On the asymptotic error expansions to finite difference methods

48

3. 4.

D'YACHENKO V.F., Basic concepts of computatinal mathematics (Osnovnye ponyatiya itel'noi matematiki), Nauka, Moscow, 1972. GODUNOV S.K. and RYABENKII V.S., Difference schemes (Raznostnye skhemy), ?Jauka,

vychislMoscow,

1973.

KHOLOWV A.S., On the construction of difference schemes with positive approximation for equations of hyerpbolic type, Zh. vych. Mat. i mat. Fiz., 18, No.6, 1476-1492, 1978. 6. KHOLODOV A.S., On the construction of difference schemes of improved order of accuracy for equations of hyperbolictype,Zh. vych. Mat. i mat. Fiz. 20, No.6, 1601-1620, 1980. 7. FRIEDRICHS K.O., Symmetric hyperbolic linear differential equations, Communs. Pure and Appl. Math. 7, No.2, 345-392, 1954. 8. SAMARSKII A.A. and GULIN A.V., Stability of difference schemes (Ustoichivost' raznostnykh skheml, Nauka, Moscow, 1973. Difference methods for initial value problems, Wiley, 1967. 9. RICHTMYER R.D., 10. TIKHONOV A.N. and SAMARSKII A.A., On homogeneous difference schemes, Zh. vych. Mat. i mat Fiz., 1, IJo.1, 5-63, 1961. 11. KALITKIN N.N., Numerical methods (Chislennye metody), Nauka, Moscow, 1977. 5.

Translated

U.S.S.R. Comput.Maths.Math.Phys.,Vo1.24,No.5,pp. Printed in Great Britain

48-56,1984

by D.E.B.

OO41-5553/84 $10.00+0.00 0 1985 Pergamon Press Ltd.

ON THE ASYllPTOTICERROR EXPANSIONS TO FINITE DIFFERENCE METHODS* TA VAN DINH Single and multiparameter formulas for the expansion of the error in Such finite difference methods for operator equations are considered. formulae are extremely useful in speeding up the convergence of the method and for obtaining approximate estimates of the error in terms of known quantities. Numerical examples are presented as an illustration.

Introduction. In finite difference methods when some asymptotic error expansion exists the Richardson extrapolation to the limit can be used for accelerating the rate of convergence of the methods. It reduces the necessary number of algebraic equations to be solved and thereby provides a The very efficient algorithm with respect to both computing time and storage requirements. asymptotic error expansions also justify the Runge principle for evaluating the approximate Many delicate investigations have been done about such expansions error via calculated values. for special as well as for operator equations (see for instance /l/--/13/ and the references in these works). We however observe that in /l/--/13/only the one-parameter formulae have We shall been presented while in many cases the multiparameter formulae are more effective. give in this paper some results about both the one- and multiparameter formulae for operator Our Theorems 1 and 2 may be considered as come generalization of /6/, /ll/, /12/. equations. For special cases we shall introduce only some numerical examples for illustration.

One-parameter formulae for operator equations. 1. Definitions and notations. Let there be given two normed spaces Consider the operator equation T :Q,cE,-E,. II111. an application 1.

T(u)=O,

u=Q,.

E,.

i=l, 2, with norms (1.1)

We consider a sequence of positive numbers Assume that it has a unique solution u'E~,. we can associate two monotonously tending to zero and we assume that to each h=(h) (A) finite dimensional spaces E, with norms \\.\l%h, two linear operators +:Ei-E,*, so that we have dim Em
h’th,

1;~ IIp,hullc\=IIuII<, IL=%.

To each

being an h=(h) we assume again that we can construct an application Th:EII+EI* approximation to T. By the abstract (one-parameter) finite difference method we want to say all the process of constructing the spaces E,,,, the operators pa,,and the appiication Th and replacing (1.1) by (1.2) T*(l&)=O, v.e=E,~.

