120024765_TT32_03-04_R1_071703
TRANSPORT THEORY AND STATISTICAL PHYSICS
Vol. 32, Nos. 3 & 4, pp. 293–319, 2003
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A Finite Element Approximation
of Grazing Collisions
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B. Lucquin-Desreux* and S. Mancini
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Laboratoire Jacques-Louis Lions,
Université Pierre et Marie Curie, Paris, France
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ABSTRACT
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In this article, we propose a finite element discretization of
the Boltzmann-Lorentz operator for which it is possible to define a
grazing collision limit. We illustrate this discretization by considering
the evolution of a system of particles in a slab, subject to collisions
both with the boundaries and with themselves. A comparison is made
with isotropic collisions or with the Laplace-Beltrami operator.
Moreover, in the case of multiplying boundary conditions, it is
proven by numerical simulations the existence of a critical value
for the absorption coefficient, which is independent on the grazing
collision parameter. Finally, the focalization of a beam is studied and
the numerical results are compared with previous simulations.
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AMS Classification:
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35B40; 65N30; 82C40.
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*Correspondence: B. Lucquin-Desreux, Laboratoire Jacques-Louis Lions,
Université Pierre et Marie Curie, Boite Courrier 187, 75252 Paris, Cedex 05,
France; E-mail: lucquin@ann.jussieu.fr.
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DOI: 10.1081/TT-120024765
Copyright & 2003 by Marcel Dekker, Inc.
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1. INTRODUCTION
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Kinetic equations, as Boltzmann (or Fokker-Planck) equations,
model the evolution of a system of particles subject to the action of a force
field and to collisions. They are applied for example in semiconductor
physics (see Markowich et al. (1994)), in the modeling of rarefied gases
or plasmas (see Cercignani (1988)), in astrophysics (see Belleni-Morante
and Moro (1996) for problems relative to interstellar clouds) or in biomathematics (see Arlotti et al. (2000) for the study of dynamics of populations). In the more general frame, the operator describing the scattering
event is nonlinear and due to its complexity, many other models have been
derived. Notably, the Lorentz operators, derived from the Boltzmann one
or from the Fokker-Planck one, model the elastic collisions of particles
against heaviest target ones (see Degond and Lucquin-Desreux (1996) and
Lucquin-Desreux (2000) for the disparate masses particles asymptotics).
In plasma physics for example, the Boltzmann-Lorentz operator represents (in first approximation with respect to the mass ratio) the effect
of collisions due to neutral particles on the electron distribution function.
On the other hand, the Fokker-Planck-Lorentz operator is the leading
order term (still with respect to the mass ratio) describing the collisions
of electrons against ions.
The Fokker-Planck operator is usually considered as an approximation of the Boltzmann collision operator when collisions become
grazing, i.e., when the angle of deviation of a particle during a collision
is very small. This essentially occurs for potentials of long range interaction, such as the Coulombian potential in the case of a plasma.
This grazing collision limit of the Boltzmann operator towards the
Fokker-Planck one has been proven in Degond and Lucquin-Desreux
(1992) and Desvillettes (1992). In Desvillettes (1992), the scattering cross
section is smooth; it depends on a small parameter " and it is localized
around the value ¼ 0 of the scattering angle when " goes to zero. On the
other hand, the Coulomb case is studied in Degond (1992) by means of a
quite different asymptotics: the scattering cross section has a nonintegrable
singularity when the relative velocity of the colliding particles tends to
zero but also when the scattering angle tends to zero. There is thus a
necessary truncation of this angle around the value zero ( ") and the
small parameter " has a precise physical meaning: it is clearly identified as
the plasma parameter. We also refer the reader to the works Goudon
(1997a, 1997b) concerning the grazing collision limit of the Boltzmann
operator when the scattering cross section corresponds to a inverse
power force (i.e., 1=rs , with s > 1) and without angular cut-off. More
recently, the convergence of Boltzmann-Lorentz operator towards the
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Fokker-Planck-Lorentz (or Laplace-Beltrami) operator in the so-called
grazing collision limit has been proven in Buet et al. (2001), both for
the asymptotics developed in Degond and Lucquin-Desreux (1992) and
Desvillettes (1992).
In this article, we are concerned with the numerical approximation of
the Boltzmann-Lorentz operator and with its grazing collision limit. Up
to our knowledge, only few recent articless have been devoted to the
study of the grazing collision limit at the discrete level: we can refer
for example to Guérin and Méléard (2002) and Pareschi et al. (2002)
concerning the full nonlinear Boltzmann operator discretized either by
spectral methods (see Pareschi et al. (2002)) or by particle methods (see
Guérin and Méléard (2002)). In our article, the considered scattering
cross section is a simplified version of the one defined in Buet et al.
(2001): it is assumed to be constant on a very small interval depending
on the small parameter " and zero outside this interval. Our goal is to give
a numerical approximation of the Boltzmann-Lorentz operator for grazing collisions for which it is possible to pass to the limit " ! 0. We first
decided to apply a finite element method (FEM) because this method is
particularly well adapted to the limit problem (we refer for example to
Cordier et al. (2000) for 3D computations). Secondly, in the FEM
described below, we decided to do exact computations of the collision
terms (instead of doing quadrature formulas): denoting by h the velocity
step, we can then choose " h and thus study the limit " ! 0 for a fixed
discretization in velocity variable, i.e., for a fixed h (which would not be
possible when using quadrature formulas).
