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Energy conservation and H-theorem in the scalar nonlinear Boltzmann equation and in its multigroup representation
Volume 148, number 3,4
PHYSICS LETFERS A
13 August 1990
Energy conservation and H-theorem in the scalar nonlinear Boltzmann equation and in its multigroup representation Georg Kugerl and Ferdinand SchUrrer Instituze for Theoretical Physics, Graz University of Technology, Petersgasse 16, A-8010 Graz, Austria Received 12 February 1990; revised manuscript received 29 May 1990; accepted for publication 7 June 1990
Communicated by A.R. Bishop
A multigroup representation of the scalar nonlinear Boltzmann equation is given. In analogy to the properties ofa well-formulated scattering kernel of the Boltzmann equation, the symmetries of the scattering coefficients, governing energy conservation and the H-theorem in the multigroup model, are discussed and illustrated by twoexamples.
1. In a recent paper [1] a moment method for solving the Boltzmann equation was proposed, which forms, as we expect, an ideal base for a multigroup representation of space-dependent nonlinear transport problems. However, there are still some open questions in this ~~p~/M method” regarding the conservation laws and the H-theorem. For practical applications the underlying series expansion of the scattering kernel must be truncated, and therefore fundamental properties of the Boltzmann equation might be lost. Thus, before applying the multigroup method, one has to specify those basic properties of the Boltzmann equation and determine how they affect the multigroup formalism. In the present paper we are concerned with a multigroup formulation of the nonlinear Boltzmann equation, in which we restrict ourselves to spatially uniform and isotropic gases. With this simplification the P0 approximation is adequate. The resulting moment equation of zero order is equivalent to the scalar Boltzmann equation. In section 2 we discuss the basic symmetries of the scattering kernel of this scalar kinetic equation, and we give two examples of fundamental interaction laws. In section 3 the multigroup method is introduced and some of its essential difficulties are illuminated. We analyse the symmetries of the scattering coefficients of the multigroup equations. These symmetries are shown to guarantee that the solutions of the multigroup equations are (i) energy conserving and (ii) tend toward a discretized Maxwellian (H-theorem). Finally, in section 4 numerical solutions are presented, in order to demonstrate the quality of the proposed multigroup method. 2. For the simplified geometrical situation considered here, the temporal evolution of the normalized energy distribution function F(x, t) is governed by the scalar Boltzmann equation [2], t)=
Jdx~Jd%P(x~,x’2xi)F~F’2—Ft$~2F2$dx~P(xi,x2x~).
(1)
2/2kT, and time t is suitably measured in units of the mean The dimensionless energy variable x stands for mv collision time. The scattering kernel P(x’ 1, x~x1) represents the transition rate for (xi, x) (x1, x + x’2 x1). Conservation of energy implies that P vanishes if x~+x~
—
158
0375-9601/90/S 03.50 © 1990
—
Elsevier Science Publishers B.V. (North-Holland)
Volume 148, number 3,4
PHYSICS LETTERS A
13 August 1990
In ref. [3] a method for deriving the scalar equation (1) from the original Boltzmann equation is given. In order to make the following symmetry properties of P transparent, we quote the results valid for two fundamental interaction laws. For instance, the scalar scattering kernel reads in the hard-sphere model /
P(x~~x’2xi)=( ~
\1/2 ,
Min(x’i~x~~xi~x’i+x’2_xi)) ø(x’1+x~—x1),
(2)
8x1 x2 and in the Krook—Wu model P(x~,x~x1)=(—~_~)arcsin(M~~~1~ x.,x1,x’1 +x’2 _x1)) ,
O(x~+x~—x1),
(3)
where 0 denotes Heaviside’s step function. In discussing the fundamental properties of a well-formulatedscattering kernel ofthe scalar Boltzmann equation we begin with the symmetry P(x~,x~x1)=P(x’2,x~x~).
(4)
This relation must not be interpreted as resulting from the “identity of the particles”. In fact, it is valid only for a class of differential cross sections [2], which includes isotropic scattering in the centre-of-mass system. However, due to the quadratic structure ofthe collision integrals in eq. (1), one can always find a symmetrized P such that eq. (4) is satisfied. The kernel P should further satisf~’the symmetry P(x~,x’2x1)=P(x~,x~x~+x’2—x1).
