Schemes of high-order accuracy for the multi-dimensional heat conduction equation

SCAEMES OF RIGE-ORDER ACCURACY FOR THE MULTI-DIMENSIONAL HEAT CONDUCTION EQUATION* A.

A.

SAMARSKII

(Moscow) (Received

In [l] tion

an economical

21

June

scheme is put forward

1963)

for

the heat conduction

with accuracy O(h” + TV), where h is the step of the space net, the time step. It is a three-layer scheme.

equa-

and T

In this paper (Section 2-4) we examine two-layer schemes of 0th” + -r2) (i.e. of order (4,2)) which are suitable for p <3. ** They are put into effect with the aid of a number of splitting algorithms or alternating direction algorithms which involve practically the same number of operations as the corresponding algorithms of 0(h2 + -r2) (see [21- [Sl). It is shown that these schemes are absolutely stable and that for any values of y = T/h2 they converge in the mean at a rate O(h4 + TV). In Section 5 we consider a three-layer scheme of order (4,2) which is more economical than the scheme of [IIend is suitable for p \<4. This scheme is absolutely stable and converges for any y (the scheme of [ll converges for y>yo

l

l

Zh.

= con&.

vych.

> 0).

mat.,

3..No.

5,

812-840,

1963.

* Note added at proof stage. After sending this paper to the printer we succeeded in proving that our scheme is absolutely stable and converges in Ly(o,J at a rate 0 (l!Q4 + ry). The corresponding proof will be published separately. 1107

1108

A. A.

A scheme of order

(4,2)

for

Samarski

i

the equation const.

asp =

is given

in Section

6.

In Section 7 we examine two-layer schemes of the same order racy for the equation with variable coefficients,

1. A

of accu-

of high-order accuracy for the one-dimensional heat conduction equation

scheme

1. Let the problem -

&i

u (0, 0 = be given

= Lu +

at

u1

where

in the rectangle

&h=

{q=

f = f (x7 t),

Ll$=$,

r=

net H = Gh X 0,

ih, O\< i
h = UN), &

=

= {ti, 0

{(ri,

To write

y’-“)lh,

5 = Yj,

Y(*l) = Y (xi f

Yx = (y(+l) -

down the high accuracy

Y)lh,

w,

scheme we use the asymptotic

of equation

(1).

0.511(7.4 + ii) = (Lu)j+‘la + ;A where

TT= rj+l, Ay =

It follows (ur -

z&‘/n = u (5, tj+a/,), tj+l/, = 0.5 (tj f

that

f) + 0 (h4) + 0 (+),

tj+l),

i.e.

y,.

expansion

-f)+o(hy,

1Zu=urr=L~+~~+O(~4)=Lu+~L(~ where u is a solution

non-uniform time of [71 putting

h, tj+l),

Yi = (Y -

tj) ET},

< i-
TV= 2').The net & is uniform, and & is an arbitrary net with step zj+l = t++r- tj. We shall use the notation Y; = (Y -

(2)

(0< 2 < I,0 < t\(2').

the difference

y = yP1 = y (Q, fj+l)r

(1)

u (2, 0) = ug (5).

u (I, t) = us (A;

V),

Let us introduce

f,

the scheme

Schemes

has fourth w.r.t. T.

(f + g Af)j””

=

order

!/I = o&

+

(1 -

is usually

u = 0.5 (1 To find

y = yj+’

zoAy -

y = -

$)

+cp,

w.r. t.

written

o) A$ + cp;

1109

accuracy

(3)

= I$ (f”1’ +

approximation

scheme (3)

The

high-order

+ &>-gAYi_

Yr =0.54 (p

of

f’-1’)

+

h and second

order

approximation

in the form

yo = ui,

YN

=

2/ (r, 0) = uo (r), (4)

u,:

= 0.5 (1 - &),

y = zlh2.

(5)

we have the problem:

F,

yo = %,

YN

=

F = c + z (1 - a) Ajl+

u,;

zcy,

or

TY i-1 -

(1 -

20~) yi + oYyi+, = -

Fi,

YO= %,

YN =

The solution of this problem for any values of y, i.e. 0.5, can be found with the known formulae of successive [91: ai+1

=

37.

--y-

a1

1 1- ST + or (1 - ai) ’

=

0,

aTPi + Fi pi+1 = Yi =

Computation

1 + ST + s-r (1 -

ai+iYi+l + with these

q

PI =

’

Pi+l~ formulae

YN

=

i=l,

(‘j)

for - co
2,.

[81,

. .,N-1,

Ul,

i=l,2

u23

i=l,2,...,N=l

is always stable,

u2’

,...,

N-l,

since

separately :liote. In the book [91 the schemes 6 and 12 are considered on PP. 108 and 109. It is not difficult to see that both these schemes are algebraically identical to scheme (4). 7. In order to discover have to iinr! estimates for zi

=

0.42 +

(1 -

the order of accuracy of the scheme (4) the solution of the problem

CT).\i + $,

where z = y - u, u is a solution

z. = Z‘V = 0, of the problem (l)-(2),

2 (5, 0) = 0,

we

(7)

y is a solution

1110

A.A.

of the problem (4)) $I= du tion error of the scheme (4).

-j-(1- a)A6 + q - uF is the

Samarskii

approxima-

We have not succeeded in discovering any published proof of the uniform convergence at a rate O(h4 t r2) of scheme (4). We therefore thought it desirable to give this proof here, despite its elementary naturgtMoreover we use a similar method in Section 4 when studying the convergence of multidimensional schemes of the same order of accuracy. In [lo], where an attempt was made to obtain the corresponding estimate, there is an error which was pointed out to us by V.B. Andreyev (the condition 27~- 1 > 0 in Theorem 5 should be 2~ - 1 < 0 or cr > 0.5 since rl =l - u). 3. Let us consider

-

the more general

qAz7 + qJ,

20 =

in (8) we obtain

ZN

=

0,

2

(8)

(z, 0) = z” (z),

and let us find the conditions for stable w.r. t. to the initial data and

where q is an arbitrary parameter, which the scheme (8) is absolutely the right-hand side. Inserting

q=l-

problem

u = 0.5(1 +&),

(9)

the scheme (4).

We need the notation

of

[Al:

N-l

(y, v) = ~YYi~fh

IIY Ilo.= max IYi 19

IIVII= I+, VI, (y, V are arbitrary ence formula

oh

i=l

i=l

functions

k/t VA = -

II v-,/J =

vlq

given on &,) together

(V,

if

Y,l?

with Green’s

y. = y,v =

differ-

0.

Lemma 1.

where v is an arbitrary For

ha II % 11”= (V -

Let us write

net function, v(-l),

v -

given on

d-1)] <

down the energy identity

oh, V. =

UN

=

0.

2 (v"+ (v(-'q2, 11. for

(8).

Multiplying

equation

of high-order

Schemes

(8)

scalarly

by

zz,-

and using

1111

accuracy

the relation

we obtain Z II27 (12+ 0.52 (11zJj”)T = (11 -

0a5) zII 47 /I2+ z ($7 zi)*

If q < 0.5 (a > 0.5), then q - 0.5 < 0 and the corresponding term can be omitted if we replace the sign = by \<. This case has been extensively studied. Let us therefore consider the case ~j> 0.5 or (3< 9.5. Inserting v = zF in Lemma 1 we find Z 11 ‘Theorem 1.

4 (r) -

yl 11zr lj2+

0.5 + $

(1 -

then scheme (8)

-+

0.52 (11 zzlj2)r <

z ($9 27).

(12)

If the condition q <

is satisfied,

0.5)

E= con&.> 0,

s),

stable

is absolutely

for

(13) any h and T:

IIzj+lllo < IIii1 II< ll 2; (5, 0) II+

(14)

where (15) If rl <

0.5 + +

(16)

,

then II zgl II< II+_ (59 0) IJ + J!f (II$j+l II -t where M is a constant

(13).

