A Variable Multigroup Approach to the Nonlinear
Boltzmann Equation Based on the Method of Weighted
Residuals
M. Galler , A. Rossani† and F. Schürrer
Institut für Theoretische Physik, Technische Universität Graz
†
Petersgasse 16, A-8010 Graz, Austria
Istituto Nazionale di Fisica della Materia, Dipartimento de Fisica, Politecnico di Turino
Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
Abstract. A multigroup approximation to the Boltzmann equation with an arbitrary partition of the speed variable combined
with a spherical harmonics expansion of the directional dependence of the distribution function is presented. Based on the
method of weighted residuals, weight functions assure the exact conservation of particle density, momentum and energy
density. A numerical example shows the applicability of this very flexible method to study the relaxation of a homogeneous
gas far from equilibrium.
INTRODUCTION
The Boltzmann equation constitutes a very powerful tool for the description of rarefied gases [1]. However, its complicated structure as a nonlinear integro-differential equation prevents a rigorous mathematical solution. Consequently,
there have been several attempts to construct approximation methods, so-called model Boltzmann equations. These
model equations allow one to obtain suitable information about the properties of rarefied gases thus avoiding many
mathematical and numerical difficulties.
One of these approaches is the concept of generating moment equations. This method is based on forming moments
of the Boltzmann equation in connection with an appropriate closure of the resulting set of equations. Here, Grad’s
13 moments method should be mentioned [2]. Since every closure method includes some intuitive assumptions, the
concept of moment equations has been extended in a way that it immediately leads to a closed set of equations. Such a
generalization can be performed by means of a discretization of the modulus of the speed variable, which ends in the
well-known multigroup approaches.
Multigroup approaches are very common and successful ways to treat the linear Boltzmann equation. As a consequence, there have been many attempts to transmit this formalism to the nonlinear Boltzmann equation as well. In a
recent paper [3], a multigroup approximation to the nonlinear Boltzmann equation was derived and its main features
were discussed. It has been shown that this approach guarantees particle conservation exactly, while momentum and
energy conservation are only approximately fulfilled. On the other hand, a conservative two group model for the nonlinear Boltzmann equation has already been established [4]. This model fulfills all the required conservation properties,
provided that the underlying transition probabilities obey certain conditions.
Another recently published multigroup model called revised multigroup approach [5] for the nonlinear Boltzmann
equation is free of the restrictions of the other models. This approximation, based on the method of weighted
residuals [8], leads to a set of integro-differential equations, which ensures exactly the conservation of particle density,
momentum and energy density without any ad hoc assumptions about the scattering process. It is the aim of this paper
to extend this conservative multigroup approach by a method similar to Grad’s method. Thus, the P2 approximation is
used for treating the angular dependence of the distribution function. Since this approximation is performed in each
speed interval, an essential improvement of the gained results in comparison to those of Grad’s method is obtained.
This procedure leads to a set of partial differential equations. In other words, this method results in a description of the
physical state of a gas in each speed interval, which corresponds with an extended thermodynamics approach [6].
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
43
THE REVISED MULTIGROUP APPROACH
In the kinetic theory, the temporal and spatial evolution of the distribution function f
f x v t of monatomic
rarefied gases is governed by the Boltzmann equation. It reads in the scattering kernel formulation [7] after the polar
v and Ω̂ v v and the introduction of the new function
decomposition of the velocity variable v vΩ̂ with v
F x v Ω̂ t : v2 f x v t :
∂F
∂t
with the collision term
JFF
∞
0
dv
∞
0
0
∞
dv dv
S2
S2
J F F vΩ̂ ∇ F
(1)
v Ω̂ Π v Ω̂ v Ω̂ d Ω̂ g vΩ̂ v Ω̂ F v Ω̂ F v Ω̂ d Ω̂
S2
d Ω̂ g v Ω̂
vΩ̂ F v Ω̂ F v Ω̂
(2)
Here, g denotes the collision frequency, while Π is the scattering kernel. In the case of isotropic, elastic scattering, g
and Π are given by
v Ω̂ vΩ̂ v Ω̂ σ vΩ̂ v Ω̂ Π v Ω̂ v Ω̂ g vΩ̂
vΩ̂
vΩ̂ v Ω̂ !
v Ω̂ v2 δ vΩ̂ v Ω̂
π v Ω̂
(3)
with the total scattering cross section σ .