Assume that the equation (1.2) has a unique solution v~' for This method is denoted by T,.. each hs(h). Then the expression e,,=vh.-p,hu’EE,A is called the error of the method and Il4ll~. the method is called to be conthe absolute error of the method. If IIehj/lh+O when h-0 vergent. H() independent of write q(h)=O,(h"), a being a real number, if there exists a constant *Zh.vychisl.Mat.mat.Fiz.,24,3,1359-1371,1984

49

h=(O,H), so that lcp(h)IfMh". the method 1s called to have If there exists a positive constant a so that ~~e&-~u(hQ) an accuracy of order a . If there exists two positive numbers 7, h (hl) and elements WJ%!L,j=m,...,n, independent of h~(0.H). (1.3)

the error is called to admit an asymptotic expansion expansion is of the one-parameter formulae. 2. The family the methods T,, having

of starting the following

methods. properties.

with respect to

Henceforth

we

Thesis 1. There exist wo positive numbers l,l (Xl) integers n, m (n>m>l), the sets Q?cEl, j-m,. , n, with Qz’, independent of &(O, H), so that for uSS& we have

Thesis 2. This Frechet we have

differentiable

h up to order

consider

n.

This

the family Q of

a positive number H<1, two F, : fi,+ S&'&p' and the operators

up to order v with

m(v+l)>n+l

and for izhr vheEth

.

where

a,=const

and

(2') denotes

(z,...,z),k

times.

Thesis 3. There exists a linear operator A(u) :D(A)cE,+E, a parameter and can be approximated by 'f&'(u) on D(A), i.e. fqr IIT,‘(~,Iu)~,IIu-P?AA when with

H),

h-+0 and at the same time each equation .Q,l~bl~+* .

T&dp,~--p,{A

Thesis 5.

(n,w+E

u=Q,

as

(a) ~Ilm+Ot has a unique

A(u)w=g+=Q,'

Thesis 4. There exist the operators c,:a,xQ,i-a;J, so that for UEQ,, WER,', i=m,...,n, we have

II

which depends on WED(A) we have

solution

w&&'cD(A)

j=t, 2,...,n-i, independentof

h”G,(u. wu) )~l,,=O&~n-i”“).

h~(0,

Gc=O.

,-I

There

exist

the operators G:: Q1x

k=2, 3,. . . , v, i!=m,. . . , n, j=O, i,=m, ....n. we have

4” x .,. x Qlik_,$$+’ ‘++‘> . . , n-km, independent of h=(O,H).

I,.

so that for

UEO,, u‘,, EC",

n--*m

!I

T?

Note 1.

(pia)

(pthwt, . . . , p,*w,*) -pmz

We rewrite

h”Gj’(U,

w,,,...,

w,,) II., ==OH(h’“--l”‘+*)‘).

j-0 Thesis

1 as

T&,,")-P.(

T(u)+ &IJV,(U)

}

=p(u,h)

,-m

Then if T.T,,.F,

are smooth

enough

and when

II

I-O we have

i

p(u+%h)-p(u.h)

IIm=O”(hc”--I+~IY)

We can deduce the property Thesis 4, and the first part of Thesis 3 with A(u)=T fU). Analoqously,withfurther assumptionsaboutthe smoothness of P(~,/L)we can deduce from Thesis 1 the property Thesis 5 as well. But in general we are given no information about p(u,h) more Therefore we must verify the properties Thesis 3-5directly in each than Ilp(~,h)((~~OH(h(n+l)7). concrete case. The exceptional case is when T is linear, then the assumptions about p(u.h) like above are always verified. 3.

Some

lennnas.

In the following

we alwayscons~derT~~~.

Lemma 1. There exist the operators UI,:n,Xo,~"X...Xn:-'-a:, so that for uES2,. WI&&', i=m, .. . , n, h=(O: H),

S,=

h’iU$

j=m,...,n,

independent

of

_. 4)

50

we have

h”[A(u)w,+F,(u)+0,(u,w,,

~u(T(n)ft

l&+1..

.? %A+

i-n

Proof.

By Thesis

1 we have ))T.(p,,u)-pu

[$“Wu)+T(4

]llM-Odh(“+‘9~

By Thesis 4 we have 11 T,,‘(plm)p,,S,,-p,,

&‘[A

(u) ra,+G,_,(u,

w,) + . . +G, (u, wi-1) 11Ih=o,(h’“+a”).