(17)

which depends only on T and 1.

The estimates (14) and (17) (14) we must use the estimate

and

rll)l

In the case of

follow

(16) we have

from inequality

(12).

To prove

1112

A.A.

Sanarskii

since

The subsequent

argument is similar

to that of

[71.

4. Let us turn to problem (6). Comparing (6) and (8) we can see that ‘I = 0.5 + l/12 y for the scheme (6). i.e. condition (13) is satisfied. Theorem 1 (for E = 2/3) gives

(1% We have proved the following

theorem.

satisTheorem 2. If the solution u = u(x, t) of the problem (l)-(2) fies the conditions for which the scheme (4) has maximum order of approximat ion

II9 II=GM (h’+ ~‘1, then scheme (3) converges uniformly at a rate 0(h4 arbitrary sequence of nets R we have the estimate

where
2.

Schemes

constant

of

conduction

1. Let us consider cients:

not depending

+ -r2),

on the net,

so that

on an

and

high-order accuracy for the heat equation with several space variables

the heat conduction

equation

with constant

coeffi-

Euclidean space, where x = (xl, . . . . np) is a point of p-dimensional Without loss of generality we can take X, 7 1, since this can always be arranged

by introducing

the new variables

z;

: x~‘\“K:.

15’~

of high-order

Schemes

G = (0 6 z, < Z,,

a = 1, 2, . . ., p}

and r its

so that

QT = c the first

boundary,

c

be a p-dimensional

t),

Lu

5 Lu,

=

QT we look for a solution

here

xi = (i,h,,

difference

a

nets.

. . .,i,h,),

of

L&=$7

a=1

Let us introduce

parallelepiped

We put QT = G x (0 < t < 2’1,

= G -k r.

X [O < t < T]. In the cylinder boundary-value problem:

8U - = Lu + f (5, dt

1113

accuracy

Let

i,=o,

oh =

I,...)

{ZiEG}

be a space net;

Nor, a=1

(...)

p,

The net &, is uniform only w.r. t. each of the space varithe steps h, and hp in general being different when a # p. Let

h, = l,lN,. ables,

oh = {riE

G} be the set of internal

boundary nodes,

T,

and yi the set of nodes of y for The net uniform, condition

0, = {tjEIO

its

step

nodes,

T = {ziEr}

the set of

the set of nodes of the boundary y for which xa = 0 which x0( = 1,.

0 < j < K}

< t < Tl, t, = 0, tK = T,

zj+l = tj+l -

only

tj>O

i

satisfying

is non-

the normalisation

z,, = T.

f=l We put CI&= {tj E(O e [III, r

=

t <

T]},

a =

[121 we use the notation Tj,l,

+ =

Tjr v(+lLz) -

V z

ohX0., V (Xi,

tj+l)

t),

v (Pa',

PQ

(i, f 1) h,, ia+ &+I, . . .) i, hp), “;, = (v -

VT= (v -

5)/z;

52 = oh X w+. Following z

V

(5,

t)

= (i,h,,

v(-qlh,,

=

Vj+l,

;

=

V3,

. . . ( La_lh,_l,

V”= = (v(+la)-

vp,,

jhj =Vy+...+h;.

2. Let us go on to derive schemes which have fourth order anproximation w. r. t. IhI and second order approximation w. r. t. T (schemes of order (4,2)). By analogy with Section 1, Para. I we have

A& =

U;&

=

L,u +

s

L&u

+ 0 (h4,).

Inserting

Lu

= g

-

2

Lpu -

$#a

f,

1114

A.A.

Samarskii

from (1) we find

It follows

that the operator

0.5 A (u + ii) - 4

5 h: A+-

+ -&- 5 hi

a=1

2

a=1

&z&ii

has approximation error 0 (1 h I4 + 9) in the class smooth solutions u = u(x, t) of equation (1) w. r. t. Therefore the difference scheme

yT = 0.511(y + i, -

& i

h2,AayT +

&

cp= (f + & i; h:A,f )j+“‘, u = u(x,

the solution

A = i

We introduce

+ cp,

A,,

(3) (4)

a=1

t) has approximation I# =

&f

of sufficiently the operator (Lu)l+‘l~.

f_ h2, gA,h&

‘X=1

for

--

PW

error

0 (I h I4 + 9).

the notation

h:Kt)

ua = + (1 and use the relation (3) in the form !I? = 5

hill2z

[a,A,y

= 0.5 -

+ (1 -

;I==-1 Y=p

for

rEY,

tEZ,;

(5)

CT,. Then we can write the scheme

a,) .I,$1 + zR,G

t (P,

Y (G 0) = uo (4

for

(6) a:EGhr

(7)

where Rpy

=

R?1/

2

KJ

a=1

P>a

(1 -

‘i:,+rh;A\,A, 0, -

y =

a,) .\,A,y,

-f- (1 -

cr, -

i

i

a=1

@=a+1

(1 -

R, Y = (1 us) ‘I,&//

+ (1 -

u, -

flp) &A,y,

cl1 -

(T2)_\&

5, -

c7.J.I,.I,y.

(8) ‘-

(9)

from (6) we obtain a symmetric second order Inserting a, = 0.5 formally, scheme. scheme, and for aa = 1 a purely implicit Below we call

(6)

the mitial

scheme.

Schemes

(if

hi’gh-order

accuracy

1115

3. Following [21- [61 we form a scheme with a split operator on the top row which has the same order of approximation as the scheme (6). We call this scheme the generating scheme. When p = 2 the generating scheme obviously has the form

where E is the unit operator,

or

(1 - u,) A& 1 + zR2jl -

yt = i baAa y + U.=l

~%~a,A,h,y~+

cp.

(10)

Comparing (6) and (10) we see that the generating and the initial schemes have the same order of approximation in the class of sufficiently smooth solutions of equation (1). When p = 3 we choose

the generating

scheme

2 fl

(E -

%A,)

y =

[E + $

(1 -

CT,)A, + .t2R, + r2 Q3 -

(11)

CC=1

cl=1 -

where 113 is given

T~~J,G,A,A,AJ

by formula

(9)

Q

and

(12) Let us rewrite

(11)

in the form

(13)

It is not difficult to see that its maximumorder of approximation Together the class of solutions of equation (1) is 0 (IhI 4 + 9). (13) we shall consider the simpler generating scheme

Thus we have put the initial difference grGiJie!fl 2,

y == IL (x,

where (14).

1) for

,J$Y2 [g] = 0

problem

[yl

5 E’{,

is equation

=

0

t EWg; (I@),

(l)-(z)

for

in correspondence

with the

(z, t)EQ,

(15)

2/ (5, 0) = uO (z) Y, Iyl = 0

in with

for z E

is equation

Ghr

(16)

(13) or

1116

A.A.

Samarskii

3. Conputational alternating direction algorithm 1. The solution of the multidimensional problem (2. 15)-(2.16) can be reduced to the successive solution of one-dimensional algebraic problems This reduction is w.r. t. the directions xl, . . . , xp (see hl- [cl, b51). usually called the splitting (factorisation) of a multidimensional operator into one-dimensional operators, or simply splitting, Several computational alternating direction algorithms correspond to one generating scheme. Thus, for instance, it is not difficult to see that the algorithms of [31 and [51 correspond to the one generating scheme which is a special case of scheme (2.13) for a, = 0.5 and, therefore, R, = 0. Let us find another algorithm for this scheme. Putting y =$ -j- ‘cyr, we write this generating scheme in the form

bayi. =&Ii+

A, = E - 0.5&.