The first step to construct the revised multigroup approximation consists of introducing a partition of the nonnegative real axis v 0 ∞ in N intervals located between N 1 speed-knots w i i 0 1 2
N. Here, wN has to be
taken high enough such that F wN Ω̂ vanishes. Moreover, w0 is set to zero.
In the second step, we represent the particle distribution function as a finite series,
"# $
∑ ϕ x Ω̂ t ψ v
(4)
%
with N shape functions ψ v and N new unknown functions
ϕ which are independent of the modulus of velocity.
Moreover, we demand that the shape functions ψ v & i 1 2 $' N fulfill the following relations
(5)
iii ψψ vv)+( 00 vv "*"* 0w ∞, w -
iii dv ψ v 1 .
F x v Ω̂ t
N
i
i 1
i
i
i
i
i
i
i 1
wi
i
i
wi 1
which allows us to obtain the new unknowns ϕi by the simple integration
.
ϕi x Ω̂ t
wi
wi 1
dv F x v Ω̂ t
(6)
As suggested by the method of weighted residuals [8], a set of approximative differential equations can be constructed in the following way: the ansatz (4) for the distribution function is inserted into the Boltzmann equation and
the result, weighted with L different weight functions λl , is integrated over N L parts of the whole variation range of
the independent variable v. This procedure leads to a closed set of conditional equations for the new unknowns ϕ i .
Macroscopic quantities, as for example the particle density, the drift velocity or the energy density are power
moments of the distribution function. Since we want to derive multigroup model Boltzmann equations obeying
the exact conservation laws for mass, momentum and energy, we choose powers of v as suitable weight functions,
λl v
vl l 0 1
L 1.
Moreover, we have to specify the intervals, we integrate over. For constructing N equations with L different weight
functions λl , N L intervals are necessary. They are determined by joining L adjacent intervals Ii
wi 1 wi i
M
12
N to N L macro-intervals Iq
wL q 1 wLq q 1 2
N L. Figure 1 displays a possible partition of the
$ / , - $$ 0 1 , 2 - $ speed axis for N 12, the macro-intervals I for L 3 as well as a possible shape function ψ .
M
q
6
44
Partition of the real speed axis, the macro-interval IqM and a possible shape function ψ6 .
FIGURE 1.
The application of the above considerations to the Boltzmann equation (1) in connection with (4) leads to a set of N
integro-differential equations. They are called the revised multigroup approach (RMA):
56 7
v
6 3 7
v
d Ω̂ g 1 2 Ω̂ Ω̂ ϕ Ω̂ ϕ Ω̂ (7)
Ω̂ ∇ϕ 8 ∑
∑
∑
% 1 , 42 3
% 1 , 243 9 %
d Ω̂ S d Ω̂ g 1 2 Ω̂
Ω̂ ; Π 1 2 Ω̂ Ω̂ < Ω̂ ϕ Ω̂ ϕ Ω̂ >=
∑
:%
with the abbreviations
6 7
v (8)
dv v ψ v &
.
dv v ψ v ?
dv ψ v g vΩ̂ v Ω̂ g 1 2 Ω̂ Ω̂
.
.
1Π 2 Ω̂ Ω̂ Ω̂ 1 2 1 dv v dv ψ v dv ψ v g v Ω̂ v Ω̂ Π v Ω̂ v Ω̂ vΩ̂&
g Ω̂ Ω̂ .
.
.
As shown below, L ( 3 is a necessary condition for the exact fulfillment of conservation of mass, momentum and
energy. This fact constitutes the lower limit for L. On the other hand, L @ 3 is combined with the emerging of high
order moments in the multigroup approach, which have a physical meaning, but does not lead to quantities which
are conserved in binary collisions. While this aspect only leads to interpretational problems, a second
consideration
explains more clearly, why L 3 is certainly a suitable choice. A microscopic quantity g v g Ω̂ is related to a
macroscopic one, G, by forming moments, i.e. g v g Ω̂ is weighted with the distribution function F and integrated
with respect to v. Moreover, a quantity G to the microscopic quantity g v g Ω̂ with v " I 0 v v , is calculated
via
G A dv d Ω̂ g v g Ω̂ F v Ω̂ (9)
Since
F v Ω̂ (10)
∑ ψ v ϕ Ω̂
% 1 , 243
is the solution to the Boltzmann equation in the macro-interval I based on the method of weighted residuals, we can
write the sub-macroscopic quantity G with v " I w 1 , 2 w and consequently the total G with v "# 0 ∞ in
the form
B
Ω̂
Ω̂
G d
Ω̂
g
dv
g
v
ψ
v
ϕ
and
G
(11)
∑G %
% 1 ∑, 243 .
