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A completely conservative scheme for Landau's equation (Rosenbluth non-isotropic potentials)
Short communications
269
Notice in conclusion that similar results can be obtained for the solutions of Eq. (1) whose fronts are at infinity, if we seek the solution as a series in decreasing negative powers of x. The authors sincerely thank S. P. Kurdyumov
for useful discussions. Translated
by
D. E. Brown
REFERENCES 1. ZMITRENKO, N. V., er al., Mets-stable and conditions for its appearance 1977.
localization of heat in a medium with non-linear heat conduction in an experiment, IPM Preprint, Akad. Nauk SSSR, No. 103, Moscow,
2. VAINBERG, M. M. and TRENOGIN, V. A., The theory of branching of solutions of non-linear equations (Teorfya vetvleniya reshenii nelineinykh uravnenii), Nauka, Moscow, 1969.
3. KALASHNIKOV, A. S., On the nature of the propagation of disturbances in processes linear degenerate
parabolic
equations?
Tr. seminam im. I. G. Perrovskogo,
4. MARTINSON, L. K., Ropagation of shear disturbances zhidkostigaza, No. 6, p. 60, 1978.
in dilatant
U.S.S.R. Compur. Maths. Marh. Phys. Vol. 22, No. 3, pp. 269-274. Printed in Great Britain
fluids,Izu.
1982.
described No. 1, 133-144,
by quasi1975.
Akad. Nauk SSSR, Mekhan.
0041-5553/82S07.50+.00 01983. Pergamon Press Ltd.
A COMPLETELY CONSERVATIVE SCHEME FOR LANDAU’S EQUATION (ROSENBLUTH NON-ISOTROPIC POTENTIALS)* I. F. POTETSENKO and V. A. CHUYANOV Moscow (Received
15 Ma)
1980)
A DIFFERENCE scheme, preserving the number of particles and the energy, is constructed for the two-dimensional Landau equation with Rosenbluth non-isotropic potentials, whereby nonstationary problems can be solved by computer without error accumulation. Several (in general, five) laws of conservation are satisfied for the non-linear Landau kinetic equation. If the difference scheme has only approximate analogues of the conservation laws, this can easily lead to error accumulation when computing non-stationary problems. Hence the principle of complete conservatism, put forward in [l] in connection with problems of gas dynamics and magnetohydrodynamics, proves to be unusually important for problems of rarefied gas and plasma dynamics [2-51. When solving the kinetic equation numerically by computer, we replace the continuous medium by a discrete model (difference scheme), and meantime have to require that the model reflect the main features of the process considered, including the laws of conservation (61. Difference schemes for the two-dimensional (in velocity space) Landau equation (the Fokker-Planck equation), retaining the number of particles, were constructed in [7-91: but *Zh. vychisl. Mar. mar. Fiz., 22, 3. 751-756.
1982.
I. F. Potetsenko and V. A. Chuyanov
270
in computations on actual meshes, such difference schemes may distort the effect in question, due to the presence of fictitious energy sources in them. By developing the methods of [2-51 we can construct a completely conservative scheme for the two-dimensional Landau equation with Rosenbluth non-isotropic potentials [lo] , i.e. a scheme which preserves the number of particles and the energy. When studying the behaviour of a plasma in open magnetic traps, the multicomponent equation [7] is of practical value; in the present paper we consider for simplicity the Landau equation for one sort of particle, in the absence of sinks and sources (e.g. ions in a neutralizing background). We take the Landau equation, written in divergence form: m2
af
4sez,i
at
--=_
1 aa(v,p) 29
au
1 a +,,j-pPw(w)l,
.I 62g___
D(L+~Lf+(l_p~) (
P(r.p)=
+
i-112 azg 1 ag +_-----
(
a% -_-ac akL (
c au
62 BP2 i
a~ aF
af v2 ak ) ap 1
ag
__v2
$f,
ag
_af L.? ap > ap P
ag
af ah --_--f 1.6~ ) au apL’
OGVC55,
-lGpG’i,
DO,
dv’f(v’, 1)jv-\.‘I, h(v, cl)= dv’f(v’, t) x Iv-v’l-i, with Rosenbluth potentials g (V. P) = s s the initial condition .f(c, P. 0) =,fO(~,~1 , and the condition that the distribution function be bounded at 12= 0, p = 5 1. As v + =, the distribution function tends to zero. We shall use dimensionless velocity! time, and distribution function [2.8] below, while retaining the previous notation. The first stage in constructing the completely conservative difference scheme is to write Eqs. (1) in a form that takes better account of the two conservation laws. To reduce (1) to the required form, we use the obvious properties of the Rosenbluth potentials: Ag=h.
