Difference schemes with a separable operator for general second order parabolic equations with variable coefficients

DIFFERENCE SCREMES WITR A SEPARABLE OPERATOR FOR GENERAL SECOND ORDER PARABOLIC EQUATIONS WITH VARIABLE COEFFICIENTS* Ye.G. D'YAKONOV (Moscow) (Received

22

January

1962)

This work is a direct continuation of the works [ll- [31 and is devoted to the design and examination of economical difference methods based on for second order parabolic equations separating the difference operator, with variable coefficients. We describe the results which were read to a meeting of the Moscow Mathematical Society in February, 1962 (see [41) and appeared in part in the first part of M . It should be noted that economical difference methods which differ from those studied here have been applied to similar classes of differential problems in several later works [d - [81. The basic apparatus used here is that of a priori difference estimates. There are similar estimates for the most simple cases in [91- [Ill.

1. Description of the difference scheme I. Initial problem I. In the cylinder find the solution of the equation

4~ =

DI (;T,, (4 D,u) +

i 1. c-1

l

Zh.

uych.

mat.,

QT =

G X

[O <

z0 Q

$j(ir,(4 D,u + 6 (4 4 + f (4,

r=1

4. No. 2. 2’78-291. 92

1964.

2’1

to

(1.1)

General

satisfying

the initial

rccond

order

parabolic

93

equations

and boundary conditions SE&!?.

u I*==0= 9’ (2’) and ZJIs = 9 @I,

U.2)

Here 3 is a closed bounded region In the space {x ‘1 = {(xl, x2, . . . , xp) 1 consisting of a finite number of p-dimensional parallelepipeds with boundaries parallel to the coordinate planes; the other notation is borrowed from [21, x = (x0, x ‘),

2, = ii”

(z) >

rl

con&.>

=

ad - =

0;

&;

0.3)

where al > 0, tjS (s = 1, 2, . , . , p) are arbitrary

real numbers.

It follows from condition (1.3) that equation (1.1) is parabolic; when p = 2 this condition is the same as the ordinary condition for parabolicity.

2. Function spaces. To formulate the conditions laid on the differential properties of the coefficients (1.1) and other functions we introduce the following function spaces:

Ho - the space of functions bounded in QT.

H[(ln) (2 = 0, 1, . . . . p) - the space of functions having In @ derivatives w.r. t. xl of order not higher than RIsatisfying the Lipschitz condition r.r.t. xl; F’“l

(s = 1, 2, . . . .

p)

-

the

space

of

functions having deriva-

tives bounded in QT which contain not more than ml differentiations w. r. t. each xl different from x0 and xS, and not more than m2 dlfferentiatlons w.r. t. x0;

A?” (1 < k < are bounded a0<

in

1, a,\
p)

- the

space

of

functions

having

derivatives

QT and of the form DFD:’ . . . DFu=D:D’u, (s

and not more than

=

which where

1, 2, . . . . p),

k of the numbers as (s = 1, 2, . . . ,

p)

are different

Ye. C.

94

D’Yakonov

from zero; xf having where

(1 <

k <

p -

derivatives

1;

s = 1, 2, . . ., p)

- the space of

which are bounded in QT and of

and not more than k of the remaining

the form

functions

DzD%,

al (2 # s) are nonzero.

The norm in each of these spaces is introduced as the sum of the norms in 1,~ of all the derivatives mentioned in the definition of the space.

be the set of space 3. The net. Let {ih) = {(ilhl, izhz, . . ., i,h,)? net points i = (i,, i,, . . . . i,), h = (h,, h,, . . ., h,), i, an integer, points if h, > 0. We shall call i”)h and i(*)h neighbouring i!l) I

-

i!“‘j

<

1

(s

=

1) 2,

. . . , p).

The set of net points belonging to 5, together with all their neighhours. is called the set of internal points and denoted by R,,. The set of net points an internal

ih E point

boundary points.

5 such that is denoted

each has at least

by lib.

We note that we shall

Ti and vectors h such that r,, & r. defined in the same way as in [21. We take

the notation

the same work, the only stead

*r

vi, vi,

of

The space-time

&vi

only those

regions

net C&v = ah x 0~ is

;r

being

which is

$, is the set of

be considering

at, vr,

difference

of h in the definition

one neighbour

Then rh = si;, \

&A’,

E,

Asvi, &vi, &

that

h,

is to be written

and Arvi.