9p,

(1)

a=1

From this lY.

we have the algorithm &r(i)

= A$ -I- cp, A+)

=

~(a+),

u>l,

V(D)= Yr9

(2)

y = j; + TV(,). In accordance

with

[51 to find

v(,)

for

z.xZT~,

1 <

CL<

p,

we must use

the formulae u(U) =

yT = @+l-

A a-+-l’ ’ ’ A&Y pa = p

for

?/Vf = (pa) y*,

“EY,.

(3)

This algorithm

is more economical that those of

it is suitable

for the solution

of the stationary

by the iterative

f21, [151. In particular, equation

method, since in this case yr z=: 0 for always obtain the homogeneous boundary condition q,) = SESya, a = 1, . . . . p. 2. For the generating [23 - [61. Let us first

consider

duce the fractional

step

scheme (2.15)

we use the algorithms

the two-dimensional tj+a,,

and put

‘z E y

0 for

problem (p = 2).

y(,,=yj+‘/z

value of the unknown function y. Let us construct algorithms (see [21, [31, k511.

and we

v(~) for

given

in

We intro-

the corresponding three one-dimensional

Schemes

A.

Y(i) -

= ~

Y,

F

z

5 =

01

[c1 = 2

of

high-order

accuracy

A,y,,, + 8’ [$ I, a,) A,i

(1 -

1117

Y(2)- Y(1)

Yr, =

z

U2&?/(2)?(4

--

a,) zA,A,~ .

+ (1 - ul) (1 -

(5)

a=1 The boundary conditions (2.16) are [51 we can find the boundary values Y(i) = A,y The order from (6), equations

yy, = qA,yf,,

+ (1 -

Yr, = %bY(2) =

yw

z1 =

xi = 0,

for

to

1,.

(6)

of the computation is as follows: we find y(I) for x E y1 calculate F [cl from (51 and then, using the one-dimensional substitution formulae (see Section 11, solve SuccessivelY (4).

successive

B.

given only for y’ and y. According for y( 1) from the formula

A,y

(~1)A,C + ?A&

+ cp,

(1 - a11(I - ““‘A2&,

=

ym

-k (I -

“)(* ” 01

“I

&$

for

q

=

(7)

y =

07

yj+i

x1 =

(8) l:

(z E TI).

(9)

The order of the computation is the ssme as for A. We can see from (‘I)(9) that the scheme has no meaning for al = 0. Unlike in A, when finding y two layers (i and y( 1)) must be used. Inserting cl = u2 = 0.5 formally in

(7)-(9)

scheme

we obtain 0 (IhI2+

a scheme

Z) (see h!,

of

0 (jhI2+ IY~),and for

al

= a2 = 1 a

[31).

Let us suppose that u G 0 for x1 = 0, x1 = 11. This can always be arranged by subtracting a function which is linear in xl and equal to t). In this case when p = 2 the U(X, t) for xl = 0, xl = 11 from u(n, problem (15)- (16) is equivalent to the problem C.

yt, =

%h/(,) + (1 - %)~%

Yi, = o‘z &Y(g) +

(1 for

Tile

r G

l,),

equation

the

(11)

value

= 0 forz, Y(l)

52) &Y,,,

+ @3

52 = 0,

can he solved

of y( 1) found

for

Y(2) =

2, =

in the

= 0, pLj+1

(11)

12,

region

x2 = 0,

x1 = I,, (IO)

(0 < x1 <

x2 = /2 being

L,,

0 _i

used

to

1118

solve

Sanarskii

A.A.

problem

(26).

The order of the computation is as follows:

on the whole net &, -, problem

(12) on the net wh -, problem

problem (10) (II)

on oh.

to see that all the algorithms Eliminating y( 1). it is not difficult A, B and C are equivalent to the generating scheme (2.15)-(2.16) for p = 2. Putting u1 = u2 = 0.5 we obtain known algorithms for schemes of accuracy

0 (I h Iz + r2).

era = 0.5 (1 -

In our case

hz/6t). The amount

of computation for schemes of order 0 (I h 1’ + 9) and 0 (I/L I4 + 9) is practically the same. The increased order of accuracy is achieved purely by a corresponding choice of the parameters u1 and oz. 3. One-dimensional alternating direction algorithms for a threedimensional generating scheme (p = 3) are constructed similarly. Let us write down algorithms A and B only. We introduce two fractional steps values of y(1) and y(2), putting ti i-‘/s and tj+l:, and the corresponding Y(3)

= y = Yi+ll, $ = Yj.

A. Yt, =

%AlY(,)f F

m

Ytcl

Y(aj -Y(--I)

=~

A

a aY(u))

z

=

a>19

(13)

3 F

1 id

=

F,

[ $1

=

2

(I-

%)&jr

+

r

(R,

+

Q3)

j/

+(p

(2.14), (14)

t;;,;Ee

. I

a=1

F [ $1 = F, [ $1 = F, Lb1- z~~~~~cT~A,A~A& for the scheme (2,13j, i’or

Y(1)-- A,Agg Y(2)

= A~J

B’.

for

Al’>(1) = Ai 1

U+.I) or

z1 =

0,

x2 = 0,

x2

f

r&ii

upa = wLv~mj, y =

t

11,

Xl = =

(16)

1,;

(17)

(P7

r = 2,3;

V(3) =

yt,

= A4 + t&ii

(19)

+ cp,

A, = E -

for v(3) = yT , This is the same as the generating can see by making obvious transformations. Algorithm B’ is a three-layer to use not only o(~_~J, but also (19)

is more economical

in its

(18)

j, + QJ(3).

The boundary conditions (16) must be added to these formulae. ing ~(1) and u(p) from (l?)-(18) we obtain the scheme

A,A2Aayy

(15)

Eliminat-

toaha,

scheme (2.13),

as we

algorithm. When finding z’(,) we have the values of ;. However algorithm (17)number of operations

than the three-layer

Schemes of high-order

scheme obtained

in

(ES = 0) we obtain 0 (lh\2 + 2%)

(see

[IIfor

o1 = o2 = U, = o.

from (1.7)-(19) [21,

4.

C151,

1119

accuracy

an algorithm

For ocr = 0.5, which gives

a =

1, 2,3

an accuracy

[121).

Stability

and convergence

1. Let us show that all the algorithms given in Section 3 are absolutely stable on an arbitrary net E and in fact have accuracy 0 (IliI’ + 22). To do this we must turn to the generating scheme (2.15)(2.16). We note that stability and convergence were considered in [II for the initial scheme. This is not sufficient to justify the method. Let y be a solution of the nroblem (2.15)-(2.16), and let u = u(x, t) be a solution of the problem (2.1)-(2.2). Putting y = z + u in (2.15)(2.16) we obtain conditions for the net function z = y - u:

21 =

5

[a&z cl=1

0,) A,i

+ (l-

Q,z~+6p,3~3~~u2u.9h~h2h3~T--;-Y,

1 + zR, i-x2

(1) 2 =i 0 for

J: E 7,

t E 0,;

= 0 for

2 (no)

2 E

oh,

63

P (l--a,-uop)~,hp,

(3)

QP= 2 2 w,Anp

(4)

R,=

2

2

a=1P>a P

a=1P>a

where &,J is the Kroneker symbol, generating scheme, ‘P = q -

c$

error

of the

7” Qpur + &,,3 T” u,u,u,h,A,A,u,,

where q~ is the approximation Let

and 7 the approximation

be the class

error

of

of the initial

functions

If u = U (z, t) E cg, (4,2) of approximation

scheme.

having derivatives

a = 1, . . . . p UP to and including the-m-th including the n-th order, bounded in Qt.

order

then the generating

Y = 0 (1h j4 + .G2)

(5)

w.r. t.

and w.r. t.

scheme (2.1.5)

X.X,

t up to and

has order

(6)

1120

A. A. Samarski

for the solution In fact, then Qpur t? (ra).

i

u = u(x, t) of equation (2. I].

