Functional properties of stable difference schemes

16

REFERENCES 1. 2. 3.

4. 5.

6. 7.

8. 9. 10. 11. 12.

13. 14.

LIONS J. L.. Optimal control of systems described by partial differential equations (Russian translation, Mir, Moscow, 1972). VAINBERG M. M., The variational method and method of monotonic operators (Variatsionnyi method i metod monotonnykh operatorov), Nauka, Moscow, 1972. KAPLAN A. A., On the use of variational methods to solve some one-sided boundary value problems, in: Mathematical methods of optimization and applications in large economic and technical systems (Matem. metody optimisatsii i ikh prilotheniya v bol’shikh ekonomich. i tekhn. sistemakh), Moscow, 1980. NOVIKOVA N. M., Stochastic quasi-gradient method of seeking a maximin, Zh. vychisi. Mat. mat. Fix., 17, No. I, 91-99.1977. FEDOROV V. V., Numerical maximum methods (Chiriennye metody maksimina), Nauka, Moscow, 1979. ERMOL’EV Yu. M., Methods of stochastic programming (Metody stokhastichcskomogo programmirovaniya), Nauka, Moscow, 1976. WASAN M. T., Stochastic approximation, Cabmidge U.P., 1968. LEVISOV L. V., On the random search method in problems of constrained minimization, Zh. vychisl. Mat. mat. Fiz., 18, No. 5, 1103-1111, 1978. ZAVRIEV S. K., On seeking the stationary points in the constrained maximin problem, Vestn. MGU, Vychisl. matem. i kibernetika, No. 2, 48-57, 1980. ZAVRIEV S. K., On a method of solving the stochastic minimax problem, in: Mathematical methods in operations research (Matem. metody v issl. operatsii), fad-vo MGU, Moscow, 1981. GERMEIER Yu. 8.. Introduction to operations research theory (Vvedenie v teoriyu issledovaniya operatsii). Nauka, Moscow, 1969. NOVIKOVA N. M., A method of seeking a constrained maximin, in: Abstracts of 3rd all-Union conference on operations research, Gor’kii, 1978. GORELIK V. A., Maximin problems in connected sets in Banach spaces, Kibemetika, No. 1, 64-67, 1981. BALAKRISHNAN A., Introduction to the theory of optimization in Hilbart space (Russian translation, Mir, Moscow, 1974).

Translated

U.S.S.R. Comput. Maths. Printed in Great Britain

Math. Phys. Vol. 24, No. 2, pp. 16-23,

FUN~IONAL

1984

by D. E. B.

0041-5553/84 %lO.OO+O.OO 0 1985 Pergamon Press Ltd.

PROPERTIES OF STABLE DIFFEREWE

SCHEMES*

YU. I. MOKIN and P. D. SHIRKOV

The stability in metrics of L,, 1 < p < m, of a family of two-hyer difference schemes for the equation of heat conduction, is studied. The stability estimates obtained are shown to be unimprovable in the class of two-layer schemes, stable in a layer. When solving differential equations of parabolic type by the mesh method, the question arises of choosing the difference approximation of the boundary value problem. Preference is given to schemes which satisfy various criteria developed as a result of computational experience, see [ 1. 21. An important criterion for the quality of a difference scheme is the satisfaction of exact a priori estimates for the solution of the difference problem. it was confirmed experimentally in {3? that this criterion is necessary from the point of view of achieving high accuracy in computations on actual schemes. The present paper studies two-layer schemes of the form [l]

f@t+&=fk

t),

(2, t) =a,

y (2%0) =a

in which operates rA and B are polynomials of certain self-adjoint, positive, and permuting We assume that scheme (1) belongs to the class of schemes, stable on a layer [l] :

B20.5rA+eE, TheoperatorAa=At

+....

+ A, and the time step r are assumed

*Zh. Q@!hiti. Mat. mat. Fix., 24, 3, 363-373,

1984

e>O. to. satisfy

(1) operators

TA,,

, TA n,

(2) the inequalftiet

17

Our

main

result is the proof

of the stability

of the family

of schemes

(2) in the sense of sattsfying

the estimate

We will show that condition (3) is necessary for stability of scheme (i) in the sense (4) in the class of schemes that the norm LPzso (w) in estimate (4) is limitinq in this family of schemes. Our results are illustrated by various difference schemes used in practice.