In order to show the efficiency of the proposed FEM approximation,
we will consider a transport equation in which the collision operator is the
Boltzmann-Lorentz operator for grazing collisions defined on a bounded
region. Thus, we equip the kinetic equation with some kind of boundary
conditions: in a first approach, we only take into account specular reflection boundary conditions. The numerical results obtained with the proposed FEM approximation are compared with those obtained considering
the Laplace-Beltrami operator (i.e., its limit when " goes to zero), with the
isotropic collisions (i.e., the scattering cross section is constant), and also
with the collisionless case. Successively, we consider multiplying reflection
boundary conditions (see for instance Belleni-Morante and Totaro (1996)
and Mancini and Totaro (1998)): each particle colliding on the boundary
of the considered region is reflected and multiplied by a factor bigger than
one. In this optic, as the region where the evolution of particles takes place
is bounded, it seems physically necessary to consider also an absorption
term in our transport equation. The existence and uniqueness of the
solution of such a problem, with the absorption cross section bigger
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than a theoretical value t , has been proven in Belleni-Morante (1996) for
multiplying Maxwell boundary conditions, and in Mancini and Totaro
(1998) in the general case. We will numerically prove the existence of a
critical absorption cross section n < t for which the total number of
particles does not sharply increase nor vanishes with time. Moreover, we
also remark that both n and the equilibrium profile of the particle density
do not depend on the initial data.
The article is organized as follows. In Sec. 2, we present the finite
element approximation for the Boltzmann-Lorentz operator for grazing
collisions and also for isotropic collisions. In Sec. 3, we present the test
problems used to validate our numerical approximation. Section 4 concerns the results of the numerical simulations. Finally, in Sec. 5, we give
some concluding remarks.
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2. NUMERICAL SCHEMES
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The goal of this article is the numerical approximation of the
Boltzmann-Lorentz operator for grazing collisions. We recall that the
Boltzmann-Lorentz operator models for example the elastic collisions of
a heavy particle against a light one (see Degond and Lucquin-Desreux,
(1996) and Lucquin-Desreux (2000)). Thus, it acts only on the velocity
direction of the particles, leaving unchanged the velocity modulus jvj
(which we will consider normalized to one). In the bi-dimensional case,
the Boltzmann-Lorentz operator is given by:
Z
Qð f ÞðÞ ¼
Bð 0 Þ ð f ð0 Þ f ðÞÞ d0 ,
ð1Þ
S1
where Bð 0 Þ is the scattering cross section which represents the
probability for a particle entering a collision with velocity direction 0
to out-go the scattering event with a velocity direction . For instance,
Bð 0 Þ may be determined from the interaction force between the
particles (see Degond and Lucquin-Desreux (1992), Desvillettes (1992),
and Goudon (1997b)); in particular, we recall that, in the case of isotropic
collisions, it does not depend on nor on 0 .
We are interested in grazing collisions, i.e., we want to take into
account also the very small deviations in the trajectories of the particles.
This fact is emphasized by introducing the dependence of the scattering
cross section B on a small parameter ". Recently, Buet, Cordier, and
Lucquin-Desreux (see Buet et al. (2001)) have proven the convergence,
when the small parameter " tends to zero, of the Boltzmann-Lorentz
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operator for grazing collisions towards the Fokker-Planck-Lorentz
operator, which is simply the Laplace-Beltrami operator.
Our goal here is to give a numerical approximation of the
Boltzmann-Lorentz operator for which it is possible to pass to the limit
" ! 0 for a fixed velocity discretization. As shown in the sequel, this is
possible when applying for example a finite element approach, where the
integral terms are all exactly computed. The numerical results we obtain
are then compared with the collisionless case, the case of isotropic
collisions, and those obtained when considering the Laplace-Beltrami
operator.
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2.1. An Implicit Finite Element Approximation
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We want to approximate by means of a finite element method the
following Cauchy problem:
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< @f
¼ Qð f Þ,
ð2Þ
: @t
f ðt ¼ 0Þ ¼ f 0 ,
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where Qð f Þ is defined by Eq. (1). We note that the dependence of the
density f on the space variable is not underlined in this paragraph
because the position of the particles does not play any role during a
collision. The angle variable belongs to the unit circle S 1 , i.e.,
2 ½, and the distribution function f satisfies periodic boundary
conditions.
We first consider the following piecewise affine approximation f N of
the function f :
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f ðt, Þ ’ f N ðt, Þ ¼
’j ðÞ fj ðtÞ,
j¼0
where ’j , j 2 f0, . . . , N 1g, are the classical ‘‘hat functions’’ of the P1
finite element approximation. More precisely, denoting by j ¼ þ hj,
with h ¼ 2=N , we have ’i ðj Þ ¼ ij (=1 if i=j and 0 otherwise), so that
fj ðtÞ ¼ f N ðt, j Þ.
The weak discretized form of Eq. (2) then reads:
Z
N
1
X
@fj
’ ðÞ’i ðÞ d
@t S1 j
j¼0
Z Z
N
1
X
fj
Bð 0 Þ½’j ð0 Þ’i ðÞ ’j ðÞ’i ðÞ d0 d:
¼
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Setting:
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Z
>
>
’i ðÞ’j ðÞ d,
< Aij ¼
ZS1 Z
>
>
Bð 0 Þ ½’i ðÞ ’j ð0 Þ ’i ðÞ ’j ðÞ d d0 ,
: Qij ¼
S1
ð3Þ
S1
we write the discretized problem more compactly as follows:
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N
1
X
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j¼0
N
1
X
@fj
Aij ¼
fj Qij :
@t
j¼0
ð4Þ
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The matrix A with entries Aij is the so called ‘‘mass matrix,’’ while the
matrix Q with entries Qij represents the collision matrix. In the next
paragraph, we give the expression of these two matrices in the context
of grazing collisions; the detailed computations are performed in
the Appendix A1. Before this, let us precise the time discretization of
Eq. (4). In order to avoid a limitation of the time step t in terms of the
angular step ¼ h, we will use a fully implicit scheme. We introduce
some notations. Let us denote by F n the N dimensional vector with
entries Fjn ¼ f ðnt, j Þ, for j 2 f0, . . . , N 1g. The implicit finite element
scheme we consider reads:
( nþ1
F
¼ ðA tQÞ1 AF n ,
n0
ð5Þ
F 0 ¼ ðFi0 Þ0 j N 1 ,
with Fj0 ¼ f 0 ðj Þ.