,-I%4

By Thesis

5 we have

IIZ

a,T:l’(p,hu)

(p,$.)‘-p,,z

*-I

where 0,N& proved. Lemma 2.

h”0i(u,

i=m,

and depends only on u and w,with There

w,, . . . , wj-,)

11 =OIZ@(“+~“)V

,-“I

exist

the elements

m+l,...,j--i

IL.~EQ~~, j-m,...,n,

~(u)w,=-F,(U)-0,(u,w,,...,wj-,)-G~-,(u, where 0, have been determined

in the Lemma

1.

at most.

satisfying

W-)--...--G,(u,

These w,are

independent

The lemma is then

equation

w,-,), of

(1.5) k(O.

H).

We shall prove the lemma by recurrence. Proof. By Thesis 1 we have F,(U)&,". Hence by Thesis 3 there exists z&,,E&"' satisfying (1.5) when Since F,(u) is independent j-m. then so is w,. Now for a fixed iQn we assume that there have existed ~,a: of h=(O,H) satisfying (1.5) for j
For the elements

It is an immediate

wt determined

consequence

of the two previous

Lemma 4. Let u‘be the solution ofthe the Lemma 2 with U-U* and (1.4) we have l/T* (pid+p,Jn) Proof.

By Thesis

Th(pd+pJn)

3. The results.

equation

(1.4) we have

lemmas.

(1.1).

Then

for the w, determined

in

jlrh=On (W’+“‘).

2 we have

II But it is clear that from Lenma 3.

in the Lemma 2 and

-T~(p,nu')-Th'(p,*~')p,hSn-

ilp,.jnlllh=OH(h'"'), hence In the following

O~(~~p~~s.;~;l’ )=o~(h”+“‘).

we assume

that

ThETh

Then the lemma follows

and U' is the solution

Theorem 1. Assume that 1) the equation (1.2) has a unique solution vn'; where M and p are positive constants, Odpp. Then there exist the lements w&Z,, j-m,...,n, independent of h=(O,H),

11 L+.-P’.~ ww, 11 ,~odh’“+*-~“).

of (1.1).

independent

of

so that we have CL.61

51 Proof.

Let

w,&,~, j-m,.

. . , n,

be the elements

determined

in the Lemma

2 with

u=u'.

We

Put " h” w,, z ,-n

&-

Now by Thesis

vn==vn', un-p&’

2 with

(1.7)

Ar=eh-p&C,.

we have (1.8)

prom Thesis

5 we get

II?')? (p,,u.)@onat

independent

112 Q&T?)(p,,Ic)

(enk)-2

a.TF’ (Pd)