(2), and

1. Formulation of the problem and some auxiliary lemmas. 1. Let R be a cube in n-dimensional Euclidean space and r its boundary. We consider in the cube n the first boundary ‘2 =(xIz=(x,,. .z,,. . . , I,], O
boundary

t),

xe2,

O
(5)

and initial conditions

u(x,

ot=o, x4,

u(f, tt=o,

p=r,

0-a~~.

We assume that f(s, t)~.&fQX(0, m)) for some p E (1, w). Hence the solution of our problem exists and is unique, and belongs to the functional space vi’ (QX (0, CO)). We introduce into the cube 0 the uniform mesh Q~=w,X . . . X0*X.. . Xo., en =(.r,=ih,; i=O, 1,. . . ,N,; form the interior nodes of mesh c&, while the points y=IJ*\tc,. are the h,-_h:-I}. The points #n=bl&Q boundary nodes. On the axis r > 0 we take a uniform mesh w, with step r. We put w = ah X w,. Let H he the set of mesh functions, defined on & and vanishing on 7, Since the space H is finite-dimensional, a scalar product can be introduced naturally into it. By using the scalar product, we can judqe whether operators acting on functions of Hare positive and self-adjoint. Denote by Ak, k = 1, 2, . , n, the operator of difference differentiation of second order with respect to the k-th variable: x=01,

AJ=&sfl,

fEH.

n. Moreover, the operators AI are pairwise It is easily seen that A,=A, ’, k=l,2 ,..., For mesh functions of H and numbers p E (1, “) we shall use the metrics ISi! j
h=h1.,

For functionsf(x.

r), dependent

permutable:

A,A,=Agf<,

. h,,

ott time t E or, we put

Definirion. We shall say that scheme (1) belongs to the initial family of schemes if the operators A and 5 have the following properties: 1) the operators ?A and B are polynomials of the operators 7Al, . , , ?A,, with constant real coefficients; 2) for some number c > 0 and all T, h,, . . , h,, inequalities (2) are satisfied with A > EE. Our aim is to study the stability of the initial family of schemes in the metrics of L,. To simplify the treatment, we wifi study the stability for difference schemes of the initiai family, approximating the one-dimensional equation of heat conduction. The extension of our treatment to the general case presents no difficulties. 2. Let us prove some auxiliary lemmas. Let A (x) and B (x) be polynomials in the variable x of degree m0 and kc respectively. We assume that the polynomial A (x), x > 0, is lower bounded by EZ, EBO: A (.T)>+Ez. z&O, while for the polynomial C(z)=B(z)-0.5.4(z) we have C(x) Z f, x 2 0. Notice that, for the polynomtals A (x) and B (x) with the properties indicated, the quantities

are finite.

Let

zE&=e[

the E of the definition,

er 9 Ct1° n r’-‘ja 1,

and y E [--II,

Q, (5, y) -r’-=z’[ Y(.r, Let us obtain

n). In the rectangle

para. 21, by the Cc of (3), and by the parameter

some estimates

(e’“--l)B(.v)+A(z)

Y)“Y[(e’Y-lI)Btt)+A(r)f-‘.

for these functions.

D=&X [-n, n), which is fully specified T, we consider the functions I-‘,

by

18

/w,

Lemma 1. Let n and I be positive x) E C2 (D), we put

integers

such that

, T2’
TZ”
For every function

Then:

2) a.(Q,)dC.e-‘S,

b,(@)=Gm-‘C,,

3) d..(O)

ZIB’I 1+2S + max 7) ad,

G ne-‘C.

Proof. Since, with 8 = sin2 (0.5~)

1(ei”-i)B(z)+A(z)

~z=A’(t)+413(B’-AB),

then I@(z, y) IGr’-“z”[min Let us prove para. 2) of the lemma.

(A(z),

(2B-A)

(z))]-‘GC.e-l

We have

a@ -= a2

atl-=za-‘[

(e’“-i)B(t)+A(z)]-’

-@[(e’Y-i)B’+A’][(e’Y-I)B(z)+A(t)]-’ From

this and para.