(5)
This interactional symmetry is suggested by energy conservation during a binary collision. Eq. (5) can be shown to be a sufficient and necessary condition for proving that the Boltzmann equation (1) is energy conserving. Next we analyse the existence of an equilibrium solution of eq. (1). To do this, we bear in mind that the six-dimensional scattering probability W describing the binary collision (vi, ‘4) (v1, v2) is the same for the forward and reverse directions [4], —~
W(vvj;v~—~v2)=W(v1---~v’j;v2—~.’4).
(6)
The corresponding symmetry of the scalar scattering kernel P will in general not have the simple form P(x~,x~x1 ) =P(x1, x~+x’2 —x1 xi), because in deriving P from W, in addition to a reduction with respect to some variables, a transformation from velocity representation to energy representation has to be made. The Jacobian of this transformation, 0(x), is introduced as a factor, so that P must show the symmetry Ø(x~)Ø(x’2)P(x~,x~x~)=O(x~)Ø(x’~+x’~—x1)P(x1, x’1 +x’~—xi; x’s) (7) For a three-dimensional gas we have 0(x) ~ ~/i Obviously the two examples given by eqs. (2) and (3) satisfy eq. (7). Note also that, as a result of eq. (4), the validity ofthe symmetry (7) implies the interactional symmetry (5). In order to examine if the solution of eq. (1) tends toward a steady state and to determine this equilibrium solution, we define an H-function by .
H(t)= JdxF(x,t)log(’~”~).
(8)
Using the Boltzmann equation (1) and the symmetries (4) and (7) one finds
159
Volume 148, number 3,4
PHYSICS LETTERS A
~5=~
Jc~x~ JdX~~
xl
13 August 1990
t)
—
F(x~,t)F(x’~±x~x:, t))
(F(xi, t)F(x’1 +x’~—x~, O(x’1)O(x’2) t))~ ‘\ og~ Ø(x1 )Ø(x~ +x~—xi) t) F(x~,t)F(x~,
(9)
It follows from eq. (9) that (10) where equality is valid only if the system has reached a steady state M(x) (H-theorem), which satisfies M’(x’~)M(x’~) M(x1 )M~(x~ +x’2 —xi)
(11)
—
0(x~)0(x~)
—
As can be seen from eq. (11), log(M/Ø) is a collisional invariant. Hence it must be a linear combination of the two independent summational invariants, 1 and x. For 0(x) x ..fx we find after the normalization of the number density to unity and energy density to ~,
M(x)= ~=~/~exp(—x).
(12)
3. The first step in the development of the multigroup method is to divide the molecular energy range into G intervals (“groups”) separated by the energies x0 = 0, x1, x2, XG. After integrating the Boltzmann equation (1) over the jth energy interval, the temporal evolution of the group density ...,
F(t)=
J
dxF(x,t)
(13)
xi_ I
is found to be governed by the multigroup equations dF dt —=
G
G
k=I!=I
G
f—I
~ A~,1F,~F1—F1 ~
G
~ A~1, i=l,2,...,G,
(14)
k=l
where the scattering coefficients are defined by
~
(15)
They represent the probability per unit time and unit volume, that a test particle of group k interacting with a target particle of group I is transferred to group i, disregarding the transfer of the target particle. In deriving eq. (14) we assumed xG to be large enough, so that in the collision integrals of eq. (1) the contributions originating from the interval (0, xG) dominate over contributions from (XG, ~). An essential difficulty in the multigroup formalism arises fromthe factthat eq. (15) involves the distribution function. Hence, in principle, the transport problem must have been solved in order to determine the scattering coefficients. This inadequacy can be avoided if we replace the integrals in eqs. (13) and (15) by the quadrature formula of first degree the “rectangular rule”: —
160
Volume 148, number 3,4
PHYSICS LETTERS A
13 August 1990
(16)
f(x)~(~(a+b))(b_a).
Using equidistant energy groups of width &, and defining (17) we find by means of this approximation for the group densities F,.(t)=F(~1,t)&,
(18)
and for the scattering coefficients (19)
A~,,=P(~k,X,;2I)&.
Energy conservation during a binary collision implies in the case of equidistant groups that the target particle is transferred to group k+ i—i. To keep this index well-defined we assume Ai1,=0, if k+l—i< 1, (20)
ifk+I—i>G.