If 0 < oa < 1/2, for any h and v we have 9 = 0 (lhj* + 3). and a,a,a,A,A,R, (u - s) are bounded; therefore Y! = II, j-

Suppose, for example, that

(h? /SZ) > 1 and, therefore,

/CT,1 <

(7%> 0, ua > 0, hf /12~.

and u, < 0,

i.e.

In this case a = 2:3,

and so on. It is easy to see that ‘4 = $) + 0 (Iiz I”), if all oLI< 0. This proves the validity of the estimate (6). We mention only that for u.a
th e derivative

condition w.r. t.

t,

since

~~u/~~~~~~~x~must satisfy

the

z%~(J~c& 12-3hfh~h~, and ht > 6%.

Below we shall always assume that the conditions for which the generating scheme (2.15) has maximumorder of approximation (6) are satisfied. 2. By analogy with the case p = 1 of Section I we examine the stability and accuracy by the method of energy inequalities. We introduce the scalar products

where op

=wh

+ r*,, and the associated norm

Lemma2. Let v be a net function defined on Whand equal to zero on the boundary y of the net oh (u = 0 for x E y). tions

Then we have the rela-

As the proof of the lemma is elementary, we omit it here. Lemma 3.

Let z be a net function defined on 3, with

z IY = 0.

Then

Schemes

The identity

(9)

follows

obtain where

(10). v =

Applying

1121

accuracy

and from the formula

ht 11v;~ If <

Lemma 1 twice

4 11 v I$. Putting

%v, v =

h:hE /I v;,~ Ip <

we obtain

= (v”),- z;&,

we

48II v I$,

z~.

3. Let us nbw derive

(1)

high-order

from (8)

From Lemma 1 we have

TV;.

of

with zero

estimates

a priori

7= 0 We rewrite

for

the solution

of equation

boundary conditions

equation

(1)

for, 2 E T.

(42)

in the form P

z,- =

0.5 A (z +’ t) - &

Multiplying that z I,,= 22

(13)

scalarly

x h&z, a=1

by

+ zR,t -

I? Qpzl +

(13)

2Zi,

using

Green’s

formula and remembering

0, we obtain the basic energy identity

(I Zi II” +

1

+

229

(Qpziv

zr)

+

26p,

3z4u1u*uS

11Z~&.$

Ip =

(14)

P

=

’ + p a=1 2 ht 11 z,,i

lr + 2~’ (Rp~y27) + 2% (Y’,

2;).

From Lemma 3 we have

(16) Then using

the relation

I-ua-uq + 2u,up=0.5[i+(l-2u,)(I-2u,)l>0.5, since

U,

GO.5, (17)

A.A.

1122

Samarskii

we find 2Ta (Q,Zi,

ZT) -

Substituting

zi_) = 0.5+

222 (R&

(18)

in (14)

i 2 [I + a=1 F>a

(18)

we obtain

P

P

Lemma 1 gives (20) 4. Let us examine the cases p = 2 and p = 3 separately. Using the estimate

22 (y9 +) < from (19)

and (20) we obtain

q?? I j’+l + ().54+1 11 Let us sum w.r. t.

IO +

$r

1 Y Iv,

the energy inequality

112 < Ii’ + ___ h;”; ht

j ’ = 0,

Ij+l <

$ IT/I Zi 112 +

Let p = 2.

Zj*+l

([I Zglz

lr)i_

+

+

Zf+l

11Y’+l

/12.

1, . . . , j:

F

zg 112+ $ (pg2,

(21)

I’)“*.

(22)

11

where

Ipq

=( lil’zj, 11y’ j’=l

We now use Lemma 1, which gives (h:! +

It follows

from this,

@

II%,;, II”<

u.

and from (21) that Jj+l <$I”

+

$ ( (Ilyj+l(12.

Schemes

of

htgh-order

accuracy

1123

Using Lemma 2 we find

where -1

( )

!I!, =+

;

+

a

a=1

This proves Theorem the initial

the following

*

theorem.

3. For p = 2 the generating scheme (1) is always stable w. r.t. data and w.r.t. the right-hand side on any sequence of nets

n, so that the solution satisfies satisfies

12 12 12 = 4(1;+1;)

of equation

estimate (23). the estimate

(1)

For p = 2 the solution

of problem

zly = 0

(l)-(2)

-vMoI/yj+’ /I.

/Izj+l (I< $ 5, Let p = 3. In this another way (see Section

with boundary conditions

case the term 1):

22 (Y, .zi-) can be estimated

always

(24) in

(25) 2T (UT,zi_) = 22 (Y, Z)c - 22 (Y’,, f) Q 22 (Y, z)i_ + c,zi + ‘? z 11Yt 112, where CO is an arbitrary positive shall use the obvious estimates:

We rewrite

(19)

constant,

Y (2, 0) = Y (2, zJ.

in the form

wnere

Q = 0.5

i

;

a-=1p=z+1

[I + (1 -

20,) (1 -

2cgl Il~~,,~ll~> 0.

We

1124

A.A.

The coefficient particular, x/k: ahd out the

oa = 0.5 (1 -

we can have

Samarskii

h36z)

ocr < 0,

can be positive

which corresponds

or negative.

In

to the condition

< ‘lo or hE/r > 6. Therefore the product O,U,U, can have any sign the third term on the left-hand side of (27) cannot be omitted withinvalidating the inequality. Let us transform this term. Consider factor

where

We find

for

an estimate

the expressions

DI, = 0: 11Z;~~;JS112, Applying

Lemma 1 we obtain

Thus for

any

and therefore

k=

1,2.

u,u,u,

the inequality

j+l Ij+l

\ c0 2 N

q,$‘-1

+ 10 + 2 [(yr, z)j+l -

(Y,

z)Ol + $Ij+l

+

jL1

(29) +!$

which is obtained from (271, (28) 2(X, 0) = 0, using the estimate

(@qP, and (29)

is always satisfied. cb = const.> 0,

Writing

Schemes

and choosing

of

accuracy

high-order

1125

1/6tj+I, ci = l/6, we find

c,-, =

j+1 P+l < &

z

3+1

Now applying

ZjP

+

12Mo [ max

1) Y’j’

11’ +

tj+l

(30)

(m/)‘].

i
j’=z

_ _ L’71 we obtain

Lemma 4 of

12M,e [ 1~~+111 yj’ II2+ fj+l

P+lQ

\

(*I()“]

and therefore

11 2j+l [I < liil where

(

max l
II F’

II + VG

MO=+ +)‘, a=1

(31)

IFyH9.

Ml=2)f3iMo.

a

On any sequence of nets n the generating scheme (1) is absolutely stable w.r.t. to the initial data and the right-hand side. The a priori estimate (31) applies to the solution of problem (l)-(2) for p = 3. The solution of the homogeneous equation (1) (Y = 0) with boundary conditions z = 0 for n E y satisfies the estimate Theorem

4.

The a priori

estimate

(32)

follows

from (26)

which gives I’+’ ,(31°. Thus, the scheme (2.13) w. r. t. the initial data in the norm 11zG II.

and from inequality is absolutely

(29)

stable

‘q’ote. For the second generating scheme (2.14) we have only succeeded in Proving estimate (31) with the additional condition 015203 > 0, for the quasi-uniform

net

0, (1~rl<

m*z, (see

[Al).

6. After the a priori estimates (23) and (31) have been established it is not difficult to show that the generating scheme and, therefore, all the algorithms of Section 3, have fourth order of accuracy w. r. t. Ihl and second order accuracy w.r. t. T. Theorem

order

5.