(p&Y

(1.7) weduduce

x

a--mini% ml.

~~~O.(hafllAnllrd,

(P,ru’)eh+

k aJ.?(PA

11 -o,(~lehll,:+‘)+o,(h=‘llAnll~h). Y.h

(PJn)’

k-*

(1.9)

(1.7) we have

T*’ (plhd)eh+ 2

Then

from

(1.8) yields

11 T~(p,ru’)+Tn’ From

Therefore

*-t

I-1

Then

h=(O,H).

of

(1.9) and Lemma

(P thus) (p,&Y-[T~(PL&

+

3 yield TV’

Therefore

GT?

by the assumption

AP%(~‘“+“~) +&(IIehII:h+’) +O,(h”Tl~A.l~,h). 2) of the theorem we have

~~An~i,~-O,(h’“+‘-‘“‘~+O,(lte~~~:,”

h-P’)+O,,(h’a-p”)

a=min (q,m).

IlA&),

(1.10)

Starting with (1.7) we can obtain the first evaluation of Then the repeated Ilehll,h and I\A,,li,h. applications of (1.10) yield IIA,li,h=,H(,'"+'-"') and the theorem is proved. Note 2. For p-_o,*Cm we can refind the conclusion of Theorem 1 in /ll/. For p=O,q=m= 1 we can refind even the conclusion of Theorem 1 in /6/. Besides that our assumptions about the smoothness of T,Th are fewer than in /ll/ 5. Theorem 2. Assume that 1) as in 1) of Theorem 1; 2) there exist two constants we have w=E,,, Then there exist the elements the expansion (1.6). Proof.

Let

w&a,’

c and

p,c>O,p&O,

independent

of

llT~(v~)-Th(u~)II~h>ch~'IIv~-uhll,h. wjeE,, j=m,.... n, independent of

be the elements Vh=vh~,

Uh==

determined

by the Lemma

hE(0.H).

so that for

hE(O,H),

2 with

u=u'.

so

vh,

that we have

We put

h’Twj. PlhU~+PlhSn. S,,=f: J-m

Then by Lemma 4 we have (ITh(Vh’)-Th(p,,u’+p,hSn) Then by the assumption

Ilth=&(h(“+“‘).

2) of the thoerem we have lbh*-P ,hu’-p,hS.(l,h=c-‘h-PlOH (h’“+“’ )

and the theorem

is proved.

Note 3. For ,,-o we can refind the conclusion of Theorem 2.2 of /12/. our assumptions about the smoothness of T.Th are fewer than in /12/. which depends 6. Now let I$, be a linear operator: Eu+E,h where M are p are constants, M>O, pa0 independent of Mh-P’ we consider the iterations h= (0, ff) Vh‘l+“=Vh(” -&T&J:”

).

Besides

that

on h=(O, H) so that ljBJl< h=(O,H). Then for a fixed ;l.li

52 Theorem 3. Assume that the operator that is, for v,UEE,,, we have h=(O, W,

m(")=~-&Z'~(v),"~~~~,

is constructive

for each

II~~~~--cp~~~ll~n~PII~-~ll~~, where m.

OO, ~20, independent of h=(O,H). Then the equation (1.2) has a unique solution v,,* and there exist the elements w,=E,, ( n, independent of h=(O, H), so that we have

j=

(1.12)

Note 4. The existence of vh'may be considered we want is the existence of (1.12). Proof. Since 'p is constructive solution vh' and I,$ -+uh* when 1-a. it is not difficult to verify that

as an extra

result,

the main conclusion

it is obvious that the equation u=q(u) has a unique It is also obvious that v,,' issolution of (1.2). No"

(I+*) IIIh. ll"h -"h(i'~),I~p~~"n("-":l-~' Hence

we have IIvh.-v:' II

From

(1.11) we have

Hence

(1.13) Now we choose n

(0) “A

h"wj. -Pihu'+p,n z I-=

u=u*. Then by Lemma 4 we have where w,,j-m,...,n, have been determined in Lemma 2 with ). On the other hand, by the assumptions about&and p we have llTa(vlp' )IJzh=Os, (h'"+'" Il&ll=G(h-'9, Then

(1.13) yields

and the theorem

is proved.

7. Theorem 4. Let independent of h=(O,H),

Assume

Eu=E,h. so that

that there exist constants

II[Z’h’(“h) 1-'/
to a neighbourhood

o,, of

M,p,g:

M>O, pao, q>o

IIT/(~h)llGfh-"

.plhu* which contains the element

the u:,=E,,j=?n,..., n, have been determined in the Lemma 2 with "=a'. Then for n large enough and h small enough the equation (1.2) has a unique solution and we have the expansion

where

Note

5.

Proof.

Anaogous

to Note 4.

we write down the Newton'smethod

for the equation

(‘+“=Uhtl)_[Thl(y~l’)]-iTh(“~‘)( Uh and choose

Then for

the first approximation

hE(O,H)

we have

as

(1.2) :

c,,'

53

Hence by Lenrna 4 we have A'&&(A-oP+~II