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2.2. Grazing Collisions
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In the case of grazing collisions, the scattering cross section B is
localized and depends on a small parameter ". In the nonCoulombian
case, B has the following expression (see Buet et al. (2001) and
Desvillettes (1992)):
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B" ðzÞ ¼ 3 Bðz="Þ S1 ðz="Þ
"
where denotes the characteristic function (i.e., S1 ðzÞ ¼ 1 if z 2 S 1 and
0 otherwise). We will here consider a simplified model for which the
scattering cross section is given by:
n
B0 if jzj < ",
B" ðzÞ ¼
0 else
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with B0 ¼ 1="3 . We then have:
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Proposition 2.1. Setting ¼ "=h and assuming that 1, the grazing
collision matrix Q is given by:
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0
1
1
1 12
0 ... 0
1 12
2 þ 34
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8
C
B
..
.. .. .. ..
..
1
C
B 11
.
.
.
.
.
.
2
8
C
B
C
B
..
.. .. .. ..
..
C
B 1
.
0
.
.
.
.
.
C
B 8
C
3B
.
.
.
.
.
.
.
C
B
.
.
.
.
.
.
.
.
.
.
.
.
.
0
.
C:
B
Q¼ B
C
.
.
.
.
.
.
.
3h B
..
.. .. .. ..
..
..
C
0
C
B
..
.. .. .. ..
..
C
B
1
C
B
.
.
.
.
.
0
.
8
C
B
C
B
.
.
.
.
.
.
..
.. .. .. ..
..
@ 1
1
12 A
8
1
1 12
0 . . . 0 18
1 12
2 þ 34
8
269
ð6Þ
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Proof. The proof is detailed in the Appendix A1.
By means of the same arguments applied in order to compute the
matrix Q (see Appendix A1), we have that the mass matrix A reads:
1
0
2 12
0 . . . 0 12
C
B
B 1 ... ... ... ... 0C
C
B2
B
.. C
.. .. .. ..
C
hB
. . C:
.
.
0
.
A¼ B
ð7Þ
C
B
.
.
.
.
.
3B .
.. .. .. .. 0C
.
C
B
.
.
.
.
C
B
@ 0 .. .. .. .. 1 A
2
1
1
0 ... 0
2
2
2
We remark that if we let " ! 0 then Qij ! ðh3 Þ=ð3Dij Þ, where Dij
is the classical finite difference discretization of the Laplace-Beltrami
operator f ¼ @ f :
1
0
2 1
0 ... 0
1
.
.
.
.
.. .. .. .. 0 C
B 1
C
B
B
.. C
.. .. .. ..
C
1B
.
.
.
.
0
.
C:
D¼ 2B
.. . . . . . . . .
C
B
h B .
. 0 C
.
.
.
C
B
.
.
.
.
@ 0
.. .. .. .. 1 A
1
0 ... 0
1 2
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Moreover, the coefficient 3 =3 is strictly related to the second order
moment of the scattering cross section, which in our case is equal to 3 =3
(since B ¼ 1).
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2.3. Isotropic Collisions
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In the following simulations, in order to compare the evolution of
the system of particles with grazing collisions and with isotropic
collisions, we will not compute the exact solution for the isotropic
Boltzmann-Lorentz operator (although it is possible), but we will approximate the isotropic operator following the same discretization as before.
Recalling that for isotropic collisions the scattering cross section B is
a constant (which we will assume equal to one), we get the following finite
element discretization of the isotropic Lorentz operator. The gain matrix
G is given by: G ¼ ðGij Þ0i, jN 1 , with Gij ¼ h2 , whereas, the loss matrix
L is such that: L ¼ 2A. Then we can apply the implicit finite element
scheme (5), where now the matrix Q ¼ G L represents the P1 finite
element approximation of the isotropic Lorentz operator.
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3. THE TEST PROBLEM
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We illustrate the finite element approximation for the
Boltzmann-Lorentz operator with grazing collisions on the following
transport equation:
1
@t f þ cos @x f þ f ¼ Qð f Þ:
ð8Þ
This transport type equation describes the evolution of a system of
particles moving between two parallel plates (slab), subject to absorption
by the host medium and to elastic collisions with other particles present in
the slab. We recall that f ¼ f ðx, , tÞ represents the number of particles
which at time t 0, are in the position x 2 ½L, þ L with velocity
v ¼ ðcos , sin Þ, 2 ½, . Moreover, Qð f Þ is the Lorentz collision
operator defined by Eq. (1), 0 is the absorption cross section, and
is the relaxation time; this one is assumed to be equal to one, except in
the last numerical test concerning the defocusing of a beam (see Sec. 4.4
below and Cordier et al. (2000)).