We shall

vi,

from in-

sometimes write

instead of VT, Asvr, &VT &v? to save space; higher S are defined with the help of the successive application of which have already been defined. We shall denote mixed differ-

0, r.Q, u;,. v;; differences differences ences (s

w. r. t.

= 1, 2,

the space variables

. . . , p) bv

vu (I v 1 = j,

A:' A:.

. . A~v,

in which

a,) : when necessary

a, <

we shall

1 write

instead of v, and so on. We shall always denote constants which do not depend on the net by i! and 6, different constants often being denoted

VP2

second

General

order

parabolic

equations

95

by the same letter. The scalar product of the net functions and O(W) C: &, respectively,

the norm in &

v and w defined

on o(u) c ah

is the expression

of the net function v then being

11Y Ilgs = [u, v]“’ =

[US, 1]“* 4. The difference prob len Ih. We shall solve problem I approximately, using the net method. To do this we construct the net &v (see Para. 3) and introduce the notation

i,vy= i,“vy = ii,

(.;t

(tn, rh) *

A&’

+ b,* - (t,,, ih) &VT +

+ -c, (t,,, ih) v; = (a, - v:, )-% + b,v;8 -

+ csvn,

are known functions obtained from the coeffiwhere a,, a,, ._ b,, b,, c,, G cients G8,&, F8 by the method indicated below. In addition. we consider the difference operators omit the index n:

A:, A”, B”,

ATvn+r= _4$+z = (E _ v&) $+I;

in which we shall also frequently

Anvn+rz.zzA@1

= fi A,$‘+‘; S=I

On the net Qhr we define a sequence of net functions v satisfying initial condition u;= the boundary

cp. 1

(ih E Qt,),

(1.5)

condition

and which for ill E 2, satisfies

the following

the

system of finite

Ye. C.

difference

D ‘Yakonov

equations:

$A??;+’ =fBv;+

fl.

(1.7)

Relation (1.7) and conditions (1.5), (1.6) define the difference problem Ih, the correctness of which will be proved In Section 2. We merely note here that since the operator A Is separable, the transition from vn to untl in the problem Ih can be realized with the help of the algorithm in [31 and requires in all only x (hlhg, . . . , !I~)-’ arithmetic operations (we recall that the region 3 Is composed of a finite number of parallelepiped with boundaries parallel to the coordinate planes). We shall also give the conditions under which problem Ih approximates to problem I in the norm !& with order of approximation 0(-r + (hi 2), where

In fact,

for such an approximation it is sufficient

iT,= %, 4 E

Ho (Ql

IJE

4 + -a,,

6 = b, +

cr,E H, (2); Dl=l, E Ho;

At:‘, u

E

D&a E f;‘, H8

to have

; = c8 + c,,

b,,

b,,b,, ~8,c,, f &Ho; c, E flt” ;

b,Efl;‘t

(1.8) (f.8’)

(a = i, 2; I = i, 2, . . . , ~1; (1.9)

(31, Zu~&(i)

(1.9’)

UEA'dO,

where u is the exact solution of problem I (assuming that it exists). in addition we assume that

r/h:<

(s = 1, 2, . . . , p,

2 < 00

If

I> 0

is an arbitrary number), then there will be an approximation of the seme order if conditions (1.8’) and (1.9 ‘1 are replaced by the weaker condit Ions: a8

E

#I$ ,

D&8 E fl$

(

b,Eq$,

c8

E

#I;

(r-i.2

UEA:‘O.

t....p);

(1.8’) (1.9’)

2. 'lbecorrectneam of the problem I,, Attempting to obtain a difference a priori estimate for the solution of problem Ih with zero boundary conditions, we first establish, in Paras. 1 and 2 of this section, the necessary auxiliary results for this

General

second

order

equations

parabolic

97

purpose. 1. Scalar product transformatim of rh lying

on the side boundaries

Then, on the basis

of the

N-1 x ur, kf,+1 -

we can establish

the validity

[z, vg*l = for

any net ,function

function

to the hyperpliIne

by parts”

5 (at, is =I

formula

(See

&,-A vi, + U#N -

x,=0.

