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The use of a continuous analogue of Newton's method to solve the equations of a non-selfsustaining gas discharge
U.S.S.R. Printed
comput.Maths.Math.Phys.,
0041-5553/89 $10.00+0.00 01990 Pergamon Press plc
Vo1.29,No.l,pp.220-221,1989
in Great Britain
THE USE OF A CONTINUOUS ANALOGUE OF NEWTON'S METHOD TO SOLVE THE EQUATIONS OF A NON-SELFSUSTAINING GAS DISCHARGE* *jt A.I. SHERSTYUK,
M.S. YUR'YEV
and N.S. YAKOVLEVA
In the well-known approximations /l/, the spatial structure discharge can be described by the following system of equations:
-=d(fh) dE with the boundary
-
e-
Wh)
dh
= s+afh-pff,
6-
4
-
4
of a non-selfsustaining
gas
f-s
conditions
,
f (0) - eyg(0)
I
g(l)
=o.
s
(Ib)
h d:==i.
Here, !=n,/no. g=n,/no.L=E/E",no=(S/Bo)'". E,=[i/L,n.,nt are the electron and ion densities, respectively, field U is the applied potential, the L is the length of the discharge column, E is strength, S is the power of the external ionizer, p=p,,/h* is the dissociative recombination a=A rxp (-B/h) characteristizes the impact ionization, constant, y~l is the coefficient of secondary emission, e=tit/lr.1O-3 is the ratio of the ion mobility to the electron mobility and 6=E,1(4nen,L)~lO-h, The problem is a non-linear and stiff one and the solution has boundary layers (b.1.) with a complex structure. Previously /l, 2/ in physical applications, the problem has been reduced to an approximate treatment of the precathode region with the formulation of a boundary condition on the boundary of the cathode layer and the positive column (the region of an electrically neutral plasma). In this paper the problem is formulated for the whole of the discharge gap as without any artificial clamping of the boundaries of the pre-electrode layers. System (1) can be y=h= reduced to a single second-order equation for the function which is proportional to the energy of the electric field:
with the boundary
conditions v'(1) -21/8+,-o,
v'(0) +21/(e6)=+)*0,
s
(I/)“’ d&-l+')=0
L
(2b)
,
where
i=h(f+eg)=const is the total current density. Hence, we obtain a problem involving the eigenvalues of the total current density in the case of a second-order differential operator. The spectral parameter occurs non-linearly in (2a). Furthermore, the boundary Conditions (2b) depend on j. The integral condition emerges in the role of a normalization. It should be noted that the normalization of the solution is solely dependent on the applied potential while the eigenvalue j which is equal to the total current density in the discharge column is independent of the potential. A modified, continuous analogue of Newton's method (m.c.a.N.m) is used to solve problem Problem (2) is treated as a non-linear functional equation defined in a (2) (see /3/). Banach space with the elements (j,y)~~x(c~[O, i]) and a defined set of operators &=(~(~I),1=l, 2, J, 4. The above-mentioned method enables one to reduce the solution of Problem (2) to the solution of the evoluton equation dyld~--_[~(yp,l-l~(~),
Y/1-o=yo.
Il
(3)
+~Zh.uychisZ.Mat.mat.Fiz.,29,2,312-3i4,1989 **The complete text of this paper is deposited Technical Information (VINITI), 7700-B88, 1988
in the All-Union
220
Institute
for Scientific
and
221 where
go
is the initial approximation, iimy(r)-_v‘is the solution of Problem (21, and *-XX
-, e(y) is a Frechet derivative. By using discretization in accordance with Euler's method Y*+l=Yn+Tkar, ir+i=i*+rkn, (4) where (W is the time variable mesh for Eq.(3) and taking the solution in the column (y=l= const), as the initial approximation, it is possible to obtain the correction I& in the following convenient form:
r*-A,& erpf-k&f
+A$erpi-hz(l-&)
I c ”
i
t
+ +
-
[l-expf-U)
(5)
lx
U (~-~xp~-2~(~-~)I~erp~-h~(~-&)l~~"(~)~~~i-exp(-2;lt)]t t f (f-erPI-bfl-E)lf
fIi-erpf-thri)lerpl--h*(e-li)l~~’:L)frl)drl 0
and rh are determined by the where h=to[(~a12e)a+2B01"*. h,=h-Bo/2e. ~z=h+~,12e. The numbers ii?,A$ (p;'W, $1 errors of closure cp;cz,, at each step t from the system of three linear algebraic equations. The error of closure Gil is equal to zero everywhere apart from the boundary layers. Hence, the problem is reduced to a treatment of just the electrode regions without any artificial anchoring of the boundaries. Problem (2) was solved numerically using the technique which has been described. The constants used for the gaseous medium corresponded to molecular nitrogen at a pressure P=-20' torr and a length of the discharge gap, The calculations were carried out on a L-10 cm. BESM-6 digital computer. About five seconds was required for a single iteration. On the average, maximum error of closure of the equations has a value of -10-s after lo-15 iterations. The spatial distribution of the electric field, the electron and ion densities and the voltage-current characteristics of the cathode layer are obtained as a function of the parameters used in the problem. The rate of convergence of the iterative process is studied as a function of the choice of the discretization parameter in the m.c.a.N.m. A comparison is made with the results of calculations by other authors /2/. REFERENCES 1. RAIZER YU.P., Fundamentals of the Modern Physics of Gas-Discharge Processes, Nauka, Moscow, 1980. 2. ALEKSANDROV V-V., KOTEROV V.N., PUSTOVALOV V.V. et al., The space-time evolution of the cathode layer in electro-ionizationlasers, Kvantovaya Elektronika, 1, 114-121, 1978. 3. ZHIDKOV E.P., MAKARENKO G.I. and PUZYNIN I.V., A continuous analogoue of Newton's method in non-linear problems of physics, in: Problems in the Physics of Elementary Particles, Atomic Nuclei, OIYaI, Dubna, 4, 1, 125-166, 1973.