Ah=-SSf?
(2)
Then, in the notation
Eq. (1) takes the form (3) where L:(r-, p) =
aG(L’,IJ) au
+
v2f(v. p, t)H(v, p).
271
Short communications
The conservation laws for the number of particles and the energy are dn -=dt The
JJ
dml dt
dE =; dt
!(I.. p, t)v’ dr dp=O.
0 -1
da’
p. JJ f(v.
t) v( dv dp=O.
0 --I
unique stationary solution of Maxwell’s type is fy=n
($)’
(4)
erp ( -$vz).
Notice that the second conservation law is connected with the satisfaction of the equation m
i
JJ
=‘iG
JJ
C(v,. p)dv dp=
-
0 --I
0 --I
since
dc
JJ
(5)
fv=H dv dp=O,
0 -1
(v’_ p’, JJ v’?f
m
H(r, \I)=
-’
dv dp+
1
t) K(v, v’) dv’ dp’,
0 -i
where
K(V,
Y’)
=?(c?-~‘2)
is the antisymmetric “kernel.”
Iv-~‘(-3
To compose the difference scheme, we replace the infinite interval by the finite interval [0, L] , which is chosen so as to take into account the high-energy particles. The order of decrease of the distribution function can be estimated from distribution (4). In the domain of space considered we introduce the space-time mesh 0
o=OhXO~=(v~+i=L.hi-lik-,~ j=i,Z
=!_Lj+hr+*,
We define the mesh functions pj)t
V(Ck.
pj).
,iV-i,
JLi=-1,
. . ,.?‘-1, kx=i,
Vi=O,
vN=L,
pj+$
tn+l
n=O. 1, *. . ( P=O}.
=P+‘l,
U(L’k.
,...
Ii=l,Z..
.
.
fkjn?
L'kj-
1;;.
...
corresponding to the functions
f
(
vk.
pj,
t”).
We shall also use the notation
hb=O.5(h+,+h~).
~k_,;=O.j(~ri!frk).
hlv=0.5hz’,
The second stage in constructing the completely conservative difference scheme consists in applying the integro-interpolation method of [ 1 I] to Eq. (3). As a result we obtain the difference scheme f,=L,!+Lz!,
fkj’“fO; Uk.L’,‘: j --uk;“. ,.
uk-
(2 j
Vk-I.‘>
(6) k=2.
3.. . . ..1’. j=2. 3.. . . Jf-1.
I. F. Potetsenko and V. A. Chyanov
272
The scheme is written in the form of a conservation law for the number of particles, while the approximation of the function uk+l/3,j has to ensure satisfaction of the energy conservation law. On approximating the integrals by the trapezoid formula, and the derivatives by the central differences, we obtain
where
Kw,;, j, et-‘,;,
i=-Km;,.;,
i,
r;,i, j.
From this expression we obtain the mesh analogue of Eq. (5):
ensuring satisfaction of the energy conservation law for difference scheme (6), In fact, the difference analogues of the conservation laws are obtained by summation over the mesh of Eqs. (6). The variations over a time step of the number of particles and of the energy are
L..q
v~+i!Ji+~,j+v_W2fNj
+-
Hence
us-
2
Hx+j;;.j
1 .
it is clear that we can arrange for the conservation laws to be satisfied imposing the respective boundary conditions: L’3‘1,j=o.
U.Y,.. ,=C’.
G.Y+,. ;=Gsj=@,
pjEO!,.
f,?EO7.
Notice
that, at the right-hand end vn: = L we can impose the simpler boundary condition ; then the degree to which the energy is not conserved is proportional to tJ’=or. the rounding errors. fxl=O,
pj=~,
Scheme (6) is constructed on a symmetric pattern, and has the second order of approximation with respect to the velocity space, and the first order with respect to time. The divergence properties of the scheme enable higher order of approximation with respect to time to be obtained. For the computations, the methods of variable directions, of stabilizing correction, or of splitting [ 11, 121, may be used.