Lq
l
i Lq 1
1
∂ ϕi
i
∂t
l 1
N
Lq
i
i
2
l
jk
2
jk 1 S
l
l
ij
wi
i
wi 1
wi
l
l
wi 1
l
jk
i Lq 1
l
ij
2
j 1 S
1
il
jk
j
i
j
k
i
wj
i
j
wj 1
wi
il
jk
N
wj
l
wi 1
wk
j
wj 1
k
wk 1
1
1
2
I
1
I
I
2
S2
1
2
1
2
2
Lq
q
i Lq 1
Lq
IqM
i Lq 1
i
i
M
q
M
q
IqM
1
Lq
Lq 1
N L
wi
1 wi 1
1
i
S2
i
2
q 1
IqM
Thus, the smaller we choose L, the more sub-macroscopic quantities G I M are obtained, which leads to more information
q
about the macroscopic quantity G. It should be noted that an individual ϕ i does not have an immediate physical
meaning, but only quantities related to ϕi via (10) and (11). To sum up, we state that L 3 is certainly a proper choice
for a realistic description of a fluid dynamic situation.
45
C
Finally, we try to transcript some macroscopic quantities into our multigroup scheme by splitting up the whole range
of v into a sum of intervals. Thus, we get for the particle density n, drift velocity u, momentum flux , energy density
K and energy flux Q
ED d v f vF ∑% dΩ̂ ϕ Ω̂
6
n u ED d v v f v ∑ v 7 d Ω̂ Ω̂ ϕ Ω̂ %
6
CG m ED d v v H v f vI m ∑% v 7 dΩ̂ Ω̂ H Ω̂ ϕ Ω̂
6
m
K
2 ED d v v f v m2 ∑% v 7 dΩ̂ ϕ Ω̂
6
m
Q
2 ED d v v v f v m2 ∑% v 7 dΩ̂ Ω̂ ϕ Ω̂
n
N
3
3
3
i
i 1
i
S2
N
3
3
3
i 1
N
2
3
3
2
2
i 1
N
2
3
(12)
i
2
i 1 S
N
3
i
i
3
i 1
i
S2
i
S2
i
i
S2
THE P2 APPROXIMATION
In this section, we describe a method for treating the angular depending functions ϕ i Ω̂ in the revised multigroup
approach. This P2 approximation [9] consists of the expansion of these functions in spherical harmonics and the
truncation of these series after the third term. Hence, we obtain an ansatz for the N unknown functions
ϕ x Ω̂ t i
This ansatz implies that
ni
S2
1
4π
d Ω̂ ϕi Ω̂
5 n x t i
Ji
S2
J Ω̂
3 Ji x t
15
2
K x t : Ω̂ H Ω̂ 8 K
d Ω̂ Ω̂ ϕi Ω̂
i
(13)
i
S2
L &
d Ω̂ Ω̂ Ω̂ ϕi Ω̂
(14)
L 12 a H b b H aJ 13 M a b with
a b
K
(15)
We remark that i is a symmetric traceless tensor.
As shown in the previous section, the revised multigroup approach leads to N evolution equations for the N unknown
functions ϕi depending on the position x, the direction of velocity Ω̂ v v and the time t. The P2 approximation yields
an ansatz for these ϕi with 9N new unknowns ni , Ji and i just depending on x and t. The evolution equations for these
9N unknowns are obtained by projecting the multigroup system onto 1, Ω̂ and Ω̂ Ω̂. This means that one inserts the
P2 ansatz (13) into the multigroup approach (7), multiplies the result with 1, Ω̂ and Ω̂ Ω̂, respectively, and integrates
it with respect to Ω̂ over the whole two-dimensional unit sphere S 2 .