1) of the lemma,

we obtain

a0 -

Iaz I +max Integrating

We integrate

this inequality,

this equation

we obtain

[

toy

1

,

28-A

Ild_ (

between

k(‘D)GZ;_y

a

zl (B-A)‘1

mar

the first estimate

with respect

1


max-

ZIA’I

s.d, A

11)

.

of para. 2). Next,

the limits (72S- 1, 72$), and substitute

B(r)r’=to

j

I = tg (0.5~). We obtain

[A’+(2E-A)‘P]-Vi’ 11

=~.~~~~~(~~-.I.I~-‘(.);[~B(~)-A(=)]-~]]. m From this and the inequalities para. 3), WC note that

A (z) b&z,

28 (I) -A (z) 22~

, we obtain the second estimate

of para. 2). To prove

Integrating this inequality, first with respect to y between the limits (~2~-t, 72s). then with respect limits (r2”-t, r2”), and using para. 2) of the lemma and also the inequality sl (e’“-l)B’(z)

+A’@)

I

I(e’v-l)B(z)+A(~)I zlA’(z)

A’ we obtain

the estimate

of dsn (a).

This proves

the lemma

!

~IWCQ-A(dl’l t-2;

B(z)-A(z)

I ’

to x between

the

19

Consider

the function

1{2B(1)-A(Z)

A((+)=[B(z)-B(0)

Wf--B(O) + A(=)-2E(z)A(r) The estimate

]-‘A-‘(t).

It is easily seen that

we-B(O)

(6)

sWt2B(+)-A(z)]

*

of the first term on the right is

D(r)-B(0)


I

%B(s)Afz)

B@)--(o)

lim

=.

*

fim E(;i;;la;(*) - B’(O)&‘(Of,

’

zB(z)

=-a.

&3(z)

2s

r-0

then

For the second

Collecting *(x!

term on the right

of (6) we have

these inequalities,

we conclude

Y). Lemma

2. Given any integers

that sup lb (x)! < m. This fact is important -0 22’
n and s such that

for estimating

the function

r5L-“a, ~2’“55n, we have:

max I Y (3, y) I< Ohe-‘; t=.r,ro 2) an(Y)aJ.Sne-‘(S-i), b.(Y)Co.sns-‘fl+xs-‘B(O)]

i)

+x’suplA(z)i; 00

3) d.,(Y)

~O.Sne-‘[S-l+O.Sne-‘SB(O)

1-h’

[

XsupiA(n)i. 00 Proof. For the modulus

of 111(x, y) we have IY (r,

y) I’--y*[A’+4(B’-AB)9”]-‘,

8=sin

y/2.

Hence

j Y (2, g) 1*drr%*fA*f4(B*-AB)9t]-‘~ bet us prove para.

Hence, from para.

2) of the lemma.

1) of the lemma

Differentiating

* (x, y), we obtain

i?Y

Ivi I (ei”-I)B’+A’I

1az= I

I (e’Y-l)B+AI*

and the definition

aY I-al:I

d O.Sne-‘a+

(~‘24’.

_,ylt(e’Y--I)B’+A’i I (e’Y-I)B+AI

of the number max

max

Id,

’

S, we have

z(l?.B-A)‘I 2&-A

XlA’i

’ ::;A

1

Integration of this inequality with respect to the variable x leads to the first inequality of para. 2). The second inequality of para. 2), and para. 3), are proved in the same way. Lemma 2 is proved. 3. To study the accuracy of estimate (4) on the initial family of schemes, we require a refinement of the mesh analogue of the multiplicator theorem [4]. For this, we denote by ff~} the Fourier coefficients of the periodic continuously differentiable function f(x):

fl = 2 j f(r)e’““dSc,

Iklc m.

0 Let gk denote the trigonometric

Fourier

series constructed

from

the coefficients

IJJ~:

20

Lemma

3. Let

h,=+(h),

/khlGl,

be a multiplicator

in L,(w).

p~(l,

mar sup Ih,(h)lc l:I**,.Z~MZA~I Proof. Let f(x)

be a continuous

differentiable

periodic

-.

function,

of period one. We put

Af)=O , Ikhl>i. Consider the function P(z)= I.:“t. By construction a trigonometric polynomial of degree m G nh-1. Hence [S]

Since the numbers

Xk = hk (h) represent

in L,, (ok),

a multiplicator

m). Then

of the numbers

we have, for a constant

At) =A.

, (khl
kk(l), the function

C. independent

F (x) is

of h and

f (x)r IIF(t) II

There is a trigonometric

polynomial

under

(8)

the norm

sign on the right of (8). Hence [6],

ll,&‘~~el”ylI‘p~“~~ c (i+n)ll,& Hence, we conclude

from

f~e-lLp (4,).

(7) and (8) that

I1‘p~o.o g (l++G@

IIF

iG.““ll‘P~O*‘l.