The interactions of the particles with the two plates of the slab may
be described by means of a linear, bounded, and positive operator
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: out ! in which relates the incoming and outgoing flux of particles
as follows:
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f in ¼ f out :
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in ¼ fðL, Þ, cos > 0g fðL, Þ, cos < 0g,
out ¼ fðL, Þ, cos > 0g fðL, Þ, cos < 0g:
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We define the norms on the incoming and outgoing sets as usual:
Z
Z
k f kout ¼
f ðx, Þj cos j d , k f kin ¼
f ðx, Þj cos j d ,
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ðx, Þ2out
kk ¼
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ðx, Þ2in
and we consider the norm for the operator to be given by:
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out
are the traces of the distribution function
In Eq. (9), f and f
respectively to the incoming and outgoing boundary sets:
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ð9Þ
in
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max
k f out kout ¼1
kf out kin :
We will consider multiplying boundary conditions (see Mancini and
Totaro (1998)), i.e., the operator is such that kf out kin ckf out kout ,
with c > 1. In other words, a multiplication of particles takes place on the
boundaries. For instance, the boundary operator may represent reflection, or diffusion boundary conditions, as well as a linear combination of
them. In the following simulations, we will consider multiplying reflective
boundary conditions:
f ðL, Þ ¼ L f ðL, 0 Þ , for cos > 0,
ð10Þ
f ðL, Þ ¼ R f ðL, 0 Þ ,
for cos < 0,
with 0 ¼ , where L and R respectively denote the reflection coefficients on the left and right boundaries of the slab (i.e., on x ¼ L and
x ¼ L respectively). As the region where the evolution takes place is
bounded, it is natural to consider also an absorption term f in order
to avoid the growth of the number of particles (see the results of Fig. 7
below). Summarizing, the problem which we approximate numerically
reads as follows:
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1
>
< @t f þ cos @x f þ f ¼ Qð f Þ,
ð11Þ
f in ¼ f out ,
>
:
f ðx, , 0Þ ¼ f0 ðx, Þ:
It is clear that the efficiency of the absorption coefficient will
depend on the ‘‘power’’ of the multiplication coefficient (i.e., the norm
of the operator ), on the modulus of the particle velocities jvj (here
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equal to 1) and on the dimension of the considered region (here the
distance 2L between the two plates). It has been proven in Mancini
and Totaro (1998), by means of the semi-groups theory and for a fixed
", that if kk 1, then problem (11) admits a unique solution whatever is
0; whereas, if kk > 1, then problem (11) admits a unique solution
provided that:
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>
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jvj ln kk
¼ t ;
2L
ð12Þ
t is called the theoretical critical absorption cross section. In the
particular case of multiplying reflective boundary conditions (10), the
theoretical value for the critical absorption cross section is given by
(see Belleni-Morante and Totaro (1996)):
t ¼
jvj lnðmaxf
2L
L,
R gÞ
:
ð13Þ
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408
We will numerically compute a critical value n for the absorption
cross section and we will precise the behavior of the solution of problem
(11) in terms of the absorption coefficient : we will see that, if < n the
particle density significantly grows in a finite time, for > n the particle
density rapidly goes to zero in a finite time, and for ¼ n it remains
bounded without vanishing (see Sec. 4.3 below). This numerical critical
value n does not depend on "; moreover, it is smaller than the theoretical
one t . This fact does not contradict at all the results proven in
Belleni-Morante and Totaro (1996) or Mancini and Totaro (1998); in
fact, it just shows the existence of a solution also for absorption cross
sections 2 n , t ½ (since n < t ), which was not possible to prove by
means of the semigroup theory.
409
4. NUMERICAL SIMULATIONS
410
411
412
413
414
415
416
417
418
419
420
+
In order to simulate the evolution problem (11), we apply a first
order splitting in time method. First, we approximate by means of an
upwind explicit finite difference method the following transport problem
(see paragraph 4.1 below):
8
@f
@f
>
< þ cos þ f ¼ 0,
@t
@x
in
out
>
: f ¼ f ,
f ðx, , 0Þ ¼ f0 ðx, Þ:
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421
422
423
424
425
426
427
428
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430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
303
Second, we apply the implicit finite element scheme of Sec. 2.1 to the
pure collision problem:
(
@f
¼ 1 Qf
ð15Þ
@t
f ðx, , 0Þ ¼ f0 ðx, Þ:
At each iteration, the program solves separately the transport and
collision parts. Our whole numerical scheme is first order in all variables.
We have done this choice in order to simplify the presentation of
the method, but it would be naturally possible to improve the accuracy
of the method by using appropriate second order schemes for each
step (convection, collision) coupled with a second order (of Strang
type) splitting algorithm.
We will here perform essentially three different numerical tests. In the
first two ones, developed in paragraphs 4.2 and 4.3 below, the relaxation
time is equal to one. In paragraph 4.2, we essentially focalize on the
discrete collision operator itself: the boundary conditions are classical
specular reflection, and there is no absorption; the grazing collision
operator is then compared to the Laplace-Beltrami operator (i.e., to its
limit when " goes to zero), to the isotropic operator and also to the
collisionless case. The dependence of the solution on the grazing
parameter " is also numerically investigated, first for " h=, but also
for bigger values of ". The influence of multiplicative boundary conditions is then studied in paragraph 4.3: in particular, the existence of a
numerical critical absorption cross section, which does not depend on "
(for sufficiently small values of ") nor on the initial data is shown.
Moreover, we underline the existence of a unique profile of equilibrium
(independent on " and on the initial data) for the particle density. Finally,
in paragraph 4.4, we study the influence of the time relaxation for a
photonic type problem already considered in a previous work (see Cordier
et al. (2000)). This last test is further investigated in Cordier et al. (2002).
Let us firstly precise the numerical scheme used to approximate the
convective part of the equation, i.e., problem (14).
454
455
4.1. Transport Problem Approximation
456
457
458
459
460
461
462
+
We consider Nx (=50 in simulations, unless otherwise specified)
discretization points xi , i 2 f1, . . . , Nx g, for the space variable
x 2 ½L, L (with L =1) and N (=50 in simulations, unless otherwise
specified) discretization points for the angle variable 2 S 1 . We introduce
the following symmetric discretization of the circle S 1 : j ¼ þ hj with
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463
464
465
466
467
468
Lucquin-Desreux and Mancini
h ¼ 2=N and j ¼ 0, . . . , N 1. We have that if cos j is a discretization
point, then also cos j is one. Moreover, if j and j 0 are such that j 0 ¼ sð jÞ,
where sð jÞ ¼ ðN =2 jÞmodðNÞ (see Fig. 1), then: cos j0 ¼ cos j .
Considering the above discretization, we approximate the transport
problem (14) by means of an upwind explicit finite difference scheme as
follows:
F1
469
470
.
471
If cos j > 0 then,
f1,nþ1
j ¼
472
nþ1
L f1, sð jÞ
473
and for i ¼ 2 . . . Nx
474
475
476
n
fijnþ1 ¼ ð1 ðt=xÞ cos j tÞfijn þ ðt=xÞ cos j f i1,
j:
477
ð16Þ
478
479
.
480
If cos j < 0 then
fNnþ1
¼
x, j
481
nþ1
R fNx , sð jÞ
482
and for i ¼ 1 . . . Nx 1
483
484
n
fijnþ1 ¼ ð1 þ ðt=xÞ cos j tÞfijn ðt=xÞ cos j fiþ1,
j:
485
ð17Þ
486
487
488
489
490
491
492
493
494
495
We recall that fijn ¼ f ðxi , cos j , tn Þ is the value of the distribution
function f at the position xi ¼ L þ xði 1Þ, i ¼ 1 . . . Nx , and
x ¼ 2L=ðNx 1Þ, with velocity cos j , j ¼ 0 . . . N 1, and at time
tn ¼ n t, n ¼ 1 . . . Niter .