Let the conditions

approximation

be satisfied:

for

r!hich the scheme (2.15)

/i \Y 1;= 0 (1 k I4 -j- 2’)

has maximum

for p = 2, 3

1126

A.A.

and, in addition (2.15)

Sanarskii

for p = 3. Then the scheme

/IYY, /I = 0 (I’h I4 + r2)

converges

in the mean on any sequence

of nets at a rate

0 (I h I4 + T2> so that for any h, and T we have the estimates I

/lp _ uj-11 II< M (I h I4+ II9 lIj+J

for p = 2,

IIY j+l -

for p = 3,

zd+l IJ <

M

/I r It, j+l)

u is a solution

vy

where 11 f (1 0,j+~=l<~:jl

(I h I4 +

of the problem

(2. l)-(2.2),

y is a solution of the difference problem (2.15)-(2.16) and J!! are positive constants which do not depend on the choice of nets. In order estimates

to prove the theorem it (24),

(31)

is sufficient

and the conditions

to use the a priori

of the theorem for

IIvy,IINote. Theorem 5 still net U, is quasi-uniform.

holds

for

scheme (2.14)

if

I/Y jl and

QQQ > 0,

and the

5. Three-layer schemes of high-order accuracy 1. We have only succeeded in justifying the two-layer high-order schemes (proving their unconditional stability and convergence) for p<3. In this section we consider a three-layer scheme (connecting values

of

yj+l, yj and ~7-1 on three

unconditionally

stable

Let us consider

time layers)

and has accuracy

which,

for p <4,

the is

0 (Ih I4f z2).

the problem

(1) in the cube h,

hp = h

=

0 Q xa < =

1,

const.,

a = 1, 2, . . . , p.

Let &, be a square net,

a, p = 1, 2, . , . , p and let

i.e.

the net &, be uni-

form. In

[II

y;=

the initial

schemes

+A(y+j;+J-g

_Yii f-g-

i a=1

2 bG, P>a

P d

4,

(2)

Schemes

were proposed

for

Y = $+I,

this

of

problem,

j, = #,

scheme (3)

T > The alternating

(E -

A,)

ytl)

direction {&E

=

?Jf = (y -

has the necessary

+

To = const.> algorithm

(E'- Aa)Y@, = Y(a-l) -

for

the scheme

‘$

(2)has the form

5 x A& a=1

} j, +

P>a

+ $A] 2,

A,,k>

-GE,

accuracy

condition

0.

(A - 2Rl) +

+

+[(I -&)E A, =+A1

b)/2t= 0.5(YF+ jl,). (4)

with the supplementary

n (I = 0 (h’ + +?

1127

accuracy

where

g = yj-l,

It was shown that the initial II y -

high-order

cx >

17

$Ay.,

A,=

Y(P)

I

I

} (5)

= y = yj+l, I

cl> 1.

j

The generating scheme was not given, the authors undertaking its convergence later. The question of the boundary conditions (see [51) was not discussed.

to prove for a < p

2. Our aim is to construct a three-layer scheme for equation (1) which is unconditionally stable and convergent at a rate 0 (h4i- T") .<4 . Ry (foranyy)forp . analogy with Section 2 we introduce the parameter 0.5

(6)

Q =1+1/127*

If

we put u = 0.5

two-layer

formally

scheme of

Thus, let

in the scheme described

below it becomes a

0 (h2+ 't").

us consider

problem

(1)

on the same net as in

We put

P

h, = h and replace

2 h,yT = CC=l

scheme (2.3).

Then we obtain

*

consider

We can

also

it

given

is

on the

the

the following

scheme

previous

AYT by the expression

(7)

layer

initial

as a two-layer (for

t =

Yii

in the

scheme*: scheme

if

tj ) as well as ;.

we assume

that

112R

A.A.

Samarskii

(7)

It is easy to see that it

has approximation

error

0 (h4+ z").

We must fix initial data for the scheme (7) not only for t = 0 but scheme of accuracy also for t = -r. To find y(x, T) we need a two-layer 0 (h’ j- za)

(see Paragraph 4).

We shall

start

from (‘7).

Using the rela-

a2u ii%

tion

(

ats

=

1

ii-1-b 0 (z), we replace

(Ui -

~

!$

(2, T)

by the ex-

pression f

f

Yi-

$

As a result

(5, 0) + f

we obtain

ijj

(2, 0) = f

the initial

Yr = 0.5 fi (Y + ii) -

&YF

Yi -

fLu,

(2) +

$

LL UO (5).

scheme

+ CPI

t = T,

Y I, = IL, Y (G 0) = uo (49

(8) h2 ‘p=,,

for y(x,

[”

Luo

7

$LL&

-

+ 2

i a=1

p=ct+1

T).

3. We rewrite

(7)

and (8)

(E - arh) yi = al CD [& = F [?jl =

2&j

in the form

r&

Y

+ (1

-

I., = PL,

y

(9)

(z, 0) = uo(4,

20) & +

\ j

+2 (I-

2a) z i x A&jr, a=* P’ i+

1>

T,

for t = t.

@ [;I = F, [u,,] = 2a [Au, j-91

(IO)

II

After substituting for the operator fi - (T~A the product of onedimensional operators Al, . . . , A, = \ we obtain the generating scheme A?& =

l? A,yi- = (1> 151, CL=*

9 I.{ .z- 11,

1/ (.I., 0) = %I (x)9

(11)

where dl

= I:’ -

UT.\,.

(I”)

Sckewes

of high-order

We obtain at once the alternating

accuracy

1129

computing algorithm

direction

Y= i + (z?& = 0,

&+I, . . . A, (pa)pl for x E ra.

u(a) =

where @ [ij

TV(P),

t

>

r,

(13) (14)

xa = 1)

is given by the formulae (IO).

4. Let u be a solution

of problem (l), For the error z = y - u we obtain

y

8

Azi- = F [tl + 2oY for t>

IT,

21, = 0

2 (2, 0) =o,

for

tE

0,;

solution

of problem (II).

AZ, = 209 for t = 7, 2 E WI,

1

(15)

where Y and u, are the approximation error of the scheme for t > T and, correspondingly, for t = T for the solution u = u(x, t) of equation (1). It is clear from the construction of the scheme that

Y = 0 (It* + x2), if the solution

u = u(z,

9 = 0 (h* + q,

t) of equation (1) is sufficiently

smooth.

Let us go on to derive a priori estimates from which the stability and convergence of our scheme will follow. The argument is similar to that of Section 4. It must be borne in mind here that u > 0 always. Multiplying (15) scalarly by zzi- and using the relation x(Aq,

xi-) = +;-I?

foxa

5

f..

z I 13q Ii” -

(1 -

26) (+

IIx;Jp

+ (Jar8a$1 Bz~n %,&T- IP+ * *.

a=1 : + @@+I 11z_ _ r, xz...;Epi- 112*

zj-11 = 243z 11.q p + (0.5 - c) 9 (/Iq /i”)i-j+ (0.5 - a) T* IIzfi Ii”,

2aTT(At, zi_) = - (TzIi- + a+ 5 11ZJ;DJ 112,

1 = Ii Ilk,;, I$,

a=1 2r

(q;9

zi_)

=

-c (II q

I&

+

9

II zii

IIS,

a=1 2%

(Yv

q-)

we obtain an energy inequality and solving this, ing, we arrive at the following estimates:

<

z

II q IP + -T II y IF”*

after the usual reason-

1130

A.A. Samarskii

(17)

It follows

from (16), j+l

(17),

(4.26)

zf

and the condition

z 11 (12 + g I/ $1 112 + q

2

Ij+l <

11 ?p112 +

z(x,

0)

= 0

that

(pq)?