The numbers e*= form a multiplicator

in L, (0, l), see [6].

Hence a constant

Ikhldi,

;l

Ikhl>l,

Consequently,

Co exists, independent

It war shown in [6] that our lemma

1,

of h and f(x), such that

then holds. The lemma

is proved.

2. Stability of schemes of the initial family. When examining the stability of schemes of the initial family we shall make use of familiar facts from the theory of difference schemes and functional analysis; for completeness, let us state these facts. 1. Let y (x) be a mesh function defined on the mesh WJ,. We denote by &%(k-l, 2,. . , N-i, Nh=l) the coefficients of the expansion of y (x) in the system of functions sin (A kx), see [7]. Thus

A0 : H--c H, acting according to the rule Aoy=-y,-. IE~,, has N - 1 distinct eigenralues: ,I*=-4h-*sin2 is the eigenfunction corresponding to the eigenvalue A,. The function sin(nkt), z=w, (nkh/2), k--i, 2,. . . , N-l. If f (x) is a polynomial, then eigenvalues pk of the operator f (Ag) are given by pA=f&), k=i. 2,. . . , N-l. Let the mesh functiony (t) be given for f > 0. We continue it by zero for f < 0. Let FJJ denote the Fourier mesh transform of y (t), see [8]. Hence

The operator

(FY) (B) = 2

y(G) e-‘#” T,

IprlGL

h-0 For the functionsy

Denote Theorem

(t). vanishing

the inverse Fourier 1. Let scheme

at t = 0, we have

transform

(1) belong

by F-1. to the initial

so that

family,

F-1

(Fy)

= y (t).

and for some p E (1, -),

let

21

Assume function

k=i,

that, for some constant M,, we have ~k;%kf,, y (x. f), (x, f) E w, we have the estimate

2,. __, a,O
Then schema

(I) is stable, and for the

Proof. The proof will be given for the one-dimensional case. The arguments are only slightly more complicated the general case. We first assume that T > 0 exists such that f (x. r) E 0 for I > T. For each t E w,. we expand the For the functionsyk (t) we have functionry (x, I) and/(x. r) into a sum with respect to the system {sin(xkx)). k--i,

s(7~.)fil,+r-‘A’tr)~~;-f~(t),

2,. . .

for

, N-f,

(9)

with initial condition yk (0) = 0. In (9), A’ denotes the operator zA, which, by definition of the initial family of schemes, is a polynomial of the operator ‘A*. To both sides of (9) we apply a Fourier transformation with respect We obtain

Consider

to f.

the function

By Lemma

1 and the inequality

we find that the function

&(fi)=@(v&,

is a multiplicator

r$)

il Y I+ We multiply both sides of (11) by @and note that Lemma 2, we obtain the estimate

0 tm>=G &C,

the function

Combining the two estimates of y (x, r), we obtain the required to the variable f. If f(x, I) is a mesh function of L,(o), p~(l,

will consist

of functions

finite

in the variable

I and

in space L, (u).

Ilf(t, t)-

If (e---f)

Hence [4]

IL, Wf

(r@)-’

is a multiplicator

inequality for the function then the sequence -),

f.(t,

t) jIL.lr)+O

f(x,

in .CP (61) [S].

By

f), finite with respect

as n + -. By what has been proved,

On passing to the limit, we obtain the theorem in the general case. The theorem is proved. 2. Let us consider the accuracy of estimate (4) and the necessity of condition (3) on the initial family layer difference schemes.

of two-

Theorem 2. Assume that a constant hf.=M(E, p), p~(l, -), eG(O, i), exists, such that, for all schemes of the initial family, inequality (4) holds. Then estimate (3) holds, in which the constant C, is solely determined by M. 2. Let p~(l, -), 220. Assume that a constant Mn exists, such that, for all schemes of the initial family, we have MJP
Then r C Q. Proof.

From

(10) and inequality

(4)

the function

has the property

is a multiplicator Hence the function %(rp), k=i, 2,. . . , N-l, and B, forming a scheme of the initial family, we have the inequality sup sup mar IQ,(+) rr(*.,, e: ,s.,.=n l
in L, (w). By Lemma

I< m.

3, given any operators

A

22

We here choose operators A and B in the form B-E+O.SrA., A=A, , and put @r = x, k = N - 1. As a result we obtain the inequality T G Mlh*Q. It is equivalent to (3). Assertion 1 of the theorem is proved. To prove assertion 2, we note that, from (11) and Lemma 3, we have sup sup max. ITA,‘[ (P rrrr.t) P: w,
-f)B(rl,)

t A’]-‘I<

m.