Setting ¼ t=x, the CFL condition writes (for any 0): 1.
In the simulations below, is taken equal to 0.8 (but ¼ 1 would give
exactly the same results). Moreover, we choose L ¼ 2 R ¼ 4 and
" ¼ 0:001, unless otherwise specified.
496
497
498
499
500
s(j)
j
j=0
501
502
503
Figure 1.
504
+
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305
4.2. Grazing Collisions and Comparison with
Other Collision Terms
505
506
507
508
509
510
511
512
513
514
515
In this section we present numerical results concerning the grazing
collision limit at the discrete level and also a comparison with different
types of collisions. We take into account only specular reflection boundary conditions (i.e., L ¼ R ¼ 1), the absorption cross section is given by
¼ 0 (i.e., the number of particles is constant along all the evolution),
and the relaxation time is ¼ 1. We introduce the density function:
Z
nðx, tÞ ¼ f ðx, , tÞ d
516
517
518
519
and the number of particles inside the slab at time t:
Z þL
NðtÞ ¼
nðx, tÞ dx:
521
522
523
ð18Þ
L
520
In Fig. 2, we show how the behavior of the density nðx, tÞ changes when
changing the value of the grazing parameter ". We have first considered
values of " such that " h=, but also larger values of ": in this last case,
F2
524
525
526
0.46
527
0.41
528
0.37
529
0.32
530
+
0.23
533
534
0.19
535
0.14
+
+
+
+
+
536
0.10
537
+ +
+ +
+ + +
+ + + +
+ + + +
+ + + +
+
+
+ +
+ + +
+ + + + + + +
538
0.05
539
0.01
-1.0
540
541
542
545
546
-0.8
+
-0.6
-0.4
Laplace-Beltrami
grazing 0.01
ε =h/ π
543
544
+ + +
+ +
+
n(x,T)
532
+
+
0.28
531
+
+
-0.2
0.0
x
0.2
+
+
0.4
0.6
0.8
1.0
ε =2h/ π
ε =3h/ π
Figure 2. The density nðx; TÞ, with ¼ 0, R ¼ L ¼ 1, for different values of ":
" ¼ 102 ; h=; 2h=; 3h= and for the Laplace-Beltrami operator, with initial data
given by a Dirac, t ¼ tNiter , with Niter ¼ 100, Nx ¼ 50, N ¼ 10:
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547
548
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551
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560
561
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567
568
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Lucquin-Desreux and Mancini
the collision terms have been computed by use of standard first
order quadrature formulas. In particular, we note that for " < 102 the
difference between the grazing collision case and the Laplace-Beltrami
operator is too small to be observed. In fact, the difference is of order "h,
so that it is rapidly very small for small values of " (and for small h as
well). In order to precise this behavior, we have computed in Fig. 3 the
L1 norm of the relative error between the density n" of the grazing
collision problem and that of the limiting one, denoted by nLB , i.e.,
n ði, TÞ nLB ði, TÞ
max "
,
i
nLB ði, TÞ
in terms of " and for different values of h (with T ¼ 50t). We first
remark that this error is very small for " ¼ 103 . But as expected, we
observe a difference between the case " ¼ h= and " ¼ 102 for large
values of h (h ¼ =2), while this difference decreases for smaller values
of h. Figure 2 is related to the ‘‘medium’’ case h ¼ =5 (i.e., N ¼ 10).
In Fig. 4, we compare four types of collisions: the collisionless case
(Q ¼ 0), the case of isotropic collisions (with B ¼ 1=4 in order to fit the
first nonzero eigenvalues), the grazing collisions case (with " ¼ 103 ) and
its limit when " ! 0, i.e., the Laplace-Beltrami operator. We trace the
evolution of the density nð, TÞ for four different times of computation
T ¼ tNiter , with Niter ¼ 50, 100, 150, 200. We observe that the grazing
collisions case and the Laplace-Beltrami operator give the same results
(up to a difference of order 104 which is not noticeable on the figure).
Naturally, in all cases, we have checked numerically that the total
density NðtÞ defined by Eq. (18) was constant with respect to time.
F3
F4
573
574
4.3. Absorption Cross Section and Multiplying
Boundary Conditions
575
576
577
578
579
Still assuming ¼ 1, we now take into account multiplying boundary
conditions, in particular L ¼ 2, R ¼ 4. We consider two initial
580
581
582
583
584
585
586
587
Figure 3.
588
+
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+ + + + + +
+ + + + + +
+ + + + +
+ + + + + + + + +
[Page No. 307]
+
+
+
+
+
+
+
+
+
+ + + + + +
+ +
n(x,T)
0.4
0.6
0.8
1.0
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0.00
1.0
+
0.8
0.4
0.2
LaplaceBeltrami150
grazing150
0.6
0.0
x
0.2
0.4
0.8
isotrope150
no collisions150
0.6
1.0
0.04
+
0.4
0.2
LaplaceBeltrami100
grazing100
0.6
0.0
x
0.2
0.4
0.8
isotrope100
no collisions100
0.6
1.0
+
0.8
0.4
0.2
LaplaceBeltrami200
grazing200
0.6
0.0
x
0.2
0.4
0.8
isotrope200
no collisions200
0.6
1.0
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
0.00
1.0
0.02
0.06
0.04
0.02
0.08
0.06
0.10
0.08
+ + + + + + + + + + + + + + + + +
+ + + + + + +
+ + + + + + + +
+ + + + + + + + + + + + + + + + + +
n(x,T)
0.14
0.12
0.12
0.10
0.16
0.16
0.14
0.8
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ + +
+ + + + +
0.00
1.0
0.20
isotrope50
0.2
no collisions50
0.0
x
grazing50
0.2
0.18
+
0.4
LaplaceBeltrami50
0.6
0.02
0.04
0.06
0.08
0.18
0.8
0.12
0.10
0.20
0.00
1.0
0.02
0.04
0.06
0.08
0.10
Finite Element Approximation of Grazing Collisions
Figure 4. The density nð; TÞ, with ¼ 0, R ¼ L ¼ 1, for the grazing collisions with " ¼ 0:001,
the Laplace-Beltrami operator, the isotropic collisions case, and the collisionless case, with initial
data given by a Dirac, T ¼ tNiter , with Niter ¼ 50 (top on the left), 100 (top on the right), 150
(bottom on the left), 200 (bottom on the right).
n(x,T)
n(x,T)
0.14
0.14
0.12
0.18
0.16
0.16
0.20
0.18
0.20
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590
591
592
593
594
595
596
597
Lucquin-Desreux and Mancini
data: first, a constant function both in space and velocity; second, a Dirac
mass both in space and velocity.