1121):

Wo,

of the formulae

I&*,

VI,

u with region

z with region

Let TS denote the subset

of ii parallel

“station

vi3 = -

f,==O

formulae.

Iz, v;“ = of definition

of definition

I%,, VI

o(u)

o (z) 2 (&

2

U I’,)

Rh and any net which becomes

on Ts. We recall that the definition of a scalar-product is given in Para. 3 of Section 1. In addition, we have the following formulae for differentiation of a product (see [121):

zero

We note also

that

[U +

v,

zl = lu, zl 4- fv, zl, if either z on

or u on

0 (4 \ (0 @) fl 0 (al o (u) \ (O (u) fl o (2))

0 (u) \ (0 (v) n 0 (2)).

or u on

becomes zero.

2. Lemmas. zebu P =P,

2. Let the symmetric p-th order matrix P = faii)

have the few

+P,

and suppose that for the arhitrarv jp.Elbj< Then the 2p-th

order

matrix

(1 P =

real a&“f:,

vector

$ = Ccl.

{r,

. . . , fj,) (2.1)

raea>o* is positive

definite,

i.e.

Ye.C. D'Yokanav

98

where q = (?I, Proof.

qP)

is an arbitrary

Instead of the vector

vector (2.1)

7-12, . . . ,

2p-dimensional real vector.

& = (El, E,, . . . , &,) we consider the

E’ = (&i, &;, . . . , &) =(VZ

El, ‘t/C

men

Es, . . . , ‘t/G&)-

is rewritten in the form

This condition is the condition for the matrices (1 - o)E - P’ and (1 - u))E + P’ to be positive, where E is the p-th order unit matrix, 0

a;,

. . . sip

ai1 0

pt =

i \a;1

tii

. .

l

a;P . \

‘.‘.‘.‘O’ /

We conclude from this that all the eigenvalues of the matrix P’ satisfy the inequality to verify *A‘(F) p=

1 A, (P’) 1 Q 1 -u

that the eigenvalues of the matrix

(i-a)

rl’) = PC

where q’ = (q;, q;, . . . , q&)

rl’) -

0 P

P’ G

1

are equal to

,

is an arbitrary

real 2p-dimensional vector.

q = (qr,qa, . . . , %%A

qw‘=&”

‘1, =-&,

(2.3)

(J (q’* rl’) > 0,

Going from the vector TJ’ to the vector

we obtain

(

not difficult

are not negative and

(0”:) (W,

from (2.3)

p’ =

is

Hence the eigenvalues of the matrix

(s - 1, 2, . . . . p).

p+

It

(s = 1, 2, . . . , p).

(r=i,2....,p),

(?%I, q) > ay (q, q).

Lemma2. If the operators such that

L,

oal,(5) > r = const.>

0;

and L,, defined by formula (1.4) a,,, b.,, cakE A):

(k = 1.2),

are

(2.4)

Genetcl

second

order

parabolic

then for any net function y defined on Qh and satisfying yi = 0 for

99

equations

the ~ndition

th E ‘I;(,

(2.5)

the inequality

will hold,

where

s1 #

If we write

[g,, L,,L,,yl,

We transform the first from Para.

We make similar

out we obtain

term on the right-hand side,

?hJ,xJ,' (d,~),.,j:~,l

transformations

we obtain

products

of the form

which depend on the coefficients

+

on the other

the sum of scalar

which depend on y and are only ties

ask, bBk,cIlr (k = 1, 2).

using formulae

1:

+ “,I

As a result

is a constant which depends only

of the coefficients

on the norms in flit Proof.

s2, nz\< T; ikf

f(a,,Xh+;ti, @J,)XJ,*y):;,l.

terms.

For example,

which contain

yX,rJ,, &,,

yX,,, y,

(u~*),~~~,(bJK)XJl,(cJk)+,

quantities and quanti-

uJk, b,,

cJk

(k = 1, 2; 1 = 1, 2; k # Z>. We can find a lower estimate for all the products not containing yxJlxJ, with the help of the inequality

scalar

fa, hl>

-

f {[02, 1) + p,

I)}.