*a(%0

Putting here B=E+O.SrA, A-A,, BT-R, k-N-i, we arrive at the inequality h”“-*‘Gf&~-‘. If I > o, we obtain a contradiction. inasmuch as step h is arbitrarily small. Hence .z < cr. The theorem is proved. 3. Let us draw some conclusions from the assertions proved about schemes of the initial famiiy. The family of two-layer schemes, approximating the equation of heat conduction, can be divided into three classes: stable in a layer [ 11, weakly parabolic, and parabolic [2]. The first class is distinguished by the condition B > 0.5 rA + 4. the second by the inequality ff,0.5r(i+~)A, E>& and the third by the inequality E-‘B~((B--TA)~. The most acceptable class for performing computations is’that in which the difference schemes have functional properties similar to the equation of heat conduction. In riew of Theorems 1 and 2 and the results of [Z], the class of weakly parabolic schemer is to be preferred. The class of stable-on-a-layer rchernes is not suitable for computations of generalized solutions of the equation of heat conduction, or of its classical solutions with Iarge gradients in the neighbourhood of certain points or surfaces. The class of parabolic schemes is narrow and may not include schemes that reflect other important functional characteristics of the equation of heat conduction (e.g., the property of asymptotic stability [l] ). These conclusions ate confirmed both by theoretical studies and by the results of model problem computations (31. On the other hand, Theorem 1 and the results of [l] indicate that stable-on-a-layer schemes can be used if it is known o priori that the solution behaves smoothly. Theorem 1 has two important special cases. The first refers to a = 0. Condition (3) is then satisfied for C, = 1. Inequality (4) and the estimates of the constant in the multiplicator theorem f9] Lead to the inequality

in,which

M depends only on the parameter E. With the inequality and the imbedding L,(o) -C(o), p>mar[fn(l/~), we can study the uniform convergence of schemes of the inital family. ln(f /h)l, The second special case refers to o = 0.5. There then arises the condition r 6; Ch (see [ 1 j ). familiar in the theory of the asymptotic stability of difference schemes. As a corollary, we obtain from Theorem 1 the inequality

3. Applications of Ao. Consider

Let A0 be a seif-adjoint

the Cauchy

problem

of finding

and positive

matrix

of order

the vector

u(t) =(u,

(6).

d?zJdt--Auff(t),

c-0,

N - 1, and let the matrix A be a polynomial

. . , UK-~(~) ): u(0) =up.

(12)

This is the problem at which we arrive when approximating the Lapiace operator in the equation of heat conduction by a difference scheme. Two-layer schemes are obtained from (12) by replacing the function U’ (I) by a difference expression of the type Bu,, in which E is a positive and symmetric matrix of order N - 1. Consider some examples. . The scheme with these operators A and B is known as a two-layer scheme Example I. Let B-E+oTA, A=A+ with weights. Condition (2) holds when E = 1 and o > 0.5. Hence, for the symmetric scheme D = 0.5 with s < Ch, inequality (4) holds, in which OL= 0.5. The schemes for which 0 > 0.5 are weakly parabolic. Their unconditional stability with respect to the right-hand side was studied in [S]. 0,. at>O_ The scheme with these operators A and B Example 2. Let B=E+(u,+O.~)TA,+U~~A~, A-A.+o,TA~‘, , to third order. Condition (2) holds if E = 1 ‘dnd oz > 0.5 ol. approximates (13) to second order, or, if 0, =O.~{U,+‘/~) Hence all the conditions of Theorepr 1 bold for this family of schemes. Notice that, with o=u,+O.5, ug=O.250’ , this family was studied in connection with aspects of asymptotic a unique fourth-order scheme in this two-parameter stability in [l] _To the values 01 = 0, 02 = l/6 there corresponds family of schemes. In the next examples we assume that the operator A in (12) can be written as A = A, + A?, where both A, and A2 are positive, self-adjoint. and permutable. Example 3. Let B-(E+u,TA;) (Ei-crrrdl), A-A,fA2. The scheme with these operators A and B approximates (12) with first order for (rl, 172 > 0.5. If oi, 02 = 0.5, the order of approximation is two. Condition (2) holds for uI, 02 > 0.5 and E = 1. Thus the present rcbeme is stable in the sense of (4). Example 4. Consider the factorized scheme with operators L?=(E+mA,) (ECmA,), A=A,+A,+uTA,A,. It is easily seen that B = E + orA, so that, formally, this is a scheme with weights for (12). The condition for its stability in the sense of (4) ls o > 0.5. Example 5. Consider the family of schemes in which the operators A and B commute, though A, and A? are not permutable. We put