In order to give examples for the scattering coefficients under the present approximation, we use eq. (2), and eq. (3), respectively, in eq. (15). In the hard-sphere model this results in 1/2
A~.1=[g(k’~/~) Min(k—~,l—~,i—i, k+1_i_~)] 0(k+I—i—~)0(G+i—k—I+~) .
(21)
In the Krook—Wu model one obtains A~1((k~(/~
)
2o(k+1il)0(G+ikl+~)
~
(22)
I—~,i—~, k+l_i_~))”
In both cases a second 0 term was added in order to fulfill the requirement of eq. (20). As illustrated by eqs. (21) and (22), the rectangular rule transforms the symmetries of a well-formulated scattering kernel into the multigroup model. In analogy to eqs. (4), (5), and (7) we find Al _A’ 23 k.1—
I.k,
(24) 0k01A~,l=0l0k+,_lA~k+,_l,
(25)
with
Also in the multigroup model the last of these basic symmetries, eq. (25), implies the interactional symmetry (24), because of eq. (23). Concerning the conservation laws in the multigroup model we first note that the normalization constant G
N= ~ F ,~ I
1(t)
(26)
is indeed constant, aresult which can be derivedimmediately from eq. (14). In accordance with the rectangular rule we define energy by 161
Volume 148, number 3,4
PHYSICS LETTERS A
13 August 1990
E=&~ (i—~)F1(t)
(27)
.
Multiplying eq. (14) with (i— ~)& and summing over i we find for the time derivative of E, after some transformations, dE GGG =~L~x ~ ~ (A~t’’—A~,)(k+I—i—~)FkFl, dt k=I 1=1 i=1
(28)
—
which shows that, in the present case of equidistant groups, the interactional symmetry (24) is a sufficient and necessary condition for energy conservation. In order to examine the existence of an equilibrium state in the multigroup model we define an H-function by H(t)= ~
Fk(t)
log(~~)).
(29)
Using eqs. (14), (23), and (25) we find after some algebra dJJ
G
—
G
G (FkFI FIFk÷/I\ (FIFk+II ~ A~Ø~Ø~—— og1 \OkO! OiOk+1—i/ \OIOk+I—j
i~I k=1 1=1
OkOl’\
~,
~ kFl,
(30)
so that
~
(31)
Eq. (31) proves that the solution of eq. (14) approaches a steady state (M1) (H-theorem), which is characterized by MkM,
—
MIMk+,l
OkO!
—
OIOk+l—i
(32)
The quantity log(M1/Ø,) is a collisional invariant, so that it must be a linear combination of the two independent summational invariants 1 and (i— ~)&. One obtains M1=AO1exp[—B(i—~)&].
(33)
In the present case, Ø1cx (i ~)I/2 the equilibrium solution takes the form of a discretized Maxwellian 2exp[—(i—~)fi~.x](flzS.x)3”2. (34) M1= (i—~)~ The parameters a and fi are determined by the normalization constant and energy of the initial state F 1(0), given by eqs. (26) and (27). In the terminology used in the theory of nonlinear differential equations [5], H(F~( t)) defined by eq. (29) is a Liapunov function for the finite dimensional system (14), provided the scattering coefficients are wellformulated, in the sense that they obey the symmetry (25). The equilibrium solution (Mk) is asymptotically stable. 4. Finally we present numerical solutions to the proposed multigroup model. As a test-case we choose the 8KW mode FBKW(x,t)=
162
~~512exP(_x/K)(5K_3+2~~x),
(35)
Volume 148, number 3,4 R 1.2
PHYSICS LETTERS A
~
0.4
I~~” 20
30
50 40
0
13 August 1990
0
4
8
12
16
20
24
28
Fig. 1. Relative distribution function R(x, t) =F(x, t)/F(x, 30) versus time t for various energies x. Solid curves: BKW mode; dashed curves: multigroup solutions (x 0=50, G= 128).