(2.7)

100

Yc.C.

D’Yakonov

To estimate the scalar products of the form [yzhz+, al = [r, a] the inequality --

; (B [P, i] +-& [a’, ll}<

where E > 0. Since sufficiently contain

b, al&{s

[(Y,,;)‘.

small in (2.8)

[rat 11 +f

%=~,I > 7’ I(y,+!‘,

11,

we use

I=‘, il}p

(2.8)

by taking e

we can ignore all the negative terms which

[(y,. x, )*, 11 in the lower estimate for

[L,L,,y, y]

I

parison with ’ f rB [(y *Jr

11.

in com-

We can find a lower estimate for the

remaining negative terms by replacing differences of the coefficients ark, b,, 9 ~8~ (k = 1, 2) by the maximumvalues in QT of their absolute values for all possible values of h. Since these maximumvalues depend only on the norms in Ho of the corresponding derivatives, the lemmais proved. Lemma 3.

If the operators

L,,,

L,,,

. . ., L,,,,,

defined by formula (1.4)

are such that conditions (2.4) are satisfied for all k = 1. 2, . . . , m, then for any net function y, defined on Rh and satisfying condition (2.5) we shall have (-

l)mTm-l

[L,L,*

(2.9)

. .

- MI ~ ,g 1 [Pw,

419

where the constant M depends only on the norms in cients

Ati

of the coeffi-

a,,, brk, elk (sl < s, < . . . < s,,,; Sk = 1, 2, . . ., p; k = 1, 2,

. . ., m; m < p). Lemma3 is a generalisation similar.

of Lemma2, and so its proof is exactly

3. First a priori estimate. We consider the net function y satisfying the difference equation*

l

The notation

rlx,I Is taken from [21.

General

. .

order

parabolic

+2

101

equations

~L,yn+

1)pTp-1LlL2 . . 4yn+l=f+

(-

+

second

EC1

(0 < nz < T, ik

&al&

Q,).

=I

I+# the boundary condition

Yin = 0 ior ih E and the initial

rh7

0 Q nr \< T,

(2.11)

condition

(2.12)

(UE Q,). We assume that the conditions

(2.13)

a,EH,(O), are satisfied. a, E H, ((0, by a suitable

It is not difficult all

the other

choice

of

to see that,

conditions

of

apart

(2.13).

a, =&a,

and

as

(2.14)

II_a# lip < +

a,>+=r;

from the condition

(2.14)

can be obtained

from conditions

-

Theorem 1. Suppose that conditions (1.8). (2.13), (2.14) ditions of Lemma 3 are satisfied. Then for the net function the conditions (h: >

O),

difference

we can find

(2.10) - (2.12)

such that

for

all

T and h

estimate

a priori

will

to>0

(r <

(1.3). and the conr satisfying

and ha = (hy,@,

z,,, h, <

hz)

. . .,I$)

the following

hold:

k

[(Yk)2, 11 +

2 1, (y”) < n=,

MK

(2.15)

(f, q~),

where 11

(Y")

=

t,

vIx
II

[(Vyn)2,11,

lVl
k-l K

(fv

cp)

=

T

Iz

n=o

W”)“,

11

+

by,

11 + I, (cp),

* - *

Ye. G. D ‘Yakonov

102

~T
and the constant M does not depend on T, h, k.

Proof. We make a scalar multiplication of equation (2.10) by -ryntl and sum the resulting scalar products over n from 0 to k - 1. We obtain

5

P (y) = z

Iy”,, yn+ll +

TZo {Pl (yn+l, y”) +z 8, X8* [L,L,Yn+l~ yntll -

n-0

-_r

yn+q+ * * - +

[L,,L,,L,,yn+l,

y,*

l)P

(-

v-1

[L,L, . . . Lpyn+l, y*"l

=

where p, (yn+l,

yn)=

$)[L*yn+1,

yn+11-

lF8bl,Y$

r=1

It is not difficult

8 ‘;;l

v

yntl I-

#,f(..Y:, 1; y”+’I* ’

I

to see that

k-l

z B [YL,yn+ll> + [(yk)Z,11 - + TIC0

(2.17)

[(p2,Il.

