and it was shown that the original Nanbu

scheme [25] is just one of infinitely many realizations of a very general approach. In

particular, there exists a surprisingly simple (and therefore easily implemented and fast)

simulation scheme. The presentation of this scheme is the main topic of Paper III. It is

also demonstrated that it is possible to derive this method directly from the Boltzmann

equation without using the general approach of [4].

In Paper IV, we study the linearized Boltzmann collision operator related to the classical

problem of transition from a kinetic models to hydrodynamics, in other words the limit

where we go from the microscopic particle description to a macroscopic fluid descrip-

tion, is related to asymptotics of (7) in the scaled form

ft +v ·∇x f =1

εQ( f , f ), ε → 0+,

where ε is the so-called Knudsen number. We are interested in the most basic but still

physically realistic case of interacting particles, the model of hard spheres. If we assume

that

f = M(v)(1+ εϕ) , M(v) = ρ (2πT )−3/2 exp

[−|v−v0|2

2T

],

12

where ρ(x, t) denotes the density, T (x, t) the absolute temperature, and v0(x, t) the bulkvelocity of the gas, the formal limit at ε = 0 yields the following linearized equation for

the unknown function ϕ(x,v, t)

Lϕ = M−1 [Q(Mϕ,M)+Q(M,Mϕ)] =(

∂∂ t

+v ·∇x

)logM (10)

under the assumption that ρ , T , and v0 are known. This linearized operator L was

studied by many authors, including great names as Boltzmann himself, Hilbert, and

Grad. Many historical references can be found in the books [7, 8] by Cercignani. In

paper IV, we study some open questions related to L. We also study similar problems

for the linear collision operator L(1), given by

L(1)ϕ = M−1Q(Mϕ,M). (11)

3.2 Summary of Paper III

A DSMC method for the Landau-Fokker-Planck equation

In order to explain how our Monte-Carlo simulation method can be derived, we write

the spatially homogeneous Boltzmann equation (9) as

∂ f∂ t

=∫R3×S2

g(|u|,μ) [F(U, |u|ωωω)−F(U,u)] dwdωωω, (12)

with F(U,u) = f (v) f (w) = f (U + u/2) f (U−u/2), and consider the case of grazing

collisions.

We assume that g = g(|u|,μ;ε) ≡ 0 if − 1 � μ < 1− εa for a small ε and a bounded

positive function a = a(|u|), meaning that scattering occurs only for small angles 0 �θ � arccos(1− ε supa). By an expansion of the integrand in the Boltzmann collision

integral for small angles, we formally obtain, in the limit ε = 0, the Landau-Fokker-

Planck equation

∂ f∂ t

=1

2

∫R3

g1(|u|)[−2u j

∂∂u j

+(|u|2δi j−uiu j

) ∂ 2

∂ui∂u j

]F(U,u)dw =

=1

8

∂∂vi

∫R3

g1(|u|)(|u|2δi j−uiu j

)( ∂∂v j

− ∂∂w j

)f (v) f (w)dw,

with g1(|u|) = limε→0 2π∫ 1−1 g(|u|,μ;ε)(1−μ)dμ .

13

To find a simple simulation scheme for this equation, we choose

g = gε(|u|,μ) =1

4πεδ (1−2εbε(|u|)−μ) , −1 � μ � 1,

with

0 � bε(|u|) � 1

εand lim

ε→0bε(|u|) = g1(|u|)

and study the Kac N-particle system related to (12). Then the Kac Master equation [20]

reads

∂FN(V, t)∂ t

=1

N ∑1�i< j�N+1

∫S2

gε

(|vi−v j|, (vi−v j) ·ωωω

|vi−v j|)[

FN(V ′i j, t)−FN(V, t)]

dωωω

where V = {v1, . . . ,vi, . . . ,v j, . . . ,vN} is the pre-collision state and the collision of par-

ticle i and particle j results in the post-collision state V ′i j = {v1, . . . ,v′i, . . . ,v′j, . . . ,vN}.With first order time discretization and time-step k = 4ε/N, we get

FN(t + k) =2

N(N−1) ∑1�i< j�N

(gtotε )−1∫

S2gε

(|vi−v j|, (vi−v j) ·ωωω

|vi−v j| |)

F(V ′i j, t)dωωω.

By the choice bε(|u|) = min{g, 1/ε}, we get a simple realization where, in each time

step, we need to generate only one random number to form the unit vector ωωω = (θ ,ϕ).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.5

2

2.5

3

3.5

4

v4,6,

8

time

LFP equation (2,4,6) and DSMC results (1,3,5); N=250, M=20, ε=0.05

1

3

2

4

56

Figure 5: Verification for Maxwell particles and for Coulomb particles

The algorithm is first verified for Maxwell particles, where we have exact analytical

expressions for the moments of f (v, t) to compare with. The left hand side plot in

Figure 5 shows one such comparison between

Mn(t) =1

N

N

∑i=1

|vi|2n and mn(t) =∫R3|v|2n f (|v|, t)dv

14

for n = 2,3,4. Secondly, the algorithm is compared with numerical results for the Lan-

dau equation from a completely conservative finite-difference scheme from [36] in an

isotropic case, considered here to be an exact reference solutions. The resulting curves

are shown in the right hand side plot of Figure 5. The method is also tested for an

anisotropic case.