where K( t) = 1— ~exp ( t/6). Eq. (35) is an exact solution of the nonlinear Boltzmann equation (1) with the scattering kernel (3) [61. To obtain the appropriate initial values we discretize eq. (35) at time t= 0, according to eq. (18). The multigroupequations (14) with (22) were solved numerically by means of the Runge— Kutta method. A comparison of these multigroup solutions with the exact solution of the Boltzmann equation is shown in fig. 1. An excellent agreement is evident in the whole energy spectral range. This conformity is remarkable, especially if one remembers the smallness of the distribution function for high particle energies. —
5. The proposed multigroup method does not only have the nice property’that it guarantees energy conservation and the H-theorem, but the resulting numerical solutions are also in excellent agreement with an exact solution of the nonlinear Boltzmann equation. It is indispensable for any numerical method that it preserves the basic properties of the Boltzmann equation. Otherwise there is a tendency for small errors to accumulate and therefore the numerical solution becomes unstable in the course oftime. For instance, energy conservation is not fulfilled in the proposed formalism, if one uses the scalar velocity representation for the Boltzmann equation instead of the energy representation. Indeed we have found by numerical experiments performed in velocity representation, that the multigroup solutions become unstable within only a few mean collision times. We point out that the proposed quadrature procedure for the Boltzmann equation does not require any artifices in order to stabilize the numerical solution. In the methods introduced by Tjon and Wu [7] and by Weller and Weller [8], for instance, it was necessary to renormalize the solution after each time step. Though such a trick is expedient, it is a rather arbitrary action. We conclude that the experiences outlined in this paper are not only essential for the numerical solution of the transport equation for the special case ofhomogeneous and isotropic gases. Without substantial alterations, the multigroup method can also be applied to the case of gas-mixtures which in the frame of the scalar Boltzmann equation has only been touched vaguely in the past. Moreover, in conjunction with the PN~P’method [1] it is a first step toward space-dependent nonlinear transport problems. This work was supported by the Fonds zur Forderung der wissenschaftlichen Forschung, Vienna, under contract No. P7008-PHY. References [I ] G. Ktigerl and F. Schurrer, Phys. Rev. A 39 (1989)1429. [2] M.H. Ernst, Phys. Rep. 78(1981)1. [3] 0. Ktigerl, J. AppL Math. Phys. (ZAMP) 40 (1989) 816. [4] M.M.R. Williams, Mathematical methods in particle transport theory (Butterworths, London, 1971). [5] H. Hochstadt, Differential equations (Dover, NewYork, 1975). [6] R.M. Ziff, Ph~.Rev. Lett. 45 (1980) 306. [7]J. TjonandT.T. Wu, Ph~.Rev.A 19(1979)883. [8] R.A. Wellerand M.R. Weller, Radiat. Eff~60 (1982) 209.
163
PHYSICS LETFERS A
13 August 1990
Energy conservation and H-theorem in the scalar nonlinear Boltzmann equation and in its multigroup representation Georg Kugerl and Ferdinand SchUrrer Instituze for Theoretical Physics, Graz University of Technology, Petersgasse 16, A-8010 Graz, Austria Received 12 February 1990; revised manuscript received 29 May 1990; accepted for publication 7 June 1990
Communicated by A.R. Bishop
A multigroup representation of the scalar nonlinear Boltzmann equation is given. In analogy to the properties ofa well-formulated scattering kernel of the Boltzmann equation, the symmetries of the scattering coefficients, governing energy conservation and the H-theorem in the multigroup model, are discussed and illustrated by twoexamples.
1. In a recent paper [1] a moment method for solving the Boltzmann equation was proposed, which forms, as we expect, an ideal base for a multigroup representation of space-dependent nonlinear transport problems. However, there are still some open questions in this ~~p~/M method” regarding the conservation laws and the H-theorem. For practical applications the underlying series expansion of the scattering kernel must be truncated, and therefore fundamental properties of the Boltzmann equation might be lost. Thus, before applying the multigroup method, one has to specify those basic properties of the Boltzmann equation and determine how they affect the multigroup formalism. In the present paper we are concerned with a multigroup formulation of the nonlinear Boltzmann equation, in which we restrict ourselves to spatially uniform and isotropic gases. With this simplification the P0 approximation is adequate. The resulting moment equation of zero order is equivalent to the scalar Boltzmann equation. In section 2 we discuss the basic symmetries of the scattering kernel of this scalar kinetic equation, and we give two examples of fundamental interaction laws. In section 3 the multigroup method is introduced and some of its essential difficulties are illuminated. We analyse the symmetries of the scattering coefficients of the multigroup equations. These symmetries are shown to guarantee that the solutions of the multigroup equations are (i) energy conserving and (ii) tend toward a discretized Maxwellian (H-theorem). Finally, in section 4 numerical solutions are presented, in order to demonstrate the quality of the proposed multigroup method. 2. For the simplified geometrical situation considered here, the temporal evolution of the normalized energy distribution function F(x, t) is governed by the scalar Boltzmann equation [2], t)=
Jdx~Jd%P(x~,x’2xi)F~F’2—Ft$~2F2$dx~P(xi,x2x~).