)ilr,(V:"'))lul=OH(h'"+*-rp-.").

So for n large enough to get ni-h-2p-q>O

and then for h small enough to have KBc'], the Kantorovitch's Theorem (Theorem 6 in /14, p.632/) is available, that is, the equation (1.2) has a unique solution v,,'in uh and ,]r;-r:D, Illl=Otl(h(l+b-'~-.)r). Therefore

of

... . ho):

the theorem

is proved.

2, Multiparameter formulae for operator equations. H
<

For a quantity cp(h)=cp(h,,...,h.) we write v(h)=0,(]]hll"), a being a real number, if there exists a constant M>O independent of h&8, so that (m(h)(
E***h'
lim Ilphull~=llu(l~, u=E,. lW-0

Assume yet that to each k(h) we can construct an application T,O:E,o-E,h being an approximation to T. By the abstract (multiparameter) finite difference method we want to say all the process of constructing the spaces Eo, the operators p,&the application Th. and replacing (1.1) by Tho(uh) =O, (2.1) W=E,h. This method is denoted by The. Assume that the equation (2.1) has a unique solution u*' for e.ach h=(h). Then the expression ekPnr'-p,,,n~~Elh is called the error of the method and ]]eh]lln the aboslute error. If (]el]],h+Owhen ]]A\]+0the method is called to be convergent. If there exists a positive number a so that ()e,Jl,p-OX(Ilhl~) the method is called to have an accuIf there exist two numbers r,h (~>O,O
c i: w,,...,.\\ ,h==OX(Ilhll’“+*~‘)

11 eh-P,hxz 4-m

(2.2)

,,+...+,a-i

the error is called to admit an asymptotic expansion with formulae. n. This expansion is of the multiparameter

respect

to o parameters

up to order

In order to simplify the writing of the formulae we introduce 2. Some abbreviations. some abbreviations. Let us denote by I the set of a-dimensional vectors with non-negative integral components, i.e. ifl if t-(i,, . . . . i.},i,-integer>O. If &I we write If] for denoting the sum of all components of i: Ii]++... +i. amd we write [i] for denoting the index i,...i, for instance mli,means w,, ,0. Now for &ml and h=a that is i={i,,...,i.),h--(h,,...,h.} we write h- for denoting the product A,'1 ...h >. So we have (h*')r-(h,'t...h.'~)~-h,"T...h~. With these abbreviations

the formula 11Ch-Pth2

(2.2) can be written

simply

as

(A")%,, 11,h=6dllhll’“+““).

c

I-n 111-i Henceforth 3. The family of starting methods. methods Th. having the following properties. Thesis 1'. There integers a,m (n>m>l),

Q!‘, lil -m..

of

he,%,

11 Tm(p,,u)-pu[

so that for

The is Frechet

a*-constants

a positive number and the operators

two Hcl, FI,,: a-+

we have

111-c

differentiable

and (2") denotes

UEQ,

rhO of the

T(u)+% r, W’)‘F,I,(~)]I)~ =~x(llhll’“+‘“). ‘-In

where

only the family

exist two real numbers r.h (y>O,Occk
, n, independent

Thesis 2'. FA,h we have

we consider

(s,...,s),k

up to order v with m(v+l)~n+l

and for uh,UIE

times.

A(u): D(A)cE,+E, Thesis 3'. There exists a linear operator a parameter and can be approximated by ThO'(u) on D(A), i.e. for

which depends on II-ED(A) we have

ue&,

as

54

il~~.‘(~,~u)~,~t~-p,A(u)c~Il,~-O when JI~(J-_o i=m,...,n, has a unique solution

and at the same time each equation n(n)~=g&,', with Q,'zQ:tl.

Thesis 4’. hdf,

so

There exist the operators that for II'%&,wES&', i-m,. .., n,

Cu,:Q,X~,'+62 :A{,Ii]-1.2, . . . . n-i, we have

u.%?,'cD(.A)

independent

of

II n-i

c c

(h@‘)‘G,,, (4 WI

111-t

i-1

Ill

Ih=OX(llhll(“-‘+~“),

G,,,=O.

Thesis 5'. There exist the operators r,+...+c+ul

* G,,,: Q,XR,'~X... k=2, 3,. . , v, Isj=m, .. . , n,

i,=m,. . . . n, ljl=O,

l,...,

n-km, izzzkt

of

'k%.

uq&,

so that for

w,.~EQ,“I,

we have II

T2 (p,rd (PthW [,,I, . . 1plh~I.,I 1n-m

cc

(ha’) q:, p-0 111-s

PM Note 6.

Analogous

(4 W[#lll. . . >w