The revised multigroup approach linked with the P2 approximation constitutes a system of evolution equations
for the quantities ni , Ji and i , which are moments of the distribution function. Hence, it is not surprising that the
connection between these quantities and other speed moments, namely macroscopic quantities, is quite a simple one.
The comparison of the multigroup versions of particle density n, drift velocity u, momentum flux , energy density
K and energy flux Q (12) and (14) leads to the following expressions for macroscopic quantities in the RMA-P2
formulation:
K
L
L
K
n
∑% n N
i 1
i
nu
C
6
6
∑% v7 J N
CG m ∑% v 7 O K n3 ?M P N
i 1
N
i i
i 1
2
i
i
i
46
K
6 7
%
m N 2
v i ni
2 i∑1
Q
6 7
%
m N 3
v i Ji
2 i∑1
(16)
Q*ACR H
1 3 tr Q and the temperature T p n k 6 7
67 67
n
SQ m ∑% T v O K 3 M?P v ∑% v J H J nUV
(17)
6 7 67 67
m
p
3∑
% 9 v 6 n v 6 ∑% v 6 J J n=W
m
7
7 7
T
3k n ∑
% 9 v n v ∑% v J J n=W
Moreover, we find for the stress tensor
ρ u u, the pressure p
separating parts of ordered motion and peculiar velocity
N
2
i
i 1
N
2
i 1
i
j i
j 1
j
N
i
N
B
N
i
i
B
i
i
j i
j 1
j
N
2
i
i 1
i
i
j i
j 1
j
As a consequence of the used multigroup concept, our model provides not only average values for macroscopic
quantities to the whole velocity space but it also contains the information about the sub-macroscopic quantities to
different speed macro-intervals as well (cp. (11)). Thus, the particle number n̂ q and the drift velocity ûq due to particles
wL q 1 wLq , q 1 2
N L are obtained by
in the macro-interval v IqM
" 1 , 2 $$ 67
1
n̂ % 1 ∑, 2X3 n û n % 1 ∑, 243 v J Lq
Lq
q
i Lq 1
1
i
q
i Lq 1
(18)
i i
1
The above equations indicate a simple and elegant way to compare results of the RMA-P2 approximation with
exact solutions f A v to the Boltzmann equation. The RMA-P2 formulations of macroscopic quantities (16) have their
counterparts in the first terms of (12). The particle densities ñq and drift velocities ũq referring to the speed macrointervals for the given distribution function can be calculated by the integrations
ñq
IqM
dv v2
S2
d Ω̂ fA vΩ̂
ũq
1n IqM
dv v3
S2
d Ω̂ Ω̂ fA vΩ̂
(19)
CONSERVATION LAWS
(
(20)
D d v h vΩ̂ Π v Ω̂ v Ω̂ vΩ̂ 12 h v Ω̂ h v Ω̂ >-
when h v is a collision invariant [7]. The multiplication with g v Ω̂ v Ω̂ f v Ω̂ f v Ω̂ and the integration of
the result with respect to v and v ends in
dv dv dΩ̂ dΩ̂ Y dv dΩ̂ h vΩ̂ g v Ω̂ v Ω̂ Π v Ω̂Z v Ω̂ < vΩ̂ F v Ω̂ F v Ω̂ (21)
dv dv dΩ̂ dΩ̂ 12 g v Ω̂[ v Ω̂ h v Ω̂ h v Ω̂ F v Ω̂ F v Ω̂ &
When the integrations with respect to v, v and v are split up into sums of integrations over speed intervals I w , w , we obtain