(1)
2/2kT, and time t is suitably measured in units of the mean The dimensionless energy variable x stands for mv collision time. The scattering kernel P(x’ 1, x~x1) represents the transition rate for (xi, x) (x1, x + x’2 x1). Conservation of energy implies that P vanishes if x~+x~
—
158
0375-9601/90/S 03.50 © 1990
—
Elsevier Science Publishers B.V. (North-Holland)
Volume 148, number 3,4
PHYSICS LETTERS A
13 August 1990
In ref. [3] a method for deriving the scalar equation (1) from the original Boltzmann equation is given. In order to make the following symmetry properties of P transparent, we quote the results valid for two fundamental interaction laws. For instance, the scalar scattering kernel reads in the hard-sphere model /
P(x~~x’2xi)=( ~
\1/2 ,
Min(x’i~x~~xi~x’i+x’2_xi)) ø(x’1+x~—x1),
(2)
8x1 x2 and in the Krook—Wu model P(x~,x~x1)=(—~_~)arcsin(M~~~1~ x.,x1,x’1 +x’2 _x1)) ,
O(x~+x~—x1),
(3)
where 0 denotes Heaviside’s step function. In discussing the fundamental properties of a well-formulatedscattering kernel ofthe scalar Boltzmann equation we begin with the symmetry P(x~,x~x1)=P(x’2,x~x~).
(4)
This relation must not be interpreted as resulting from the “identity of the particles”. In fact, it is valid only for a class of differential cross sections [2], which includes isotropic scattering in the centre-of-mass system. However, due to the quadratic structure ofthe collision integrals in eq. (1), one can always find a symmetrized P such that eq. (4) is satisfied. The kernel P should further satisf~’the symmetry P(x~,x’2x1)=P(x~,x~x~+x’2—x1).
(5)
This interactional symmetry is suggested by energy conservation during a binary collision. Eq. (5) can be shown to be a sufficient and necessary condition for proving that the Boltzmann equation (1) is energy conserving. Next we analyse the existence of an equilibrium solution of eq. (1). To do this, we bear in mind that the six-dimensional scattering probability W describing the binary collision (vi, ‘4) (v1, v2) is the same for the forward and reverse directions [4], —~
W(vvj;v~—~v2)=W(v1---~v’j;v2—~.’4).
(6)
The corresponding symmetry of the scalar scattering kernel P will in general not have the simple form P(x~,x~x1 ) =P(x1, x~+x’2 —x1 xi), because in deriving P from W, in addition to a reduction with respect to some variables, a transformation from velocity representation to energy representation has to be made. The Jacobian of this transformation, 0(x), is introduced as a factor, so that P must show the symmetry Ø(x~)Ø(x’2)P(x~,x~x~)=O(x~)Ø(x’~+x’~—x1)P(x1, x’1 +x’~—xi; x’s) (7) For a three-dimensional gas we have 0(x) ~ ~/i Obviously the two examples given by eqs. (2) and (3) satisfy eq. (7). Note also that, as a result of eq. (4), the validity ofthe symmetry (7) implies the interactional symmetry (5). In order to examine if the solution of eq. (1) tends toward a steady state and to determine this equilibrium solution, we define an H-function by .
H(t)= JdxF(x,t)log(’~”~).
(8)
Using the Boltzmann equation (1) and the symmetries (4) and (7) one finds
159
Volume 148, number 3,4
PHYSICS LETTERS A
~5=~
Jc~x~ JdX~~
xl
13 August 1990
t)
—
F(x~,t)F(x’~±x~x:, t))
(F(xi, t)F(x’1 +x’~—x~, O(x’1)O(x’2) t))~ ‘\ og~ Ø(x1 )Ø(x~ +x~—xi) t) F(x~,t)F(x~,
(9)
It follows from eq. (9) that (10) where equality is valid only if the system has reached a steady state M(x) (H-theorem), which satisfies M’(x’~)M(x’~) M(x1 )M~(x~ +x’2 —xi)
(11)
—
0(x~)0(x~)
—
As can be seen from eq. (11), log(M/Ø) is a collisional invariant. Hence it must be a linear combination of the two independent summational invariants, 1 and x. For 0(x) x ..fx we find after the normalization of the number density to unity and energy density to ~,
M(x)= ~=~/~exp(—x).