(18)

3'=1 For the estimate

of

[I .zj+l 11 for p = 4 we need the following

obvious

1emma. If

Lemma 4.

z(x,

0) = 0, then

(1% 5. Using (17), Theorem

7.

for

theorem.

the conditions for which scheme (11) has maximumorder for the solution u = u(n, t) of the problem (I) are

/j Y !j = 0 (h” -1 t’), then it

the following

any

If of approximation satisfied, i.e. Theorem

and Lemma 3 we obtain

The generating scheme (15) is absolutely stable for p \<4, of the problem (15) satisfies the h and T the solution

6.

so that for inequality

(18)

converges

any values

The proof tions (21).

at a rate

ii $ ,I = 0 (h4 +

27,

PI)

0 (IL“ -t t”) :

of y for p SG 1.

of Theorem 7 follows

immediately

from Theorem 6 and condi-

of high-order

Schemes

/Vote. In [ll

an estimate

1131

accuracy

of the form ~23)

II zj+l II < C(r) II 2 ($9 @ 111

where C(y) is a constant which does not depend on y = -r/h2, was obtained for the initial scheme (2). 6. For simplicity and convenience in our comparison with [II we have considered the scheme on the square net oh. If oh is the net of Section 2, so that h, $1 he, instead of (‘7) we have the scheme y- = 0 5 A (y + y*) f *

5

+

lh12 Y-12

(24)

tt

hy*,hpy,

2

Ih12 =

h;+ .** +h;.

a=1 P>a It is not difficult

A

1..

to construct .

A,yr = F &I,

the generating A,

=

E

scheme

- T(T&, (25)

and we can see that Theorems 6 and 7 remain valid

in this

case also,

for

P<4. The scheme for a non-homogeneous equation (1) is written by analogy with Section 2. The problem of finding schemes of accuracy O( 1hl4 + 7') which are absolutely stable for p > 4 is of interest.

6. A scheme of high-order accuracy for an equation with mixed derivatives 1. Economical

schemes of

[cl,[II]for the parabolic

accuracy

0(1~’ + T) were described

Let us show that we can construct a scheme of accuracy for the case p = 2, when a,p = const. Without loss

of

generality

in [I41,

equation

we can take

a,, = az2 =

i,

0 (h4 + .t”)

aI2 = azl+ so

1132

A.A.

Samarskii

b1aI

<

that

Thus in the region the problem a4 - = Lu,

0 < xa < 1,

(a = 1, 2),

a = 1,

LU = (L, + L, + ~+&L,J a1

2. Let Oh be a square net with step !I, 4 we use two difference

To approximate to L,,u

After

calculation A&

0 < t < T we consider

2;

(3)

Ll,U= as,

a uniform net with step T. operators:

a;*u = 0.5 (us;; -I- z&J.

= 0.5 (up& + u,z),

A&

(2)

Cl.

u (x, 0) = ug (x),

u Ir = CL(29 G,

at

1 -

we ‘have = L,u -

2 Lfdz + ;L1s

(L, + L&J + 0 (h’),

Putting Ay = (A1 + A, + 2ar,AZ)

y,

AaY = Y;;,,a?

we find Au = Lu + $ [Lf + Li + 4aI& Using equation

0.5 (A, + &)

(2),

after

&

+ L,) ‘F 6a,&J

a number of calculations

u + 0 (h’).

we obtain

- g uii i- F(i -k 2aT, f 3a,,) hIA,;

(u -I- L) + 2a,&;

=

= Liz + 0 (h’ + .G~). The operator

3.

Now

mation:

let

AZ is chosen depending on the sign of a12:

us write

A,, = A,,

if

a12<0,

A,,

if

au>O.

=

A:2,

down the initial

scheme of order

(5)

(4,2)

of approxi-

Schemes

of

high-ordrr

0.5 (A, + A,) (y + ;) + 2a,,&

y; =

1133

accuracy

(6)

- ; yTil + ; b&A&

where

b = 1 + 2& The generating Ayy = A&y,-

3 1aI, 1,

scheme will

clearly

= F Ii, &I,

Ayi- = F, [q,l,

yp = (y -

(7)

have the form

&IT= p

Y IY= P (% $9

?&22 = 0.5 (y7 + &).

for t>z,

y (5, 0) = 7x0(2)

fort

= T, 1

(8)

where

A, = E -

azAa,

a = l/(1+&),

(9)

F [i, iI = (1 - 2a) & + LS A (i + G) + 4m,,Alz~ + 2 (1 - a) zbA,A.$. We shall

not write

out the expression

for

F, [u,] (see Section

(IO)

5).

The alternating direction algorithm is written by analogy with Section 5. Let u be a solution of problem (3) and y a solution of problem (8); for their difference we have

AZ, = F [i, :I + 2aY, 21, = 0,

T I: 2~ +

(11)

scalarly

r) = -p

It is not difficult

2 (2.0)

= 0,

by 2 (zF f

t=r, I

(11)

u = u (z, t) E Cp’.

zi)

we obtain

the energy

+ 0.5 at4 (11 z;;,;~ lj2)T+ 0.5 az3 11 zG;r;i+ &--jr -

iF 11”+ z (I + &

_- 4Q.i.2 (AlzvZ,2 -

Az,=2a$,

II, = 0 (h4 + TV), if

where Y = 0 (h4 + r2),

4. Multiplying identity

t>z;

z (11Zi112)i+ ~ bZ (z~,

Z;,~)r + 2~ (Y, Zi + ~~).

to see that

(A&

Z -

i)

=

-

0.5

(A:$,

Z -

i,

=

-

0.5 z (Q&,

Z (QL2)p

9i2

=

(i;,,

Z;,)

+

(Z;,,

Z;,),

Q:2 = (ix,, z;,) + (Z&, i;,,.

1134

A.A.

Following

Section

5, we find

We use the identity

(v, ~)=0.5(11~1~-j-112, r)-0.5r21/

b > 0. Then the term

0.5 t2~bI/z~~T/l

-

can be used to estimate

(I-

expression left.

$b)

2a,,&

& =

>

$(I

As a result,

By analogy

2;:,

-

u#,

v=G,;;.

can be ignored,

h2j zGGj/“. As a result

Let

and Lemma 1

on the left

we obtain

the

(I’+’ + Ii) = [I al2 I + $ (1 - ai2)] (I’+’ $- I’) on the

For definiteness

Writing

Samarskii

let ka =

d2)

us consider zj;.,

(G

+

inequality

we find

the case

AI2 = A,,

&I

[I cl2 I + + (1 -

83 > + 8 (Et +

k.3,

i.e.

since

al2 CO.

C.“,+

1 -

42

%J >

+ 6.

(12) takes the form

with Section

5 we find

and therefore I’+l + I’ < j$j [//$[I2 + (I(Yj+l)‘]

1

Since

b > 0 for

/ ~12) < 0.5 estimate

Let b < 0. Then we can ignore

in (12)

and take the term with we obtain

any T = j$

is valid

for

(15)

cl = 0.5.

the term

0.5 ; b (II2;;:

the left

(15)

for

the expression

II2+ 1zix, l12),

I/ z$$/12 I

to the left-hand

side.

Then on

The coefficient

0.50

r>Tro=

;[++1/1+6]

(i.e.

I aI2 I >

for

-

0.5)

b=

the condition

(16)

1 +&&--3~Ua,,I<0

holds.

$ ,

5. We have thus proved 8.

if

0,

for

Since max 16 1 = arrive at (15).

Theorem

h2 ) b 1/6z >

1135

accuracy

of high-order

Schemes

The solution

[I + 1/fil/48.

max To =

the following of problem

Using (16) we again

theorem. (11)

satisfies

the a priori

esti-

mate

if condition (16) is satisfied. true for any y = T/h2. Theorem

9.