We find a lower estimate for (2.18) +

x [yz l+s

( (qyn+1)_

] Xl

=I

$ {[b,y!s,

yn+l I -

)-

k8, (Y”“)“])

XI

#=l

Noting that

lY!L x,,

(al‘y”“);‘l>

$ qyy + y;,yy + YZlYy -

e [(Y;~Y,11 -

where E > 0 is a constant from (2.8).

+ Yh y;y, %I

-

+ [(y*+q2, 11,

we arrive at the inequality

General

Due to conditions

order

second

(1.81,

for

parabolic

sufficiently

equations

small

103

IhI we have the inequal-

ity

(2.20)

Furthermore,

it

follows

from (2. I4) that

I_a:I 1<

$ i

(2.21)

[(Yy” + (y:,)‘, 11.

r=1

Therefore,

choosing

Lemma 1 we obtain

TO so that when from (2. 191,

k-l

r

Y”) >

(2.20),

Z,

We make a lower estimate

for

the scalar

of

j&,1;,, . . . Lrmy*+l

p (9) >

which contain

f

using

MykE 11 +

(2.17),

6 5

(2.22)

1, (y”) -

it

hi.

We

of

2 w2,

n=O

on the left-hand 2),

(m >

[(y”)“,

side

using Lemmas 2 and

is not difficult

to obtain

the

(2,23)

11 +

which do not depend on T and h for

now find an upper estimate for P(y), (2.16). It is easy to see that

k-1

F (Y) < r

products

n=o

with 6 > 0 and M > 0 constants h,\(

using

(2.22)

Jr { % ;

n=1

hand side

q/8,

7 r 2 2 [(Y;y, 11 a=1

3. As a result, inequality

or-1 1 <

(2.21)

n=l

(2.16)

1 a: -

P

k

r, Pl w+l, n=o

z <

1I + er i: n=o

$j[(y;6)2,11 +

s=I

where t > 0 is the constant from (2.8). a suitable choice of E we obtain

Vl (9) = N9k)2? 11+ i, -I, (9”) <

M

considering

$T

i

cp) +

the right-

11.

(2.24)

m-0

Combining (2.23)

{K (f,

f(y)“,

T\
and (2.24)

with

(2.25)

Ye. c. D’rakonov

104

where M does not depend on r and h. From this, [121), we derive

‘[(ylr)*,

in the usual W&Y(see

11 + I, (yk) < MK (f, p), and this, together with

(2.25), also gives (2.15). This proves the theorem. It clearly follows from Theorem 1 that the problem Ih is correct in the sense defined by the a priori estimate (2.15). Note 1. It is clear from the proof of the theorem that the constants ~~ and ho depend, first, on the size of the region QT and, secondly, on the norms of the coefficients in the corresponding spaces. In a number of cases restrictions on the smallness of T and h for the second reason become superfluous. For example, if all the al, = d (I# 8) or the ab do not depend on xS, then the restriction of the smallness of h falls away and the condition u,ER,(O) becomes unnecessary; if the coefficients og do not depend on x0, the restriction on T falls away. Note 2. By altering the proof of the theorem somewhat. we can obtain the a priori estimate (2.15) for r d T ‘, where instead of K(f, 9) we have

888 [51). The restriction Theorem 1 (T’ ( TO).

Note 3. If we replace

on To in this case will be weaker than in

fn

by 7” =

$

ii, (d:F”),

Pl

in (2. lo), where d, E Ho, then instead of (2.15) exact estimate

3. bwergence

we can obtain the more

and estimates of speed of convergence

1. Proof of convergence using the first

a priori

estimate.

Theorem 2. If the solution of problem I exists and the approximation conditions (1.8’)-(1.9’) tind the conditions of Theorem 1 are satisfied,

General

then the difference has the estimate

second

order

parobot

of the solutions

[(z~)~,II +

ic

105

cquotiont

zn = un - v* of problems I and Ih

ijl1, (2”) =

0 (r + 1h I”)‘.