-_O1r( Ilhll(‘-*m+L)’1.

to Note 1.

4. The results. we can prove four lemmas analogous to the Lemmas 1-4 and after that the following four theorems. we always assume that Th.~t, and u'EQ, is the unique solution of the equation (1.1).

Theorem 5.

Assume that 1) the equation (2.1) has a unique solution vh'; 2) II[T~,‘(P,*~‘)I-‘I~~M~-~’ where M and p are constants M>O, OdpGm, 31 the method Tha has an accuracy of order pi with q>p. Then there exist the elements wlr,'E,.]j]-m,..., n, independent of

independent hd,

so

that

of h&S

;

we have (2.3)

5.

Theorem 6. Assume that 1) as in 1) of Theorem 5; 2) there exist two constant c and p (c>O,pbO), independent of M, so that for vn+ we have u~=E,h I/T~.(~~)-T~.(~~)lI~~ch~'Ilu~--l(~~~,~. Then there exist the elements h=S so that we have Q,, IjI=m,.. . , n, hkpendent of the expansion (2.3). 6. Let B, be a linear operator: .l&-+E,, which depends on h&S, so that ~~B$GWrPT where M and p are constants, M>O,p>O, Then for a fixed hm we independent of hd8. consider the iterations ll+'r_yh(') B T (s;)) -hha . Vii UEE,,, is constructive for each Theurem 7. Assume that the operator cp(u)-u-BhTn.(u), hc%, i.e. for v,n~;B,~ we have ]]cp(u) where OO, qZ0, independent of ha. Then the equation (2.1) has a unique solution II,,* and there exist the elements wu@,. so that we have ljl=m,. . . , n, independent of ha,

II e*-p,h2 r, UP9%, 11,~OX(Ilhll(~+L-~--p)T). t-7”Iii--i 7. Theorem 8. Let E,,,=EII. independent of hE.#, SO that

Assume

that there exist the constants

]][Th.'(~h)] -‘I(Gfh-P’, for any vk belonging

to a neighbourhood

oh of

M,p,

q,bO,p>O,q>O.

IIThs”(un)II
PlhU. which contains

cu,‘,=E,, ljl-m,. . . , n. Then for n large enough and (Ihl( small enough the equation WE' and we have the asymptotic expansion for the error:

the elements

where

(2.1) has a unique solution

55

of

wu, are independent

where

3.

hEa.

Some numerical examples.

In this paragraph Example

1.

we give some numerical

Consider

examples

for illustration.

the problem

Au=f(r, y), OCz
O
U(0, y)=u(i, y)--u(s,0)=u(z,1)=0,

Its solution

is

u(r, y)=sinnzsinny.

We use

z,=ih, h-l/N, k-l/M, y,=ik, where N, M are positive integers satisfying O
the grid

any discrete

I% Y,)

function

AhuI,=h-1(Ui+,,j-2u,,Su,-,,j) +k-‘(uii+,-2uij+vt.l-,). Consider

the discrete

problem

Ahvi,j=ft.hOO, k>O so that v(P; h, k)--U(zp, yp) =h’w,,o(zp,

YP)+

k’wo., (zp, YP) +h’wt.o (~3 YP) fh’f-*w,s

(G,

VP) +

k‘w,,t(zp, YP) + . . . +h*“w,o (zp, yp) +h’“-‘k*w._,,,

(zp, yp) +

..

k’“w,,.(zp, y,)+O(h;“+z+kz”+z), where n denotes

any positive

$

integer.

Form that we deduce

v(P; h, k) ==u(zP, y,)+O(h’+k’).

Then u(P;h, k;h/2,k/2) may be considered as the first ameliorated approximate value of u(fp,yp) The numerical results are presented in the Table 1 in which the last line is added only to help us to make some comparison of approximate values at p('/,,'/,).

I

ExamPle

2.

Consider

I

the problem

I

I

Table

1

/lo/:

3 yw'_ 9 Y',

o
y(O)=4,

y(l)-1.

Its solution is y(s)-4/(1+r)*. We use the grid (r,} with one parameter h: z,-ih, h-l/N, where is a positive integer. For any discrete function v defined over this grid we write 0, to denote ~(2,) . We consider the discrete problem

N

$(

3 u,+,-2u,+Uc-r)--IJ,*, 2

vr-4,

V&.=-i.

Resolving it we denote the approximate value calculated at a fixed grid point P of the grid with parameter h by u(P; h). Since here y(t) and its derivatives are smooth enough we can prove the existence of n continuous functions o,independent of h>O, so that u(p; h)--u(zr)-

where n is any positive y(rp):

u

etc.

The numerical

integer.

&P’q+O(h*‘+‘). I-I

From that we duced