∑ dv dv d Ω̂ d Ω̂ dv d Ω̂ h vΩ̂ g v Ω̂ v Ω̂ Π v Ω̂ v Ω̂ vΩ̂ F v Ω̂ F v Ω̂ ::%
Now, we prove that the revised multigroup P2 approach preserves particle number, momentum and energy density for
arbitrary choice of N and shape functions ψi v , provided L 3. First of all, we state that
3
3
∞
∞
0
∞
0
S2
∞
S2
∞
0
0
0
S2
S2
S2
i
i 1
i
N
i j k 1 Ij
S2
Ik
S2
: ∑% dv dv N
i j 1 Ii
Ij
S2
Ii
S2
d Ω̂
S2
d Ω̂
12 g v Ω̂ v Ω̂ h v Ω̂ h v Ω̂ F v Ω̂ F v Ω̂ &
47
(22)
\ 1 2 (23)
∑ d Ω̂ d Ω̂ d Ω̂g 1 2 Ω̂ Ω̂ Π 1 2 Ω̂ Ω̂ Ω̂ ϕ Ω̂ ϕ Ω̂ ::%
: ∑% dΩ̂ dΩ̂ g 1 2 Ω̂ Ω̂ ϕ Ω̂ ϕ Ω̂ Moreover, we obtain for h vΩ̂ vΩ̂
(24)
∑ d Ω̂ d Ω̂ d Ω̂ g 1 2 Ω̂ Ω̂ Ω̂ Π 1 2 Ω̂ Ω̂ Ω̂ ϕ Ω̂ ϕ Ω̂ ::%
: ∑% dΩ̂ dΩ̂ g 1 2 Ω̂ Ω̂ Ω̂ ϕ Ω̂ ϕ Ω̂ while the calculations for h vΩ̂ v 2 end in
(25)
∑ d Ω̂ d Ω̂ d Ω̂g 1 2 Ω̂E
Ω̂ ; Π 1 2 Ω̂ Ω̂ Ω̂ ϕ Ω̂ ϕ Ω̂ ::%
: ∑% dΩ̂ dΩ̂ g 1 2 Ω̂ Ω̂ ϕ Ω̂ ϕ Ω̂ Finally, we manipulate the revised multigroup approach in the following way. For l 0 we project the equations with
respect to Ω̂ onto 1 and sum the result over all the possible q. Hence, we obtain by the help of (14) and (16)
67
67
∂
∂
=
=
=
d
Ω̂
ϕ
Ω̂
∇
v
d
Ω̂
Ω̂
ϕ
Ω̂
n
∇
∂t 9 ∑
∂t 9 ∑
9 ∑%
%
% 9 ∑% v J = 0 (26)
and therefore
∂n
∇ n u ] 0 (27)
∂t since the right-hand side of this equation vanishes because of (23). If one takes l 1 in (7), projects this equations
Finally, the insertion of the multigroup ansatz (4) into the above equation and the application of the definitions (8) of
gi j Ω̂ Ω̂ and Πijkl Ω̂ Ω̂
Ω̂ yields for h1 vΩ̂
1
N
2
i jk 1 S
S2
S2
0
jk
i0
jk
0
ij
i
N
2
i j 1 S
S2
j
k
j
2
N
2
i jk 1 S
S2
S2
N
2
i j 1 S
S2
1
jk
i1
jk
1
ij
i
2
jk
i2
jk
2
ij
i
j
j
2
3
N
2
i jk 1 S
S2
S2
N
2
i j 1 S
S2
N
j
k
j
N
i
2
i 1 S
k
N
i
i 1
i
S2
i 1
N
i
i 1
i i
onto m Ω̂, sums the result over the index q and applies (24), the validity of the following equation is proved with (14)
and (16):
67
∇ m ∑ 6 v 7 dΩ̂ Ω̂ H Ω̂ ϕ Ω̂>=
d Ω̂ Ω̂ ϕ Ω̂ Y=
9 %
9 %
6
6 7
∂
7
(28)
∂ t 9 m ∑% v J = ∇ 9 m ∑% v O K n3 M P = 0 which means that
∂ ρ u
∇ YC 0 (29)
∂t Finally, we use l 2 for constructing a further equation by the projection onto m 2. Taking advantage of (25), (14)
and (16) yields
6 7
6 7
6 7
6 7
∂ m
m
∂ m
m
∂t 9 2 ∑
% v dΩ̂ ϕ Ω̂ = ∇ 9 2 ∑% v dΩ̂ Ω̂ ϕ Ω̂ = ∂ t 9 2 ∑% v n = ∇ 9 2 ∑% v J = 0
∂
∂t
N
m∑ v
i 1
N
i
i
S2
i 1
N
N
i i
i 1
N
i 1
2
N
i
S2
i
i 1
3
2
i 1
2
i
i
i
S2
i
i
N
i
i
S2
i 1
2
N
i i
i 1
3
i i
(30)
48
0.25
n̂q
n
0.3
n̂q 0.25
n
0.2
ñq
n 0.15
0.2
0.15
0.1
RMA-P2 model
0.1
0.05
0
Maxwell distribution
0.05
2
5
8
11
14
17
v
v1
20
0
2
5
8
11
14
17
v
v1
20
FIGURE 2. The initial (left plot) and equilibrium distribution (right plot) of the particle density n̂q in comparison to ñq obtained
from the Maxwell distribution.