23

B-(E+O.SorA,)

(E+mA>) (EfO.5mAJ

7

A==AI+A,+0.5ar(A,A,+A,A,)+0.2~-rA,f

Since here 8 = E + arA scheme (1) with operators A and B of this type is a scheme with weights. Both operators are are positive and self-adjoint. Inequality (2) holds with e = 1 and o > 0.5. Hence Theorem 1 holds for this scheme. In particular,

(P-i )‘Ml

IlYllLpC., GM - p’

L,,.,,

i
The above results are obtained on the assumption that operators TA and B are self-adjoint. In this case they have real spectra. However, by repeating exactly the proofs of Lemmas 1, 2 and Theorems 1, 2, we can extend the initial family of schemes, stable in the sense of (4). Take the schemes in which the spectrum of operator E is a complex-valued function. Let us define in this case the initial family of schemes and Hate its main properties. Let B = B0 + $1 and let operators rA, Bo, and E, be polynomials of the self-adjoint, positive, and permutable operators rA,, , rAa. Assume that constants E and p exist for which

B&=O.SrA+eE,

e>O,

B,GpBo,

v>O.

(13)

In addition, let inequality (3) hold for the operators A o = A, + . + A,, and time step r. Then, Theorems 1 and 2 hold for the family of schemes. Their proof, and also the proof of Lemmas 1 and 2 for the case of a complex-valued function B(z) =Bo(z)+iB,(z), will be omitted. We merely remark that all the estimates then obtained depend on the functions flu (x) and Et (x), and also on the parameter Jo. Example 6. Consider the scheme with complex

coefficients

E++A)v,+Ay--f,

lb;=;,

^y=y(t+s).

proposed for the numerical solution of the equation of heat conduction and studied in detail in [lo]. Its unconditional stability in the norm of 12 is proved in [lo]. Also, scheme (14) has the second order of approximation, and is unconditionally asymptotically stable, conservative, and almost monotonic [lo]. For it, B,=E+(r/2)A, B,=(r/2)A. This means that the scheme does not belong to the class of weakly parabolic schemes [2]. However, with constants E = 1 and /.i = 1, it belongs to the initial family (13) and hence is stable in the sense of (4).

REFERENCES 1. 2. 3. 4. 5. 6. 7. 5. 9. 10.

SAMARSKII A. A., and GULIN A. V., Stability of difference schemes (Ustoichivost’rasnostnykh skhem), Nauka, Moscow, 1973. MOKIN YU. I., Two-layer parablic and weakly parabolic difference schemes, Zh. vychisl. Mat. mat. Fix., 15, No. 3, 661-671, 1975. FESHCHENKO A. I., On the accuracy of the numerical solution of the equations of heat conduction by difference schemes, Vestn. MGU., Vichisl. matem., Ser. 15. No. 2, 57-65, 1978. MOKIN YU. I., A mesh analogue of the multiplicator theorem, Zh. vychisl. Mat. mat. Fir., 11, No. 3, 746-749, 1971. MOKIN YU. I., A mesh analogue of the imbedding theorem for type IV classes, Zh. vychisl. Mat. mat. Fiz., 11, No. 6,1361-1373.1971. NIKOL’SKII S. M., Approximation of functions of several variables and imbedding theorems (Priblizhenie funktsii mnogikh peremennkh i teoremy vloxheniya), Nauka, Moscow, 1969. SAMARSKII A. A., and ANDREEV V. B., Difference methods for elliptic equations (Raznostnye metody dlya elliptichetkikh uravnenii), Nauka, Morcow, 1976. IONKIN N. I., and MOKIN YU. I., On the paraholicity of difference schemes, Zh. vychisl. Mat. mat. Fix., 14, No. 2,402-417.1974. MOKIN YU. I., Estimates of Lp-norms of mesh functions in limiting cases, Differents. ur-niya, 11, No. 9. 1652-1663, 1975. KALITKIN N. N., and RITUS I. V.. Complex scheme for solving parabolic equations, Preprint IPMatem. Akad. Nauk SSSR. Moscow, no. 32, 1981. Translated

by D. E. B.