173
Short communications For numerical
computation,
the “kernel” K (v, Y’)is expressed in terms of the complete
elliptic integral of the 2nd kind:
When solving Eq. (3) by computer, the greatest labour lies in computing the functions g (v, cl) and h (v, /.L),including complete elliptic integrals of the 1st and 2nd kinds. Effective methods have now been devised for computing the potentials g (v, p) and h (u, cc)by means of numerical integration or from Eqs. (2) (see [8, 91). In addition, the function V (u, P) of (3) differs from the function p (v, 1) usually considered of (1) in one term, so that it is possible to employ the existing methods of computations (see [8,9] ) for scheme (6). Notice in conclusion the connection between the equation, written in form (3), and the expressions previously obtained for the Landau equations [2,4]. We expand the Rosenbluth potentials in Legendre polynomials P,, (p) (see [lo] ):
In the notation
i PI fnct.1 = J f(v,P,t)pn (II)dl.l, in(VI= fn(Y)!I-“+’ dy, J
-ia 0 P~(v,P)= J~(.Y,P.~Y+‘+‘~Y the function U (Y,cl) of (3) takes the form
+2(2nil)
J[
n
h(Y!!(4
0
P)
-E ( L’ )
- In (Y)PTI(~7@)
1
P*
dy
(7)
0
-2(2n+:)
J0
[P~(Y)!(Y.
v!-.fn(~)~m(y.
u)
>
IY”+’&/ .
Notice that, if we introduce the extra condition for symmetry of the distribution function with respect to p = 0, as is usually done in magnetic trap computations, only the terms with even II are retained in expansion (7) so that the law of conservation of momentum is automatically satisfied.
274
I, F. Potetsenko and V. A. Chuyanor
From (7), for n = 0 and an isotropic distribution function, we get t’
L.-,&!-
aL
(s
[P(Y)!(L.)--!(Y)P(~)lY2
dY
0
1
and the equation of [2] :
With n = 0 and a distribution function dependent on v and ~1(isotropic Rosenbluth potentials), we have F CC,,3 b [P,‘(Y)!(L’,~)--!o(Y)P(v,~I)lY* dY r,
-
s F
[Po(Y)!iY.PI _fi’(Y)P (Y. II) IY? dY
1
and the equation of [4], corresponding to one sort of particles. Translated lg. D. E. Brown
REFERENCES 1. POPOV. YU. P.. and SAMARSKII. A. A.. Completei!, conservative difference schemes. Zh. l:Fchisl. Mat. mat. Fir., 9. No. 4. 953-958. 1969 2. BOBYLEV. A. V., and CHUYANOV. V. A.. On the numerical solution of the Landau kinetic equation, Zh. ,:Fchisl. Mat. mat. Fi:., 16, No. 2.407-416, 1976. 3. POTAPENKO. I. F.. and CHUYANOV. V. A.. On completely conservative difference schemes for the system of Landau equations. Zh. r_Fchisl.Mat. mat. Fiz., 19, No. 2.458-463: 1979. 4. POTAPENKO. 1. F.? and CHUYANOV. 1’. A., Completely conservative scheme for the two-dimensional Landau equation.Z\t. I.)‘cltisl. Mar. mat. Fiz., 20, No. 2. 513-517. 1980. 5. BOBYLEV, A. V.. POTAPENKO. 1. F., and CHUYANOV. V. A.. Kinetic equations of Landau’s type for a model of Boltzmann’s equation and completely conservative difference schemes. Zh. vychisl. Mat. mat. Fiz., 20, No. 4,993-1004, 1980. 6. SAMARSKII. A. A.. and POPOV. YU. P., Difference schemes of gas dynamics (Raznostnye skhemy gazovoi dinamiki), Nauka, Moscow. 1975. 7. KlLLEEN. J., Computer models ofmagnetically
confined plasmas.Nucl. Fusion, 16, No. 5, 841-864,
1976.
8. CHURAEV, R. S.. and CHUYANOV. V. A., Calculation of ion leakage from a trap with magnetic mirrors, Preprint IPM Akad. Nauk SSSR. No. 33, Moscow. 1975. 9. KARETKINA, N. V., Numerical modelling of the kinetic processes in the Tokamak plasma, Dissertation for candidate of physical-mathematical sciences, MCU, Moscow, 1978. 10. ROSENBLUTH. M. N. :t al., Fokker-Planck 1-6, 1957.
equation for inverse-square force, Phys. Rev., 107, No, 1,
11. SAMARSKII. A. A., Ilheor), of difference schemes CTeoriya rarnostnykh
&hem), Nauka, Moscow, 1977.
12. YANENKO. N. N., Method of fractional steps for solving multi-dimensional problems of mathematical physics (Meted drobnykhshagov resheniya mnogomernykh zacach matematicheakoi R.&i), Nauka, Novosibirsk, 1967.