_ ^_
ûq x
ux
_ ^_
_ ^_
0.25
ûq x 0.25
ux
0.2
ũq x
ux 0.15
0.2
0.15
0.1
Maxwell distribution
RMA-P2 model
0.1
0.05
0
0.3
0.05
2
5
8
11
14
17
v
v1
20
0
2
5
8
11
14
17
20
v
v1
`
`
FIGURE 3. The initial (left plot) and equilibrium distribution (right plot) of the drift velocity ûq x in comparison to ũq x obtained
from the Maxwell distribution.
and can be written as
∂K
∇ Q 0
(31)
∂t
The equations (26), (28) and (30) are the multigroup versions of conservation laws for mass, momentum and energy.
Since they agree completely with the conservation equations of the original Boltzmann equation, our model can be
regarded as quite realistic in the sense that mass, momentum and energy conservation are described in a correct way.
NUMERICAL RESULTS
In this section, we demonstrate the relaxation of homogeneous gases from an arbitrary initial distribution towards the
equilibrium state by the help of the deduced RMA-P2 model.
Here, we define the speed-knots simply by demanding an equidistant partition of the speed axis:
wi
v1
5
i
1
28 i
1 2 $ N 1 (32)
Since we want to neglect the influence of particles with speeds higher than w N , we set wN to a sufficient large number.
Moreover, we choose delta-distributions as shape functions:
δ v v v i i 1 2 $ N ψi v
vi
i
1
(33)
Because of the definition of the Dirac delta distribution, one can easily prove that these ψ i v match with the conditions
(5) for shape functions. We assume that our gas particles are Maxwell molecules, which lead to constant collision
frequency.
We intend to illustrate the physical behavior of our model by comparing the equilibrium state with a multigroup
representation of the Maxwell distribution. For this purpose, we study the relaxation of a drifting gas with nonvanishing initial shear stress. Thereby, we choose N 21 speed intervals, which are combined to N L 7 macrointervals.
49
a
p
a
1
p
1
0
0.5
−1
0
1
1
2
3
FIGURE 4.
1
2
3
2
3
The initial (left plot) and equilibrium stress tensor
0 1 , 2 b
1
2
3
(right plot).
Q
:
The left plots in Figs. 2, 3 and 4 illustrate the initial distribution of the particle density n̂ q and the drift velocity ûq x
wL q 1 wLq and the initial stress tensor (17). The comparison of these
(18) referring to the macro-intervals IqM
initial conditions with the Maxwell distribution allows us to regard the present initial state as a far from equilibrium
state.
The evolution equations for the quantities ni , Ji and i are numerically solved. After a sufficient long time, the
variations of these quantities vanish and an equilibrium state is reached.
The right plots in Figs. 2 and 3 show the particle densities n̂q and the drift velocities ûq in the equilibrium state,
compared with ñq and ũq from the Maxwell distribution via (19). The necessary parameters for the Maxwellian are
obtained from the initial conditions of ni , Ji and i by (16) and (17). The figures illustrate the agreement of the
equilibrium solution of the RMA-P2 model with the Maxwell distribution. The particle densities and drift velocities
match within the range of numerical accuracy with the results obtained by the integration of the Maxwellian. The
right plot in Fig. 4 shows the final pressure tensor. The initial shear stress components vanish, while the diagonal
components of the stress tensor even and are equal to the pressure in the regarded gas.
The most important properties of our RMA-P2 model are the exact conservation of particle density, drift velocity and
energy density combined with great flexibility in the partition of the speed axis and the choice of the shape functions.
Thus, the RMA-P2 model is a very suitable tool to describe the properties of rarefied gases.
K
K
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