(3.4)

We can easily prove Theorem 2 using the a priori estimate (2.151, since the function z* satisfies the relation (2.10) with fa = O(T + lh12) and zero initial and boundary conditions (2.11). (2.12). We recall that the approximation conditions (1.8’). (1.9’) can be replaced by weaker ones if we assume that z/hi \< 1< 00 (s = 1, 2, . . ., p; 1 > 0 is an arbitrary number). In this case it is sufficient to have

2. Second a prior&

estimate.

‘Theorem3. Let the conditions of Theorem 1 be satisfied, and suppose that the net function y satisfies (2. lo)-(2.121, fn in (2.10) being replaced by

Then there exist T,, > 0, ho (hz> 0), such that for all the following a priori estimate holds:

r < r,, h, < kf

where

* - - + zp-’

2

[(VF;)*, I],

JVlr;P

kT <,<, Proof.

the constant V does not depend on T, h, k. With the method used to obtain (2.15) we arrive at the in-

equality

We find an upper estimate for

106

Yc.C.

To do this

we use the transformations

2 to reduce

p-l

where

each of

[Y,,,

*,... x

V,,

9 %=a, #rn

products

(-

- - - ~~Fzc, I x,,...x,,l

I VI I <

IL,

. . . L,,F;,

m-:,c,m~ IL,, - . . L,F;,

i

n=l

(3.4).

gives

M{z

*** +

2

{

(3.4).

+ . . . +

we obtain

1 (Y”) <

aIk,

u E H, (3)

ZJ,E

H, (I),

(2)

the coefficients

of

Theorem 1 are satisfied;

8 ([v ~~+l)~, 11 +

(-

ks

I(f)$

n=l

zP-l

(1)

the case when,

1)pzp-1L,L2 . . . L,F.

(3.3’)

the estimate

11 + t2 ,,F, [(VFy

x

on the basis

Theorem 4. Suppose that satisfies the conditions

of

brk, csk (k = 1, 2,

We consider

6 = 1>2, . . . t ~1;

moreover,

... (3.4’)

J

of t.he second

the solution

equation

[(VP)*, l]f

II\ .

IV-P

of convergence

of differences

NV FE?, 1 I} 9

x2 ,X,,L,,L,,L,,F”

Then, instead of

9

Ivl
and this, together with (3.5). instead of (3.3). we have -

dV,F]

of the coefficients a,, b,, c, of the theorem, and we use

&‘+‘I \< P-’

+ $

Y=ft

[Vy,

v, VI. d

of the coefficients

m

z

3. Proof

of Lemmas 1 and

x

+

. . .( ml. We note now that the differences obtained are bounded, due to the conditions inequality (2.8). As a result we obtain

[(YkJ29 11 +

l)m$“-l

I VI I < my d is the product

1,

IVI\
the type

the scalar

used in the proof

to the form

m>i,

yn+ll,

D’Yakonov

estimate.

a priori

of problem I exists

?.A E A;“;

and

(3.6)

u E igo;

(1. I) are such that the conditions a,, D,a,,

b,, c,

belong

to

A:‘:_

of Then

General

second

parabolic

order

107

equations

the sequence of net functions v which are a solution of problem Ih tend to the solution of problem I as T and lh( tend to zero in the following sense: for any k (0
(3.7) for zk = uk - vk. Proof.

With the conditions

of the theorem zn satisfies

(2.10) with

(3.8) and zero initial and boundary conditions (2.111, a priori estimate (3.4’) for zn and noting that

(2.12).

Applying the

are bounded we arrive at (3.7). It clearly follows from the above convergence theorems that in the metric Se,(Q,) the difference schemes we are considering possess first order accuracy w. r. t. T and second order accuracy w.r.t. h.

Vun+l(/VI< p)

4. Notes. 1. In Theorem 4 convergence is proved with conditions under in the usual sense. which, generally speaking, there are no approximations In those cases when, for the difference problem written in the form (u is

Lv = F

a solution

side put in “divergent” operators) the estimate

is valid, then it

where

II /I, and

seems useful

the differential problem), if

of the difference form

7 = Rf,

problem,

L and R are certain

II II2 are certain ;U = p

LIC -

Rv = ZF

the right-hand difference

norms and U is a constant,

to say that the difference

problem

Tis

problem approximates

(11 is the solution

to

of the differential

and I\F 11 2 -, 0

(3.10)

as the net step decreases without bound. With this modification of the concept of an approximation for linear problems correctness (3.9) and approximation

(3.10)

will

2. With the conditions placed

by the weaker one

scheme (l.S)-(1.71,

L:

imply convergence (1.8)

I!I‘ -

r 11,- 0.

a*, n.*EII, (2) can be reoS. yS, Din,, DSnSE 11~. if, in the difference and

the requirement

i:

are taken to mean the operators

108

Ye. G.