In Fig. 5, we show the convergence of the sequence of critical absorption cross sections towards a finite value n for " ¼ 0:001. The critical
absorption value we compute is independent (for large times) on the
initial data: in fact we observe that both the constant initial data and
the Dirac mass give the same value. Moreover, this value is smaller than
the theoretical one t given by Eq. (13). The way the computation is
performed is the following one:
F5
598
599
600
601
602
603
604
605
606
We compute the total densities Nðti , Þ for ti ¼ ði þ 1Þ50t, with
i ¼ 0, . . . , 9, and for different values of .
. We determine the intersection point of two successive curves
Nðti , Þ and Nðtiþ1 , Þ describing the evolution of the density in
terms of ; the respective value of for the intersection points
are called critical and denoted by c ðiÞ. In fact, for each c ðiÞ, the
total density is constant for the two successive times of iteration ti
and tiþ1 .
.
607
608
609
610
0.50
0.49
611
612
0.48
613
0.47
614
615
0.46
616
0.45
617
618
0.44
619
0.43
620
621
622
623
624
0.42
0.41
0.40
1
625
626
627
628
629
630
+
2
3
constant
dirac
4
5
6
7
8
9
Figure 5. The interpolation curve of c ðiÞ. Each point is given by the intersection
of the total density curves for two successive iteration times, i.e. 1 corresponds to
the intersection point of 50 and 100 iteration curves. We increase of 50 iterations
at every ordinate point, finishing by 500 iterations.
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631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
309
We observe in Fig. 5 that these values of critical absorption cross
sections numerically converge to a finite value n , which is the searched
value. Moreover, additional numerical simulations show that this value
n does not depend on " (for sufficiently small values of ").
In Fig. 6, we show the limit profile of the curve nð, tÞ for the computed critical absorption ¼ n ¼ 0:466 and t ¼ 500t. Again the initial
condition is given by a constant function or a Dirac mass. We underline
the fact that for both initial data, we end up with the same (up to a
normalization) profile which does not depend on ", for " small enough.
In Fig. 7, we show the influence of multiplying reflection boundary
conditions with respect to different values of the absorption cross section.
When time evolves, we clearly see the significant growth of the number
of particles NðtÞ when < n , and its fast shoot to zero for > n .
For ¼ n , this number remains bounded without ‘‘exploding’’ both in
the case of the initial condition given by a constant function or a Dirac
mass. Moreover, it seems to reach an almost constant value. Hence, this
numerical approximation n of the critical cross-section c done in Fig. 5
seems to be satisfactory.
F6
F7
649
650
651
652
1.6
653
1.4
654
655
+
1.2
656
657
+
1.0
+
658
659
660
0.8
661
0.6
662
0.4
663
664
+++
+++++++
666
-1.0
669
672
+
++
++
+
+
0
667
668
671
+++++
++
+
0.2
665
670
+++++++++++++++++
+
+++
+
+
-0.8 -0.6
constant
+
dirac
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1.0
Figure 6. The final profile of nð; tÞ with initial data a Dirac mass or a constant
function, both in position and velocity; ¼ 0:466, " ¼ 0:001, R ¼ 2 L ¼ 4,
t ¼ 500t:
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Lucquin-Desreux and Mancini
2.0
674
1.8
675
676
1.6
677
678
1.4
679
1.2
680
1.0
681
682
0.8
683
0.6
684
0.4
685
686
0.2
687
0
688
0
100
200
300
400
500
689
sigma=0.466
690
691
692
693
694
sigma=0.1
sigma=1
Figure 7. The total density N(t) for different values of the absorption ( > n ,
< n , and ¼ n ¼ 0:466) with initial data a constant function, " ¼ 0:001 and
t ¼ 500t:
695
696
697
698
699
700
701
702
703
Finally, in Fig. 8, we show the behavior of the density function nð, tÞ
for three different iteration times. The initial data which we consider is a
Dirac mass both in space and velocity. The absorption cross section is
the numerical critical coefficient n ¼ 0:466. It is clearly seen that the
Dirac mass is convected and diffused respectively by means of the transport equation and of the Boltzmann-Lorentz operator for grazing collisions (with " ¼ 0:001). Finally, the density is also multiplied and reflected
at the boundaries, and we can see the beginning of boundary layers.
F8
704
705
4.4. Defocusing of a Beam
706
707
708
709
710
711
712
713
714
+
Following the test performed in Cordier et al. (2000) for a photonic
type model, we now assume that between the two plates there is vacuum.
Moreover, a focussed beam enters in the slab on the left side (i.e., at
x ¼ L) and scatters. The absorption cross section is fixed to ¼ 0,
the grazing coefficient to " ¼ 0:001 and the discretization points in x
and respectively to Nx ¼ 50 and N ¼ 50. We now consider the influence of the relaxation time . The boundary condition corresponding to
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715
311
5
716
717
4
718
719
720
3
721
722
723
2
724
725
726
1
727
728
729
0
-1.0
730
731
732
733
-0.8 -0.6
1 iter
-0.4
-0.2
20 iter
0
0.2
0.4
100 iter
0.6
0.8
1.0
Figure 8. The evolution of nð; tÞ with initial data a Dirac mass (normalized) in
position and velocity.
734
735
736
737
the injection of a mono-kinetic beam in the direction perpendicular to the
planes of the slab is given by:
738
739
740
741
f ðL, cos > 0, tÞ ¼ ,
where denotes the delta measure with respect to the angle variable ; on
the right boundary we have:
f ðL, cos < 0, tÞ ¼ 0,
742
743
744
745
746
747
which means that the particles leaving the slab on the right side never
return.