If

condition

jl$ II = 0 w then the scheme (8)

If

la1,1<0.5

(16) holds

(17)

at a rate

(18)

0 (h* + r2), so that

IIY - u II< M @’ + r2), where M is a positive

constant

Theorem 9 follows

(1)

from (1’7) and (181. X: =x,/G, -CZ&~ = 1, nip = a+/ J/u,,app.

we obtain

we have used corresponds in the old variables

All the results are so chosen that

2.

If

cl

= 0,

to the variable

the steps

h = const.

Vote

(19)

which does not depend on h and T.

ivote 1. If the new variables tion

is

and /I y II = 0 @’ + T2),

+ r2),

converges

then the estimate

i.e.

section

in equa-

The square net & which Since

must satisfy

h,

of this

)ai$

xi.

are introduced,

Ax’, =

Ax=z/I/c,

the condition

still

< 1 then besides

apply if

(18)

ha/ )fc=

the steps

h,

and (19) we obtain

113s

A.A.

Samarskii

the estimates IIzjsl + zJll < v tj+l(VTll II(yj+l+yj) When deriving

(18’)

Q II+ 2 IIyj+l II).

(W

-((uj+l+.j)(IgM(h4+22).

(19’)

in Paragraph 4 we must use the estimate

+ 22 11Y j/a 22 (Y, ZT+*z,) < 0.57 11ZI +“q 112 for

2t(Y,

2; + lZT).

7. Schemes of high-order accuracy for equations with variable coefficients 1. Let us begin by constructing a scheme of order (4.2) for conduction equation for one space variable. Let it be required the problem s

at

= Lu + f (2, 2), Lu = g (k (z, t) 2)

O
U (1, t) = U2 (t),

u (0, t) = Ul@), u (x,0)

inT=(O
,

=

uo(2),

the heat to solve

o
(1)

O
r;>c,>o

O
We consider

the homogeneous difference

scheme (see

[131)

(3) AY

=

(K/;),,

a = a (2, t) =

l/A[p

(z + sh, t>l,

approximating to the operator Lu. Here functional satisfying the conditions

A [lI=l,

A [sl = -

A [p (s)] is a linear

A Is21 = $,

0.5,

The scheme (3) has second order of calculations-u;e find

approximation

i,.ZL= (au-) .\ s = IA21-f- 1; L (pLu) + 0 (It”), If u = u(x,

t) is a solution

of

equation

p=

-Ifs
(l),

A IfI > 0 w. r. t.

+

pattern

for f>o.

h. After

,

(4)

a number

p (17 t) = &j. then we can express

(5) Lu

Schemes

from (1):

of

f and substitute

Lu = du/dt -

Au = Lu+ It

follows

YF=

1187

accuracy

in (5):

2 L (pg - pf) + 0 (h4).

that the scheme (for

OS* (Y+ ii>-;

hiph-order

the notation

A 0%) +

9;

see Section

Yo = Ul,

1)

yN = u,; y (2, O)=u,(z),

(6) where Cp= [f + LA

Ay = (U (2,

@f)li+“.

tj+l/,)

g;;>,,

has fourth order approximation w.r. t. h and second order w.r.t. -r for the solution u = u(x, t) of equation (1). We can write

equation

Ys = Roy

+ fi

(6)

approximation

in the form

(I - 4 y +

To find y on the new row AMY,_, -

(7)

_A

t =

tj+l

Ciyi + &yi+r

u =

‘PI

we

= -

$(I-kp).

(8)

obtain

the problem:

Fi,

Yo = u19 YN = ua

(9)

where

Ai =

U~TU+~

Os5Ui

=

t‘r- ‘q p

)

Bi = 0.5ai+,

Ci = 1 + 0.5 (ai + ai+J(y-:j, It

is clear

aip;J

>0,

from this

that

if

is sufficiently

h
Ci = Ai f

T/h2.

Bi + Diy Di = 1 -k small,

h 1;‘Jp I< c’ < 6.

We can use successive

solution

(9).

of problem

7~

or,

(~i+lp,,~

more precisely

substitution

formulae

2. For the error z = y - u where u is the solution of problem and y the solution of problem (6) we obtain the conditions Zt = 0.5h

(z + k) -

;A

(PZT) + 9,

2 (z, 0) = 0,

-I-

zo = b.N - - 0,

for

the

(1)

(10) (11)

1138

A.A.

where y is the approjrimation

error

Sanarskii

of the scheme (6):

9 = 0 (h4 + z”).

(12) By analogy

We investigate the stability of scheme (10). 1 we write down the basic energy identit?

q--l + z (9, 2;).

cc&,-* We transform

with Section

(13)

the sum ((p'i_)T, az;;il = (aP, zzi]+ (aal-7z~ll&]*

We shall

04)

assume that the conditions 0 < c; < P <

are satisfied, have

8P &

Cl7

II

G

59

I

where c;, cl, c,, e, are positive

o
I(

1 a

;

1 I

aP aj

I

(15)

\(GJ,

We shall

constants.

then

2 \
I(

-a

r

)I

\
UfO

Using the relation +zi-19

ac, < C&l,’

p\
M = IIf (cz, c;> >

0,

and Lemma 1 we obtain T II zi_ I$ + 0.5 1 <

0.5-t* + wc;>

1 + (c*+ MIT) II21 II" + &[I$ Ii", (17)

a (5, t-l,,) = a (z,O), M, = M(c', c2,cs),

I = (a, $1, where CO is an arbitrary

positive fz (a&,

If h is sufficiently

z;-l)z;jl<

7

We have used the estimate 1127j]“.

small:

h =G h,, from (1’7) we have

constant.

where ho = ho (c,, c;, c2, CJ > 0,

(18)

Schemes

From this

we find

1 <

M'

By analogy

w

(for

with Section

1 <

1139

accuracy

= 0) M

pp+q,

M

=


M

c3)

(20)

1 we can use the estimate a (*,

Then, instead

< ($, q).

0)

high-order

11zi+1 110<

llj29

r (97 3r) = r ($7 317 for

z(x,

of

H) < of

r NJ1 3)T + 0.5%r (19),

we shall

+ &~ilo,-k

(21)

have

(1 + (c* + c0) z) i + 22 ($9 317 + &I

r

IIQ,iII2

(22)

with the condition h < ho, As usual

from (22)

h, = h, (c;, ~2, ~3).

(23)

we find

I/zitl iI0\( M ( I~~~~+llVll+ llWi$Ti)for 2 by 0) = 0, \\ liz;+’II< Ml12, (G 0)II This proves

the following

for

(24)

d, = 0.

(25)

theorem.

Theorem 10. If conditions (15) are satisfied, then for sufficiently small h \(:to and any T the scheme (10) is absolutely stable w. r. t. the initial data and w. r. t. the right-hand side q~, so that estimates (201, (24). (25) hold. Theorem 11. Let the conditions for which the scheme (6) has maximum order of approximation (12) be satisfied. Then the scheme (6) is uniformly so that

convergent for

I/ T /i. = max ‘tj tend independently 0: small h
as h and

sufficiently

Iiy where J! is a positive

to zero,

u $,’ ’ < M (h4 + II‘t I]$,

constant

(26)

which does not depend on the net.

4. Up to now we have been assuming that the functional satisfies conditions (4) only. and is otherwise completely is not difficult to see that the pattern functional

A [CL(s)] arbitrary.

It

(27)

1140

A.A.

Samarskii

satisfies conditions (4). However, this functional is not always suitable for practical purposes. The most suitable functionals in calculations are the “discrete” functionals [131 which depend on the values of the function at a finite number of points. In particular, the functional -4 [p satisfies

conditions 1

-

ai

where (6)

(s)l = $ [v (- 1) + P @)I + f I4 - 0.5)

X+J,

=

=

(4).