269
Notice in conclusion that similar results can be obtained for the solutions of Eq. (1) whose fronts are at infinity, if we seek the solution as a series in decreasing negative powers of x. The authors sincerely thank S. P. Kurdyumov
for useful discussions. Translated
by
D. E. Brown
REFERENCES 1. ZMITRENKO, N. V., er al., Mets-stable and conditions for its appearance 1977.
localization of heat in a medium with non-linear heat conduction in an experiment, IPM Preprint, Akad. Nauk SSSR, No. 103, Moscow,
2. VAINBERG, M. M. and TRENOGIN, V. A., The theory of branching of solutions of non-linear equations (Teorfya vetvleniya reshenii nelineinykh uravnenii), Nauka, Moscow, 1969.
3. KALASHNIKOV, A. S., On the nature of the propagation of disturbances in processes linear degenerate
parabolic
equations?
Tr. seminam im. I. G. Perrovskogo,
4. MARTINSON, L. K., Ropagation of shear disturbances zhidkostigaza, No. 6, p. 60, 1978.
in dilatant
U.S.S.R. Compur. Maths. Marh. Phys. Vol. 22, No. 3, pp. 269-274. Printed in Great Britain
fluids,Izu.
1982.
described No. 1, 133-144,
by quasi1975.
Akad. Nauk SSSR, Mekhan.
0041-5553/82S07.50+.00 01983. Pergamon Press Ltd.
A COMPLETELY CONSERVATIVE SCHEME FOR LANDAU’S EQUATION (ROSENBLUTH NON-ISOTROPIC POTENTIALS)* I. F. POTETSENKO and V. A. CHUYANOV Moscow (Received
15 Ma)
1980)
A DIFFERENCE scheme, preserving the number of particles and the energy, is constructed for the two-dimensional Landau equation with Rosenbluth non-isotropic potentials, whereby nonstationary problems can be solved by computer without error accumulation. Several (in general, five) laws of conservation are satisfied for the non-linear Landau kinetic equation. If the difference scheme has only approximate analogues of the conservation laws, this can easily lead to error accumulation when computing non-stationary problems. Hence the principle of complete conservatism, put forward in [l] in connection with problems of gas dynamics and magnetohydrodynamics, proves to be unusually important for problems of rarefied gas and plasma dynamics [2-51. When solving the kinetic equation numerically by computer, we replace the continuous medium by a discrete model (difference scheme), and meantime have to require that the model reflect the main features of the process considered, including the laws of conservation (61. Difference schemes for the two-dimensional (in velocity space) Landau equation (the Fokker-Planck equation), retaining the number of particles, were constructed in [7-91: but *Zh. vychisl. Mar. mar. Fiz., 22, 3. 751-756.
1982.
I. F. Potetsenko and V. A. Chuyanov
270
in computations on actual meshes, such difference schemes may distort the effect in question, due to the presence of fictitious energy sources in them. By developing the methods of [2-51 we can construct a completely conservative scheme for the two-dimensional Landau equation with Rosenbluth non-isotropic potentials [lo] , i.e. a scheme which preserves the number of particles and the energy. When studying the behaviour of a plasma in open magnetic traps, the multicomponent equation [7] is of practical value; in the present paper we consider for simplicity the Landau equation for one sort of particle, in the absence of sinks and sources (e.g. ions in a neutralizing background). We take the Landau equation, written in divergence form: m2
af
4sez,i
at
--=_
1 aa(v,p) 29
au
1 a +,,j-pPw(w)l,
.I 62g___
D(L+~Lf+(l_p~) (
P(r.p)=
+
i-112 azg 1 ag +_-----
(
a% -_-ac akL (
c au
62 BP2 i
a~ aF
af v2 ak ) ap 1
ag
__v2
$f,
ag
_af L.? ap > ap P
ag
af ah --_--f 1.6~ ) au apL’
OGVC55,
-lGpG’i,
DO,
dv’f(v’, 1)jv-\.‘I, h(v, cl)= dv’f(v’, t) x Iv-v’l-i, with Rosenbluth potentials g (V. P) = s s the initial condition .f(c, P. 0) =,fO(~,~1 , and the condition that the distribution function be bounded at 12= 0, p = 5 1. As v + =, the distribution function tends to zero. We shall use dimensionless velocity! time, and distribution function [2.8] below, while retaining the previous notation. The first stage in constructing the completely conservative difference scheme is to write Eqs. (1) in a form that takes better account of the two conservation laws. To reduce (1) to the required form, we use the obvious properties of the Rosenbluth potentials: Ag=h.