In Fig. 9, we draw the focalization coefficient, defined as the ratio
between the particles leaving the slab on the right side with a velocity
perpendicular to the plane, and the injected particles:
748
F ð, tÞ ¼ f ðL, ¼ 0, tÞ:
749
750
751
752
753
754
755
756
+
F9
The computations are done for t sufficiently large (t ¼ 150t) so that
the density has reached a stationary state.
Finally, in Fig. 10 we show for three values of the relaxation time,
¼ 25 , 1, 25 , and for four iteration times Niter ¼ 25, 50, 100, 200 the
particle density nð, tÞ evolution (with t ¼ tNiter ).
Let us note that the numerical results we have obtained for this
model are in perfect agreement with those performed in Cordier et al. (2000)
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757
758
F
1.0
759
760
761
762
0.9
0.8
0.7
763
764
0.6
765
0.5
766
767
0.4
768
0.3
769
770
0.2
771
0.1
772
k
0
773
10
774
Figure 9.
8
6
4
2
0
2
4
6
8
10
The focalization F ð; t ¼ 150tÞ. ¼ 2k ; k ¼ 10; . . . ; 10.
775
776
777
778
779
780
for a 1D in space and 2D in velocity (i.e., R S 2 ) analogous problem,
where the collision operator involved was the Laplace-Beltrami operator
on the unit sphere.
781
782
5. CONCLUDING REMARKS
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
+
We have described an implicit finite element approximation for the
grazing collision Boltzmann-Lorentz operator. This approximation
allows to pass to the limit " ! 0 for a fixed discrete scheme (with respect
to the velocity variable). This fact is due essentially to the possibility of
carrying out the exact computations of both the mass matrix and the
collision matrix with no need of quadrature formulas. Other methods
would have been possible: for example, a finite volume method based
on a piecewise parabolic operator or a spectral method (see Buet et al.
(2001), Pareschi et al. (2002)). Our choice of a FEM method has been
essentially motivated by the nature of the limiting problem and also by
previous computations done for the Laplace-Beltrami operator in 3D.
The generalization to the three-dimensional case seems to be possible: we can in fact compute the matrices A and Q, but the coefficients
we find are much more complicated. According to Buet et al. (2001),
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Finite Element Approximation of Grazing Collisions
799
1.40
800
1.26
801
1.12
802
0.98
803
804
0.84
n(x,T)
0.70
805
0.56
806
0.42
807
0.28
808
0.14
809
0.00
1.0
810
0.8
0.4
0.0
x
0.2
0.4
0.6
0.8
1.0
100 iterations
200 iterations
50 iterations
1.26
814
1.12
815
0.98
816
0.84
n(x,T)
817
0.70
818
0.56
819
820
0.42
0.28
821
0.14
822
0.00
1.0
823
0.8
824
825
826
1.40
827
828
1.26
0.6
0.4
0.2
0.0
x
0.2
0.4
0.6
0.8
25 iterations
100 iterations
50 iterations
200 iterations
1.0
1.12
829
0.98
830
0.84
n(x,T)
0.70
831
0.56
832
0.42
833
0.28
834
0.14
835
836
0.00
1.0
0.8
0.6
0.4
837
25 iterations
838
50 iterations
840
0.2
1.40
813
+
0.6
25 iterations
811
812
839
313
0.2
[293–320]
[Page No. 313]
0.2
0.4
0.6
0.8
1.0
100 iterations
200 iterations
5
Figure 10. The density nð; TÞ, ¼ 2
with Niter ¼ 25; 50; 100; 200.
[17.7.2003–10:39am]
0.0
x
(left),1 (center) , 25 (right), T ¼ Niter t
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314
841
842
843
844
845
846
847
Lucquin-Desreux and Mancini
a spectral method should be a good idea for the numerical approximation
of the 3D grazing collision Lorentz operator.
It should be also interesting to generalize the approximation
presented in this article to nonconstant grazing collisions scattering
cross sections and also to the Coulombian case (we refer to Cordier et
al. (2002) for a numerical comparison between the Boltzmann-Lorentz
operator and the Fokker-Planck one in the Coulombian case).
848
849
850
A1. APPENDIX
851
852
853
854
855
856
857
Here we perform the computation of the coefficients Qij of the grazing collision matrix Q. We first express the hat functions ’i as follows:
’i ðÞ ¼ ’h ð i Þ, where:
1
’h ðzÞ ¼ h ðh jzjÞ, if jzj < h,
0
else:
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
0
Replacing in Eq. (3) ¼ i and ¼ 0 j yields:
Z Z
0
0
0
B" ð þ Þ ½’h ðÞ ’h ð Þ ’h ðÞ ’h ð þ Þ d d ,
Qij ¼
S1
where we have set ¼ i j ¼ ði jÞh. Introducing the new hat
function
n
ð1 jzjÞ, if jzj < 1,
’ðzÞ ¼
0
else
we have ’h ðzÞ ¼ ’ðz=hÞ. Now using the new variables ¼ =h and
0
0 ¼ =h, we can write:
Z þ1 Z þ1
Qij ¼ h2
B" ðh h0 þ Þ ½’ðÞ ’ð0 Þ
1
873
875
876
1
881
882
+
1
’ðÞ ’ð þ ði jÞÞ d d0 :
877
878
880
1
’ðÞ ’ð þ ði jÞÞ d d0
Z þ1 Z þ1
¼ h2
B" ðhð 0 þ ði jÞÞÞ ½’ðÞ ’ð0 Þ
874
879
S1
Let us set Bh ðzÞ ¼ BðhzÞ, i.e., (with B0 ¼ 1="3 ):
B0 if jzj < ¼ "=h,
Bh ðzÞ ¼
0
else;
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Finite Element Approximation of Grazing Collisions
883
315
then the collision coefficient Qij reads:
884
885
Qij ¼ h
886
2
Z
þ1
1
887
888
Z
þ1
Bh ð 0 þ ði jÞÞ ½’ðÞ ’ð0 Þ
1
’ðÞ ’ð þ ði j ÞÞ d d0
Z Z
¼ h2 B 0
½’ðÞ ’ð0 Þ ’ðÞ ’ð þ ði jÞÞ d d0
889
890
j0 þðijÞj<
891
892
893
with , 0 2 ½1, þ 1 . We split Qij into a gain part Gij and a loss part Lij ,
i.e., we set Qij ¼ Gij Lij , where:
894
895
896
Gij ¼ h2 B0
897
Lij ¼ h2 B0
898
899
Z Z
Z Z
’ðÞ ’ð0 Þ d d0 ,
j0 þðijÞj<
’ðÞ ’ð þ ði jÞÞ d d0 :
j0 þðijÞj<
900
901
902
903
904
905
906
907
908
We now have to compute separately these gain and loss terms. For
this, we will assume that 1, so that " h= ¼ 2=N : it is thus possible
to pass to the limit first for " ! 0 and then for h ! 0.