(12)
3. The first step in the development of the multigroup method is to divide the molecular energy range into G intervals (“groups”) separated by the energies x0 = 0, x1, x2, XG. After integrating the Boltzmann equation (1) over the jth energy interval, the temporal evolution of the group density ...,
F(t)=
J
dxF(x,t)
(13)
xi_ I
is found to be governed by the multigroup equations dF dt —=
G
G
k=I!=I
G
f—I
~ A~,1F,~F1—F1 ~
G
~ A~1, i=l,2,...,G,
(14)
k=l
where the scattering coefficients are defined by
~
(15)
They represent the probability per unit time and unit volume, that a test particle of group k interacting with a target particle of group I is transferred to group i, disregarding the transfer of the target particle. In deriving eq. (14) we assumed xG to be large enough, so that in the collision integrals of eq. (1) the contributions originating from the interval (0, xG) dominate over contributions from (XG, ~). An essential difficulty in the multigroup formalism arises fromthe factthat eq. (15) involves the distribution function. Hence, in principle, the transport problem must have been solved in order to determine the scattering coefficients. This inadequacy can be avoided if we replace the integrals in eqs. (13) and (15) by the quadrature formula of first degree the “rectangular rule”: —
160
Volume 148, number 3,4
PHYSICS LETTERS A
13 August 1990
(16)
f(x)~(~(a+b))(b_a).
Using equidistant energy groups of width &, and defining (17) we find by means of this approximation for the group densities F,.(t)=F(~1,t)&,
(18)
and for the scattering coefficients (19)
A~,,=P(~k,X,;2I)&.
Energy conservation during a binary collision implies in the case of equidistant groups that the target particle is transferred to group k+ i—i. To keep this index well-defined we assume Ai1,=0, if k+l—i< 1, (20)
ifk+I—i>G.
In order to give examples for the scattering coefficients under the present approximation, we use eq. (2), and eq. (3), respectively, in eq. (15). In the hard-sphere model this results in 1/2
A~.1=[g(k’~/~) Min(k—~,l—~,i—i, k+1_i_~)] 0(k+I—i—~)0(G+i—k—I+~) .
(21)
In the Krook—Wu model one obtains A~1((k~(/~
)
2o(k+1il)0(G+ikl+~)
~
(22)
I—~,i—~, k+l_i_~))”
In both cases a second 0 term was added in order to fulfill the requirement of eq. (20). As illustrated by eqs. (21) and (22), the rectangular rule transforms the symmetries of a well-formulated scattering kernel into the multigroup model. In analogy to eqs. (4), (5), and (7) we find Al _A’ 23 k.1—
I.k,
(24) 0k01A~,l=0l0k+,_lA~k+,_l,
(25)
with
Also in the multigroup model the last of these basic symmetries, eq. (25), implies the interactional symmetry (24), because of eq. (23). Concerning the conservation laws in the multigroup model we first note that the normalization constant G
N= ~ F ,~ I
1(t)
(26)
is indeed constant, aresult which can be derivedimmediately from eq. (14). In accordance with the rectangular rule we define energy by 161
Volume 148, number 3,4
PHYSICS LETTERS A
13 August 1990
E=&~ (i—~)F1(t)
(27)
.
Multiplying eq. (14) with (i— ~)& and summing over i we find for the time derivative of E, after some transformations, dE GGG =~L~x ~ ~ (A~t’’—A~,)(k+I—i—~)FkFl, dt k=I 1=1 i=1
(28)
—
which shows that, in the present case of equidistant groups, the interactional symmetry (24) is a sufficient and necessary condition for energy conservation. In order to examine the existence of an equilibrium state in the multigroup model we define an H-function by H(t)= ~
Fk(t)
log(~~)).
(29)
Using eqs. (14), (23), and (25) we find after some algebra dJJ
G
—
G
G (FkFI FIFk÷/I\ (FIFk+II ~ A~Ø~Ø~—— og1 \OkO! OiOk+1—i/ \OIOk+I—j
i~I k=1 1=1
OkOl’\
~,
~ kFl,
(30)
so that
~
(31)
Eq. (31) proves that the solution of eq. (14) approaches a steady state (M1) (H-theorem), which is characterized by MkM,
—
MIMk+,l
OkO!