the ameliorated

approximate

(P;h;S;f)-~u(P;~;a)-~” ( P;h;+)-y(zp)+O(h’) results

at the point

P(0.5)

are

presented

in the Table

2.

values

for

56 Table

Profiting by this occasion we expressourgreat thankfulness Zhidkov for precious advices and remarks, and effective help.

to N.N. Govorun

2

and E.P.

REFERENCES 1.

SAMARSKII A.A., The theory of difference schemes (Teoriya raznostnykh skhem), Nauka, Moscow, 1977. 2. MARCHUK G.I., Methods of computational mathematics (Metody vychiclitel'noi matematikil, Nauka, Moscow, 1977. 3. MARCHUK G.I. and SHAIDUROV V.V., Increasing the accuracy of the solution of difference schemes (Povyshenie tochnosti reshenii raznostnykh skhem), Nauka, Moscow, 1979. 4. AIRYAN E.A. and ZHIDKOV E.P., On one method for increasing the accuracy in the numerical solution of elliptic equations. Pll-12709, Joint Institutue of Nuclear Research (OIYAI), Dubna, 1979. Improving the accuracy of approximate solu5. ZHIDKOV E.P., NGUEN MONG and KHOROMSKII B.N., tions of a non-linear singular integral equation of the Chu-Lou type, PS-12916, Joint Institute of Nuclear Research (OIYAI), Dubna, 1979. 6. ZHIDKOV E.P., NGUEN MONG and KHOROMSKII B.N., Refinement of the approximate solutions of non-linear operator equations, P5-12979, Joint Institute of Nuclear Research (OIYAI), Dubna, 1979. 7. VOLKOV E.A., Investigation of one method for improving the accuracy of the method of nets in the solution of the Poisson equation, in: Computational mathematics (Vychislitel'naya matematika), Issue 1, 62-00, Izd. Akad. Nauk SSSR, 1957. 8. WIDLUND O.B., Some recent applications of asymptotic error expansions to finite difference schemes, Proc. Roy. Sot., A323, 1553, 167-177, 1971. 9. JOYCE D.C., Survey of extrapolation processes in numerical analysis, SIAM, Rev., 13, 4, 435-490, 1971. LAURENT P.J., &ude 10. des pro&d& d'dxtrapolation en analyse nume/rique, Thesis, University of Grenoble, France, 1964. 11. STETTER H.J., Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations, Numer. Math. 7, 1, 18-31, 1965. 12. PEREYRA V., Iterated deferred corrections for non-linear operator equations, Numer. Math. 10, 4, 316-323, 1967. 13. TA VAN DINH Sur les formules asymptotiques de l'erreur en mgthodes aux diffe/rences finies, Acta I4ath. Vietnamica, 2, 1, 116-140, 1977 . 14. KANTOROVICH L.V. and AKILOV G.P., Functional analysis in mormalized spaces, (Funktsional'nyi analiz v normirovannykh prostranstvakh), Gostekhteorizd., Moscow, 1959. Translated

U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain

Vo1.24,No.5,pp.

5662,1984

by E.L.S.

0041-5553/84 $10.00+0.00 01985 Pergamon Press Ltd.

APPROXIMATE AND NUMERICAL METHODS FOR CALCULATINGTHE COMPOSITION OF AN EQUILIBRIUM PLASMA* P.D. SHIRKOV

A simple mathematical model based on approximate separation of dissociation and ionization processes is proposed for calculating the particle densities in a plasma which is in local thermodynamic equilibrium in the region of dissociation and primary ionization of the atoms and molecules. Accurate solutions of this separated model are found and "a priori" A separated model is constructed estimates of its accuracy are obtained. Comparisons are for calculating the multiple ionization of a medium. made with numerical calculations of the composition of a plasma. An efficient iterative process is proposed for calculating the equilibrium densities of a dissociated and ionizing medium.

Introduction. 1. Calculations of chemical and ionization equilibria lie at the foundation of the calculation of thermodynamic functions and gas transfer coefficients and are exceedingly tedious *Zh.vychisl.Mat.mat.Fiz.,24,9,1372-1380,1984