Let us first compute the gain coefficients Gij . As the ’ function is null
outside a domain of length 2, it is clear that for ji jj 3, Gij ¼ 0 (see
also Fig. 11). Thus, we have to compute Gij for i ¼ j, ji jj ¼ 1, and
ji jj ¼ 2. Considering also the symmetry of the collision operator, we
only have to compute the integrals for i j ¼ 2, i j ¼ 1, and i ¼ j.
F11
909
910
θ’
911
912
θ’ = θ
913
914
θ ’= θ − 1
915
916
θ
917
918
θ’ = θ − 2
919
920
−β
921
β
922
923
Figure 11. Set of definition of Bh ð 0 þ i jÞ:
924
+
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Lucquin-Desreux and Mancini
θ’
925
926
927
928
929
930
θ
931
932
I(−β)
933
934
935
936
937
938
J( β )
β
−β
Figure 12. The integrals IðÞ and JðÞ.
939
940
941
942
943
Looking at Fig. 12, we define the following integrals for a given
> 0:
944
945
946
Z
2
Z
0
IðÞ ¼ h B0
947
þ h2 B 0
949
950
Z
Z
2
ð1 þ Þ d
1
1
Z
ð1 Þ
0
Z
1
þ h B0
951
0
ð1 þ Þ
948
þ1
þ1
d
ð1 þ 0 Þ d0 d
1
0
ð1 Þ
1
Z1
0
F12
0
ð1 þ Þ d
0
d
1
Z þ1
þ h2 B 0
ð1 Þ
ð1 0 Þ d0 d
1
0
1 3
1
1
ð 4Þ þ ð1 Þð1 þ Þð2 þ 4 þ 1Þ þ 2
¼ h2 B 0
12
24
4
1
1
1
1
1
¼ h2 B 0 4 þ 3 þ 2 þ þ
8
6
4
6
24
952
953
954
955
956
957
958
959
960
961
962
963
and
Jð Þ ¼ h2 B0
Z
1
965
1
¼ h B0 ð 1Þ4
24
[17.7.2003–10:39am]
1
ð1 Þ
964
966
+
Z
ð1 þ 0 Þ d0 d
1
2
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Finite Element Approximation of Grazing Collisions
967
968
969
970
317
In particular, IðÞ represents the integral with support on the
bigger triangle and JðÞ represents the integral with support on the
small triangle (Fig. 12)
We get:
971
972
Gij ¼ Jð1 Þ ¼
973
1 2
h B0
24
4
,
for i j ¼ 2
ð19Þ
974
1
Gij ¼ Ið Þ Jð Þ ¼ h2 B0 6
975
976
977
4
þ
1
3
3
þ
1
3
,
for ji jj ¼ 1
ð20Þ
978
979
980
2
981
Z
Z
þ1
þ1
0
Gij ¼ h B0
ð1 jjÞ d
1
1 4 2 3 4
2
,
þ
¼ h B0
4
3
3
982
983
984
ð1 j jÞ d
0
2Ið1 þ Þ
1
for i ¼ j
ð21Þ
985
986
987
988
989
990
991
992
993
We now compute the loss coefficients Lij . As before, we just have to
compute Lij for ji jj ¼ 2, 1, and 0. We first remark that for ji jj 2,
the product ’ðÞ’ð þ i jÞ is null. In fact ’ðÞ differs from zero on the set
½1, þ 1 while ’ð þ i jÞ differs from zero on the set ½1, 3 . Hence,
Lij ¼ 0 for ji jj 2. Moreover, for ji jj ¼ 1 the product
’ðÞ’ð þ i jÞ is not null only for 2 ½0, 1 , and the dependence of Lij
on the variable 0 appears only through the kernel Bh which is not null on
a set of amplitude 2 . Thus, we have:
994
995
996
Z
2
1
Lij ¼ h B0 2
ð1 Þ d
0
997
¼ h2 B0 2
998
1 1 2
¼ h B0 ,
6 3
for ji jj ¼ 1
ð22Þ
999
Analogously, for the principal diagonal, we have:
1000
1001
1002
2
Z
þ1
Lii ¼ h B0 2
1003
1004
ð1 jjÞ2 d
1
2 4 2
¼ h B0 :
¼ h B0 2
3 3
2
1005
ð23Þ
1006
1007
1008
+
Thus, collecting Eqs. (19)–(23) and recalling that B0 ¼ 1="3 and that
¼ "=h, the computation of the coefficients Qij is completed.
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1009
1010
Lucquin-Desreux and Mancini
We finally remark that the computation of the coefficients Aij of the
mass matrix A may be done analogously.
1011
1012
1013
1014
ACKNOWLEDGMENT
1015
We would like to thank the referees for their very helpful comments.
The authors also thank the European TMR Network ‘‘Asymptotics
methods in kinetic theory’’ N o ERB FMR XCT 970157 for its financial
support. This research was supported through a European Community
Marie Curie Fellowship.
1016
1017
1018
1019
1020
1021
1022
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Received September 13, 2001
Revised January 31, 2003
Accepted February 4, 2003
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