Translated by E.L.S.
Comput.Muths.Math.Phys.,Vol.29,No.l,pp.221-222,~989 Printed in Great Britain
0041-5553/89 $lO.OO+O.OO 01990 Pergamon Press Plc
U.S.S.R.,
ESTIMATES OF THE ERROR WHEN THE SOLUTION OF THE BO~TZMANN EQUATION IS APPROXIMATED BY THE SOLUTION OF THE IT0 STOCHASTIC DIFFERENTIAL EQUATION" i)($ S.L. POPYRIN It was shown in /II that the solution of the Boltzmann equation for a gas with virtually any type of interaction between its molecules can be approximated with any accuracy by the solution of the Ito stochastic differential equation with respect to a Poisson measure. Algorithms based on this idea were given in /2/ for the numerical solution of the Boltsmann equation. These algorithms use an approximation of the kernel of the collision integral by a step function. In the present paper we investigate how the error of the algorithms depends on the number n of "steps". Let Sz be the unit sphere in three-dimensionalEuclidean space R3. and let li=RW?xS? be a smooth eight-dimensionalmanifold. We put ll~l!=s~P~lrp~~~/+/rg~Yl-~l~i~~l/lY-~l~.
x. Y, ire F3.
‘tZh.vychis2.Mut.mat.Fiz.,29,2,314-315,1989 )'"Thecomplete text of this paper is deposited at VINITI, 7699-388, 1988.
comput.Maths.Math.Phys.,
0041-5553/89 $10.00+0.00 01990 Pergamon Press plc
Vo1.29,No.l,pp.220-221,1989
in Great Britain
THE USE OF A CONTINUOUS ANALOGUE OF NEWTON'S METHOD TO SOLVE THE EQUATIONS OF A NON-SELFSUSTAINING GAS DISCHARGE* *jt A.I. SHERSTYUK,
M.S. YUR'YEV
and N.S. YAKOVLEVA
In the well-known approximations /l/, the spatial structure discharge can be described by the following system of equations:
-=d(fh) dE with the boundary
-
e-
Wh)
dh
= s+afh-pff,
6-
4
-
4
of a non-selfsustaining
gas
f-s
conditions
,
f (0) - eyg(0)
I
g(l)
=o.
s
(Ib)
h d:==i.
Here, !=n,/no. g=n,/no.L=E/E",no=(S/Bo)'". E,=[i/L,n.,nt are the electron and ion densities, respectively, field U is the applied potential, the L is the length of the discharge column, E is strength, S is the power of the external ionizer, p=p,,/h* is the dissociative recombination a=A rxp (-B/h) characteristizes the impact ionization, constant, y~l is the coefficient of secondary emission, e=tit/lr.1O-3 is the ratio of the ion mobility to the electron mobility and 6=E,1(4nen,L)~lO-h, The problem is a non-linear and stiff one and the solution has boundary layers (b.1.) with a complex structure. Previously /l, 2/ in physical applications, the problem has been reduced to an approximate treatment of the precathode region with the formulation of a boundary condition on the boundary of the cathode layer and the positive column (the region of an electrically neutral plasma). In this paper the problem is formulated for the whole of the discharge gap as without any artificial clamping of the boundaries of the pre-electrode layers. System (1) can be y=h= reduced to a single second-order equation for the function which is proportional to the energy of the electric field:
with the boundary
conditions v'(1) -21/8+,-o,
v'(0) +21/(e6)=+)*0,
s
(I/)“’ d&-l+')=0
L
(2b)
,
where
i=h(f+eg)=const is the total current density. Hence, we obtain a problem involving the eigenvalues of the total current density in the case of a second-order differential operator. The spectral parameter occurs non-linearly in (2a). Furthermore, the boundary Conditions (2b) depend on j. The integral condition emerges in the role of a normalization. It should be noted that the normalization of the solution is solely dependent on the applied potential while the eigenvalue j which is equal to the total current density in the discharge column is independent of the potential. A modified, continuous analogue of Newton's method (m.c.a.N.m) is used to solve problem Problem (2) is treated as a non-linear functional equation defined in a (2) (see /3/). Banach space with the elements (j,y)~~x(c~[O, i]) and a defined set of operators &=(~(~I),1=l, 2, J, 4. The above-mentioned method enables one to reduce the solution of Problem (2) to the solution of the evoluton equation dyld~--_[~(yp,l-l~(~),
Y/1-o=yo.