Lnv”+l=

p+1

a I

1

x,2‘

+

D’Yakonov

vy + c,v”+~.qvn=yz; + p, + Dp,) v;,+ yn.

(b,+ DA

ss

3. If, instead of problem I, we consider the more general problem in which equation (1.1) is reduced by the equation

P

(4 D,u =

D,

i

(;, (4Dgu)

IG (4D,u

-k i

1 ,a=1

P (2) EA’dO;

P (2) E H, (11, P (21 >

+ ; (4 ~1 +f (2, u, Dlu, . . ., Dpu),

r=1

P >

0.

P (4

E

af,HO

HO (O),

(3.11)

(c’ =

u, DIU,

. . ., $,4,

acr

then, in the difference

equation (1.71,

for A and B. The proof of these results plicated derivations. 4. We have constructed

we can take the operators

involves only slightly

and examined a class of difference

more com-

schemes with

a separable operator which possess second order accuracy w.r.t.

for the case when in equation (1.1) the details of these results schemes here: 1

T

p

JJ( a=1

aIs = 0 for

in a separate article,

($1

E-

all

-

$)

=

$+‘/a +

I#

8% We

T

and h

shall give

but give one of these

i

A:“q,

(3.12)

r,,,

(3.13)

a=1

vy= cp,,

ih E 51,;

v:= I@:and

ih E

where

+-E,((n++),r,

ih)v.

For this scheme the algorithm in [I]- [31 for the solution of a system with a operator can be applied, for instance, to find &” = ;on+l- v”,

General

second

order

parabolic

equations

109

and it is then easy to find wn+’= on d- En. Whenmachines with a large store are being used, this algorithm m&ybe preferable to an algorithm which finds

v*+’ directly.

Convergence with order

0 (9 + I h 1’) has been proved in the norm

II 2 Ilw’(Rh)= { yJ a

iv* VI )” *

IVW

5. We have also considered the case (see [41) when u, problem (1.1).

(1.2)

f, 9, y in

are K-dimensional vecotrs and <,, (~),a, (x), ;, (z)

are IY-th order iatrices

with components (1;6’(~), ztr (z), zy (s) (9, r = i, 2,

For this system of differential equations the difference scheme with a separable operator has the same form as (1.5)-(1.7); it is only . . ., K).

necessary to rememberthat now vr = (vI (nr, ih), q (nr, ih), . . ., vK (rn, ih)); 4 d i +,8s6, 1

is a K-th order matrix, and so on. To find

v”‘l

we have to

solve p one-dimensional systems in which the matrix of the unknownshas nonzero coefficients only on the 3K diagonals which adjoin the main diagonal,

Therefore

t.“+’ can be found also using only

=‘(h& . . . h&-l

arithmetic operations. Theorems 1, 2, 3, 4 can be obtained with the same conditions for the smoothness of the components of the matrices alat b,, as above, and the additional assumptions CP 488, _b**5

Ua,,--Q,UG- @Y 8

I

is a real vector. In Theorems 1-4 we must 6, = (Ef, E;, f * ** I$, then take [z? I], where I = (tit 2,. , . ., zx), to mean

where

12’. II = $ IbJ? il. I=1

If we consider a system of equations with matrices then convergence in the norm

aez- 0 for

s # 1,

110

Yc.C.

with order 0 (zt t I h I*) (3. 12)-(3.13).

D’Yakonov

has been proved for the difference

scheme

Note added on correction of proof. The question of applying difference schemes with a separable operator to systems of partial differential equations of more general form than (1. I), (1.2) is discussed in the recently published work of the author in Uspekhi Mat. Nauk, 19, No. 1, 207-208, 1964.

Tranotated

by R. Fefnstein

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