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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
Karlstad University StudiesISSN 1403-8099
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