Il
(3)
+~Zh.uychisZ.Mat.mat.Fiz.,29,2,312-3i4,1989 **The complete text of this paper is deposited Technical Information (VINITI), 7700-B88, 1988
in the All-Union
220
Institute
for Scientific
and
221 where
go
is the initial approximation, iimy(r)-_v‘is the solution of Problem (21, and *-XX
-, e(y) is a Frechet derivative. By using discretization in accordance with Euler's method Y*+l=Yn+Tkar, ir+i=i*+rkn, (4) where (W is the time variable mesh for Eq.(3) and taking the solution in the column (y=l= const), as the initial approximation, it is possible to obtain the correction I& in the following convenient form:
r*-A,& erpf-k&f
+A$erpi-hz(l-&)
I c ”
i
t
+ +
-
[l-expf-U)
(5)
lx
U (~-~xp~-2~(~-~)I~erp~-h~(~-&)l~~"(~)~~~i-exp(-2;lt)]t t f (f-erPI-bfl-E)lf
fIi-erpf-thri)lerpl--h*(e-li)l~~’:L)frl)drl 0
and rh are determined by the where h=to[(~a12e)a+2B01"*. h,=h-Bo/2e. ~z=h+~,12e. The numbers ii?,A$ (p;'W, $1 errors of closure cp;cz,, at each step t from the system of three linear algebraic equations. The error of closure Gil is equal to zero everywhere apart from the boundary layers. Hence, the problem is reduced to a treatment of just the electrode regions without any artificial anchoring of the boundaries. Problem (2) was solved numerically using the technique which has been described. The constants used for the gaseous medium corresponded to molecular nitrogen at a pressure P=-20' torr and a length of the discharge gap, The calculations were carried out on a L-10 cm. BESM-6 digital computer. About five seconds was required for a single iteration. On the average, maximum error of closure of the equations has a value of -10-s after lo-15 iterations. The spatial distribution of the electric field, the electron and ion densities and the voltage-current characteristics of the cathode layer are obtained as a function of the parameters used in the problem. The rate of convergence of the iterative process is studied as a function of the choice of the discretization parameter in the m.c.a.N.m. A comparison is made with the results of calculations by other authors /2/. REFERENCES 1. RAIZER YU.P., Fundamentals of the Modern Physics of Gas-Discharge Processes, Nauka, Moscow, 1980. 2. ALEKSANDROV V-V., KOTEROV V.N., PUSTOVALOV V.V. et al., The space-time evolution of the cathode layer in electro-ionizationlasers, Kvantovaya Elektronika, 1, 114-121, 1978. 3. ZHIDKOV E.P., MAKARENKO G.I. and PUZYNIN I.V., A continuous analogoue of Newton's method in non-linear problems of physics, in: Problems in the Physics of Elementary Particles, Atomic Nuclei, OIYaI, Dubna, 4, 1, 125-166, 1973.
Translated by E.L.S.
Comput.Muths.Math.Phys.,Vol.29,No.l,pp.221-222,~989 Printed in Great Britain
0041-5553/89 $lO.OO+O.OO 01990 Pergamon Press Plc
U.S.S.R.,
ESTIMATES OF THE ERROR WHEN THE SOLUTION OF THE BO~TZMANN EQUATION IS APPROXIMATED BY THE SOLUTION OF THE IT0 STOCHASTIC DIFFERENTIAL EQUATION" i)($ S.L. POPYRIN It was shown in /II that the solution of the Boltzmann equation for a gas with virtually any type of interaction between its molecules can be approximated with any accuracy by the solution of the Ito stochastic differential equation with respect to a Poisson measure. Algorithms based on this idea were given in /2/ for the numerical solution of the Boltsmann equation. These algorithms use an approximation of the kernel of the collision integral by a step function. In the present paper we investigate how the error of the algorithms depends on the number n of "steps". Let Sz be the unit sphere in three-dimensionalEuclidean space R3. and let li=RW?xS? be a smooth eight-dimensionalmanifold. We put ll~l!=s~P~lrp~~~/+/rg~Yl-~l~i~~l/lY-~l~.
x. Y, ire F3.
‘tZh.vychis2.Mut.mat.Fiz.,29,2,314-315,1989 )'"Thecomplete text of this paper is deposited at VINITI, 7699-388, 1988.