Ah=-SSf?
(2)
Then, in the notation
Eq. (1) takes the form (3) where L:(r-, p) =
aG(L’,IJ) au
+
v2f(v. p, t)H(v, p).
271
Short communications
The conservation laws for the number of particles and the energy are dn -=dt The
JJ
dml dt
dE =; dt
!(I.. p, t)v’ dr dp=O.
0 -1
da’
p. JJ f(v.
t) v( dv dp=O.
0 --I
unique stationary solution of Maxwell’s type is fy=n
($)’
(4)
erp ( -$vz).
Notice that the second conservation law is connected with the satisfaction of the equation m
i
JJ
=‘iG
JJ
C(v,. p)dv dp=
-
0 --I
0 --I
since
dc
JJ
(5)
fv=H dv dp=O,
0 -1
(v’_ p’, JJ v’?f
m
H(r, \I)=
-’
dv dp+
1
t) K(v, v’) dv’ dp’,
0 -i
where
K(V,
Y’)
=?(c?-~‘2)
is the antisymmetric “kernel.”
Iv-~‘(-3
To compose the difference scheme, we replace the infinite interval by the finite interval [0, L] , which is chosen so as to take into account the high-energy particles. The order of decrease of the distribution function can be estimated from distribution (4). In the domain of space considered we introduce the space-time mesh 0
o=OhXO~=(v~+i=L.hi-lik-,~ j=i,Z
=!_Lj+hr+*,
We define the mesh functions pj)t
V(Ck.
pj).
,iV-i,
JLi=-1,
. . ,.?‘-1, kx=i,
Vi=O,
vN=L,
pj+$
tn+l
n=O. 1, *. . ( P=O}.
=P+‘l,
U(L’k.
,...
Ii=l,Z..
.
.
fkjn?
L'kj-
1;;.
...
corresponding to the functions
f
(
vk.
pj,
t”).
We shall also use the notation
hb=O.5(h+,+h~).
~k_,;=O.j(~ri!frk).
hlv=0.5hz’,
The second stage in constructing the completely conservative difference scheme consists in applying the integro-interpolation method of [ 1 I] to Eq. (3). As a result we obtain the difference scheme f,=L,!+Lz!,
fkj’“fO; Uk.L’,‘: j --uk;“. ,.
uk-
(2 j
Vk-I.‘>
(6) k=2.
3.. . . ..1’. j=2. 3.. . . Jf-1.
I. F. Potetsenko and V. A. Chyanov
272
The scheme is written in the form of a conservation law for the number of particles, while the approximation of the function uk+l/3,j has to ensure satisfaction of the energy conservation law. On approximating the integrals by the trapezoid formula, and the derivatives by the central differences, we obtain
where
Kw,;, j, et-‘,;,
i=-Km;,.;,
i,
r;,i, j.
From this expression we obtain the mesh analogue of Eq. (5):
ensuring satisfaction of the energy conservation law for difference scheme (6), In fact, the difference analogues of the conservation laws are obtained by summation over the mesh of Eqs. (6). The variations over a time step of the number of particles and of the energy are
L..q
v~+i!Ji+~,j+v_W2fNj
+-
Hence
us-
2
Hx+j;;.j
1 .
it is clear that we can arrange for the conservation laws to be satisfied imposing the respective boundary conditions: L’3‘1,j=o.
U.Y,.. ,=C’.
G.Y+,. ;=Gsj=@,
pjEO!,.
f,?EO7.
Notice
that, at the right-hand end vn: = L we can impose the simpler boundary condition ; then the degree to which the energy is not conserved is proportional to tJ’=or. the rounding errors. fxl=O,
pj=~,
Scheme (6) is constructed on a symmetric pattern, and has the second order of approximation with respect to the velocity space, and the first order with respect to time. The divergence properties of the scheme enable higher order of approximation with respect to time to be obtained. For the computations, the methods of variable directions, of stabilizing correction, or of splitting [ 11, 121, may be used.