—
OIOk+l—i
(32)
The quantity log(M1/Ø,) is a collisional invariant, so that it must be a linear combination of the two independent summational invariants 1 and (i— ~)&. One obtains M1=AO1exp[—B(i—~)&].
(33)
In the present case, Ø1cx (i ~)I/2 the equilibrium solution takes the form of a discretized Maxwellian 2exp[—(i—~)fi~.x](flzS.x)3”2. (34) M1= (i—~)~ The parameters a and fi are determined by the normalization constant and energy of the initial state F 1(0), given by eqs. (26) and (27). In the terminology used in the theory of nonlinear differential equations [5], H(F~( t)) defined by eq. (29) is a Liapunov function for the finite dimensional system (14), provided the scattering coefficients are wellformulated, in the sense that they obey the symmetry (25). The equilibrium solution (Mk) is asymptotically stable. 4. Finally we present numerical solutions to the proposed multigroup model. As a test-case we choose the 8KW mode FBKW(x,t)=
162
~~512exP(_x/K)(5K_3+2~~x),
(35)
Volume 148, number 3,4 R 1.2
PHYSICS LETTERS A
~
0.4
I~~” 20
30
50 40
0
13 August 1990
0
4
8
12
16
20
24
28
Fig. 1. Relative distribution function R(x, t) =F(x, t)/F(x, 30) versus time t for various energies x. Solid curves: BKW mode; dashed curves: multigroup solutions (x 0=50, G= 128).
where K( t) = 1— ~exp ( t/6). Eq. (35) is an exact solution of the nonlinear Boltzmann equation (1) with the scattering kernel (3) [61. To obtain the appropriate initial values we discretize eq. (35) at time t= 0, according to eq. (18). The multigroupequations (14) with (22) were solved numerically by means of the Runge— Kutta method. A comparison of these multigroup solutions with the exact solution of the Boltzmann equation is shown in fig. 1. An excellent agreement is evident in the whole energy spectral range. This conformity is remarkable, especially if one remembers the smallness of the distribution function for high particle energies. —
5. The proposed multigroup method does not only have the nice property’that it guarantees energy conservation and the H-theorem, but the resulting numerical solutions are also in excellent agreement with an exact solution of the nonlinear Boltzmann equation. It is indispensable for any numerical method that it preserves the basic properties of the Boltzmann equation. Otherwise there is a tendency for small errors to accumulate and therefore the numerical solution becomes unstable in the course oftime. For instance, energy conservation is not fulfilled in the proposed formalism, if one uses the scalar velocity representation for the Boltzmann equation instead of the energy representation. Indeed we have found by numerical experiments performed in velocity representation, that the multigroup solutions become unstable within only a few mean collision times. We point out that the proposed quadrature procedure for the Boltzmann equation does not require any artifices in order to stabilize the numerical solution. In the methods introduced by Tjon and Wu [7] and by Weller and Weller [8], for instance, it was necessary to renormalize the solution after each time step. Though such a trick is expedient, it is a rather arbitrary action. We conclude that the experiences outlined in this paper are not only essential for the numerical solution of the transport equation for the special case ofhomogeneous and isotropic gases. Without substantial alterations, the multigroup method can also be applied to the case of gas-mixtures which in the frame of the scalar Boltzmann equation has only been touched vaguely in the past. Moreover, in conjunction with the PN~P’method [1] it is a first step toward space-dependent nonlinear transport problems. This work was supported by the Fonds zur Forderung der wissenschaftlichen Forschung, Vienna, under contract No. P7008-PHY. References [I ] G. Ktigerl and F. Schurrer, Phys. Rev. A 39 (1989)1429. [2] M.H. Ernst, Phys. Rep. 78(1981)1. [3] 0. Ktigerl, J. AppL Math. Phys. (ZAMP) 40 (1989) 816. [4] M.M.R. Williams, Mathematical methods in particle transport theory (Butterworths, London, 1971). [5] H. Hochstadt, Differential equations (Dover, NewYork, 1975). [6] R.M. Ziff, Ph~.Rev. Lett. 45 (1980) 306. [7]J. TjonandT.T. Wu, Ph~.Rev.A 19(1979)883. [8] R.A. Wellerand M.R. Weller, Radiat. Eff~60 (1982) 209.
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