3.3 Summary of Paper IV

On some properties of linear and linearized Boltzmann collision operators for hardspheres

We study the linearized and the linear collision operators (10) and (11) in the case of

hard spheres by looking at the integral equations

Lϕ(v) = g(v) and L(1)ϕ(v) = g(v)

for given right hand side function g. By expansion of ϕ and g in spherical harmonic

function, these equations are transformed to the corresponding sets of independent

scalar integral equations

Llϕlm(v) = glm(v) and L(1)l ϕlm(v) = glm(v), (13)

and the operators are given in explicit form for all l = 0,1, . . . .

One of the key results of the paper is that it is shown that the integral equations (13)

can be reduced to ordinary differential equations. For all l = 0,1, . . . , reductions can be

performed to obtain sets of 2([l/2]+1) first-order equations.

We use this approach to study the eigenvalue problems

L(1)ϕ =−√

π2

μϕ and Lϕ =−√

π2

μϕ

for isotropic solutions ϕ(v) = ϕ(|v|). With appropriate variable substitutions for the

eigenfunctions we get, in the first case, the equation of Schrödinger type

u′′(x)− [1+U(x;μ)]u(x) = 0,

u(0) = 0, u(x)−→x→∞

0;

U(x;μ) = x2− 2μθ(x)−μ

,

θ(x) = e−x2 +(2x2 +1

)J(x),

J(x) =∫ 1

0dt e−t2x2,

(14)

15

and in the second case the more complicated equation

u′′(x)− [1+V (x;μ)]u(x)+ Pu = 0,

u(0) = 0, u(x)−→x→∞

0;

V (x;μ) = x2−2μ +(xθ(x))′

θ(x)−μ,

(Pu)(x) = 8

xJ(x)θ(x)−μ

ex2/2∫ ∞

xdyu(y)e−y2/2.

(15)

Figure 6: Computation of eigenvalues of the collision operators L(1) and L. The eigen-values are the interceptions of the curves with the line −1.

We study these equations numerically and present some formal arguments, based on

quasi-classical approximations in quantum mechanics, showing that these equations (or

at least Eq. (14)) probably have infinite numbers of discrete eigenvalues μn, n = 0,1, . . . .For Eq. (14), most of them are very close to μ∞ = 2:

2−μ(1)n ∼ exp

(−√

2

3nπ

), n→ ∞.

The four smallest ones are

μ(1)0 = 1.638, μ(1)

1 = 1.959, μ(1)2 = 1.997, 1.999 < μ(1)

3 < 2.000.

For problem (15), we perform a similar study, and find the eigenvalues

μ1 = 1.342, μ2 = 1.823, μ3 = 1.964, μ4 = 1.994.

The numerical results are depicted in Figure 6, where En(μ), n = 0,1, . . . are “energylevels” for Eqs. (14) and (15)

16

A rigorous proof of the existence of discrete eigenvalues for both operators, gives the

following estimates for the smallest non-zero eigenvalues:

μ(1) � 1.8856 and μ � 1.5085.

The proof is based on the variational principle. Isotropic matrix elements of L and L(1)

are computed in explicit form.

The technique of reduction of the integral equations for L to ordinary differential equa-

tions is useful also when looking at the hydrodynamic limit using the Chapman-Enskog

procedure [9]. We are interested in the equation

L1ϕ(|v|) =

√2

π|v|(|v|2−5

),

and we show that it can be reduced to the boundary value problem for the second-order

equation

ddz

r(z)ddz

z3/2v(z)− e−z

p(z)v(z) =−5 e−z

p(z),

v(z) is bounded at z = 0, v(z)−→z→∞

0,(16)

where

r(z) = 2e−z

Λ(z), p(z) = e−z +

(z+

1

2

)Λ(z)√

z, Λ(z) =

∫ z

0dt

e−t√

t.

We solve Eq. (16) numerically, and the solution is used to compute

K =5

2

∫ ∞

0dz p−1(z)(v(z)−5)z3/2e−z =−4.5173,

which is proportional to the heat conduction coefficient in the Navier-Stokes equa-

tions. If we use the classical approximation of one polynomial in the Chapman-Enskog

method, we find the corresponding value

K1 = a∫ ∞

0dze−zz5/2 (2z−5) =−15

4

√πa =−225

64

√π2

=−4.4062,

so we can conclude that the heat conduction coefficient in the Navier-Stokes equations

is

λ (T ) = Cλ1(T ), C =∣∣∣∣ KK1

∣∣∣∣≈ 1.0252,

where λ1(T ) is the standard approximation. The value of C was known from [32], but

here it is obtained in an easier way.

17

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21

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