173
Short communications For numerical
computation,
the “kernel” K (v, Y’)is expressed in terms of the complete
elliptic integral of the 2nd kind:
When solving Eq. (3) by computer, the greatest labour lies in computing the functions g (v, cl) and h (v, /.L),including complete elliptic integrals of the 1st and 2nd kinds. Effective methods have now been devised for computing the potentials g (v, p) and h (u, cc)by means of numerical integration or from Eqs. (2) (see [8, 91). In addition, the function V (u, P) of (3) differs from the function p (v, 1) usually considered of (1) in one term, so that it is possible to employ the existing methods of computations (see [8,9] ) for scheme (6). Notice in conclusion the connection between the equation, written in form (3), and the expressions previously obtained for the Landau equations [2,4]. We expand the Rosenbluth potentials in Legendre polynomials P,, (p) (see [lo] ):
In the notation
i PI fnct.1 = J f(v,P,t)pn (II)dl.l, in(VI= fn(Y)!I-“+’ dy, J
-ia 0 P~(v,P)= J~(.Y,P.~Y+‘+‘~Y the function U (Y,cl) of (3) takes the form
+2(2nil)
J[
n
h(Y!!(4
0
P)
-E ( L’ )
- In (Y)PTI(~7@)
1
P*
dy
(7)
0
-2(2n+:)
J0
[P~(Y)!(Y.
v!-.fn(~)~m(y.
u)
>
IY”+’&/ .
Notice that, if we introduce the extra condition for symmetry of the distribution function with respect to p = 0, as is usually done in magnetic trap computations, only the terms with even II are retained in expansion (7) so that the law of conservation of momentum is automatically satisfied.
274
I, F. Potetsenko and V. A. Chuyanor
From (7), for n = 0 and an isotropic distribution function, we get t’
L.-,&!-
aL
(s
[P(Y)!(L.)--!(Y)P(~)lY2
dY
0
1
and the equation of [2] :
With n = 0 and a distribution function dependent on v and ~1(isotropic Rosenbluth potentials), we have F CC,,3 b [P,‘(Y)!(L’,~)--!o(Y)P(v,~I)lY* dY r,
-
s F
[Po(Y)!iY.PI _fi’(Y)P (Y. II) IY? dY
1
and the equation of [4], corresponding to one sort of particles. Translated lg. D. E. Brown
REFERENCES 1. POPOV. YU. P.. and SAMARSKII. A. A.. Completei!, conservative difference schemes. Zh. l:Fchisl. Mat. mat. Fir., 9. No. 4. 953-958. 1969 2. BOBYLEV. A. V., and CHUYANOV. V. A.. On the numerical solution of the Landau kinetic equation, Zh. ,:Fchisl. Mat. mat. Fi:., 16, No. 2.407-416, 1976. 3. POTAPENKO. I. F.. and CHUYANOV. V. A.. On completely conservative difference schemes for the system of Landau equations. Zh. r_Fchisl.Mat. mat. Fiz., 19, No. 2.458-463: 1979. 4. POTAPENKO. 1. F.? and CHUYANOV. 1’. A., Completely conservative scheme for the two-dimensional Landau equation.Z\t. I.)‘cltisl. Mar. mat. Fiz., 20, No. 2. 513-517. 1980. 5. BOBYLEV, A. V.. POTAPENKO. 1. F., and CHUYANOV. V. A.. Kinetic equations of Landau’s type for a model of Boltzmann’s equation and completely conservative difference schemes. Zh. vychisl. Mat. mat. Fiz., 20, No. 4,993-1004, 1980. 6. SAMARSKII. A. A.. and POPOV. YU. P., Difference schemes of gas dynamics (Raznostnye skhemy gazovoi dinamiki), Nauka, Moscow. 1975. 7. KlLLEEN. J., Computer models ofmagnetically
confined plasmas.Nucl. Fusion, 16, No. 5, 841-864,
1976.
8. CHURAEV, R. S.. and CHUYANOV. V. A., Calculation of ion leakage from a trap with magnetic mirrors, Preprint IPM Akad. Nauk SSSR. No. 33, Moscow. 1975. 9. KARETKINA, N. V., Numerical modelling of the kinetic processes in the Tokamak plasma, Dissertation for candidate of physical-mathematical sciences, MCU, Moscow, 1978. 10. ROSENBLUTH. M. N. :t al., Fokker-Planck 1-6, 1957.
equation for inverse-square force, Phys. Rev., 107, No, 1,
11. SAMARSKII. A. A., Ilheor), of difference schemes CTeoriya rarnostnykh
&hem), Nauka, Moscow, 1977.
12. YANENKO. N. N., Method of fractional steps for solving multi-dimensional problems of mathematical physics (Meted drobnykhshagov resheniya mnogomernykh zacach matematicheakoi R.&i), Nauka, Novosibirsk, 1967.