In this

$ (Pi-1+

xi -

0.5

h.

case the coefficient

Pi) + +

Pi-l/,

Pi-l/*,

=

l/a

a is calculated

is equal to

P (Xi-1\2, t)t

Theorems 10 and 11 are valid

in which the coefficient

(28)

for

from formula

(29)

the scheme (29).

We have restricted our attention to a scheme of order (4.2) for the to show that the simplest equation (1). Using (5)) it is not difficult scheme of high order accuracy for the equation c (x, t)

2 = -&(k (x, t) ;+)- q($7t) u +

f (x7t)

(30)

4j+‘l’!/+ cp,

(31)

has the form

c (x7 Q+l/*)yr = 0.5 A (y + ii) - ;A

(cp yc + pqy) -

where cp and Ay are given by formulae (7). With corresponding Theorems 10 and 11 remain true for this scheme.

conditions

5. Similarly, we can construct a high accuracy scheme for the heat conduction .equation with several space variables. Here we shall restrict our attention to the case of two dimensions (p = 2). see Section 2. We shall

consider

~=~L.uff(r,t), Cr=l

the problem L&=a&

u (x, t) = p (x, t) for za = 0, To simplify net

(32)

8’1 i

j7&J’

xa = Z,, a = 4,~;

the argument we can without

ah = {(i,h,;

a = I.. 2. The net

i.Jz2) E G} to be square, 4

is arbitrary.

with the help of the functional

O
A

loss i.e.


O
ZJ(x, 0) = &J (5). of

generality

(33)

take the

h, = 1,/N, = h = const.,

Let us define the net functions [p (s)] from Para. 1, putting

aa

Schemes

of

high-order

1141

accuracy

t>l,

a, = a, (zJ=l/A

[p1 (xl+ sh,

a2 = u2 (s,t)=l/A

[pz (21,22 + Sk $1,

x2,

t= t - 0,52,

A = $)A~.

&?I =%W&

a=1

By the argument of Paragraph 1 we can see that the initial order (4,2) of approximation has the form

yr = 0.5A (y + 5) - ;$ Aa (P& a=1

+;

scheme of

[A,(p,A&) + A, (~,A,$)ltcpv (34)

cp= fj+'lr + g 5 (A,p,f)j+'l~. a==1 6. Let us find

the generating 6,

=

scheme. Introducing ;

(1

-

g

(35) the notation

pa)

(36)

2

and replacing 7ba,

E -

we obtain

y = p for

z x A,Q~ a=1 tne generating

by the product

A,A,,

where

A,

= E -

scheme

5 E 7, t E Or;

y (CT-,0) = u. (2) for

5 E

Oh+

(35)

The reduction of the generating scheme to one-dimensional alternating direction algorithms is achieved by analogy with the case of constant Here we shall just give algorithm ‘4 (see Section 3): coefficients. ”

Y(,) --

z

Y

= ‘2ps&)

+ 0,

Y(2)- !/Cl) =A+y(2), -_ t

yj+l =

Y(2),

(39)

CD = i A, (1- a,)c + z [A,(0.5- 01)A2?j + A, (0.5- a,)&~] + cp. az-1 7. Vow let us discuss the basic problem, that of the stability and convergence of the generating scheme. Let u be a solution of the initial problem (32)-(33). y a solution of the difference problem (37)-(38). For the difference z = y - u we have

1142

ZT-

A.A.

Samurskii

0.5 AZ + T2A,6,A2G2Zt = 0.5 hr - ~a$l*L7

(p,z,-> + (40)

+ zAl (0.5 -

z=o for Multiplying 22

(40)

a,) A,t + zA, (0.5 -

z E 7, scalarly

&IL

t by

02) A,s + Y,

z(x,O>

22z,-,

we obtain

“:,]a + 23 (A,qbw,,

II27II2+ &b*

=o

b, 4 E for XEOh.

the basic

zi) = i,(~,

QT, (41)

energy identity

fz;olla+ (42)

+ g r ~UP&,~ We shall

a,~~~~]~+

~(AlplA,~

+ Ati,,@,

q)

+ 2% Vu,

zi-).

assume that

Constants

which depend only on

c;, cl, c2, cs, c4,

8. As in the one-dimensional

are denoted

by .J!.

case we find

(44) Now let Consider

us estimate

the other

terms in (42).

the expression A2

=

‘;(&p,A,%

Lemma 5.

If conditions

Estimate

(46)

follows

q)

(43)

=

-T

((plA2z),,

are satisfied,

from the expression

a,~;,~).

then

(45)

Schemes

which is obtained variable x2. 9. Lemma 6.

of

from (45)

high-order

after

For any a,,

1143

accuracy

the use of Green’s

formula

for

the

a = 1, 2 we have the estimate

where

Using Green’s we find

The underlined

To estimate (a)

formula

expression

for

a number of transformations

is maximised as follows:

a)

< M,z

b)

<

the last

x1 and x2 after

(I + i) for u= > 0,

(49)

Mi dz /I zt Ija for o, < 0. two terms in (48)

(50)

we must examine separately:

6) ua < 0 and, therefore,

0 Q CL < 0.5,

a = 1,2;

Ij.ucr[I,,<

NcJl22.

We also need the estimates 2@‘)o,

+

Mh3x2

1-

o;-11’ -

&l”)=

0.5 (1 + (I--2cp’) > 0.5 - M,h3/z,

(I-2u,)]

+ (u,-up))> (51)

/Iz;;,;~lr”Q ~4;’ 12IIz;,,~IIII2;II< w3 IIz,,;;,? Ii”+ Md2 I/q/12.(52)

10. Now, collecting together all the estimates (441, (52) and choosing CO we obtain from (42) for sufficiently h < &I,

r <

(46). (47), (51), small h and T,

To,

(53)

the energy identity:

(1 - M9zj+I) I’+‘< Mloj&.li~*l

+ ; (ala2 (pi-“’ + P;-~~)), z:~;)~+‘+ (54)

+ (1 + M,otJ

I(O) + Mu (II)‘)“.

A. A. Sanarskii

1144

For sufficiently inequality

We arrive

small

at the following 11 i+l II< M,,

where c4,

M12,.M,, are

11,

we have a,p,

h < h,

(-1,)

<

We use the

1 + Mlah.

inequality:

II 2; (5, 0) II + &+lij positive

constants

for h -< hot

f <

~0,

which depend only on ci,

cl,

....

12.

11. We have thus proved

the following

theorems.

Theorem 12. If conditions (45) are satisfied, then the generating scheme (34) is unconditionally stable w.r.t. the right-hand side ‘f’ and the initial data, so that for sufficiently small h and T we have the estimate II zj+’ II + II 2; If+’ <

c2r

0) II +

M’

TO, M and V’ are positive

where ho, Cl,

M II Z;(G

c3,

7’heorem

C4r

ll,

constants

ho,

II t IJo <

~0,

which depend only on ci.

for

which

Ii Y /I = 0 ( 1h r + r2),

(56)

and conditions (43) are satisfied, then scheme (34) tend independently to zero so that, for sufficiently have the estimate IIyj+l -

uj+l II<

where ,\Iare positive net. For a hyperbolic

also

we can write

M ( I h I4 +

constants equation

[It2 /Ij+l)

for h <

converges as h and T small h and T we

ho,

z <

=

3$

-

down economical

(,? +

0.5

T2)&

r.

which do not depend on the choice

high accuracy

rl = ii&,

of the

schemes.

Using the above methods we can show that the generating Ay

(55)

12.

If the conditions

13.

h <

r-1,

rl, = E -

schemes uzzz2ilz,

uf high-order

Schemes

1145

accuracy

a=1

Ayi-= (E - 0.5 TEA) ii-+ zA$ are absolutely stable and converge in the mean at a rate 0 (X2 j- Ih I’). shall examine the question of economical schemes for multidimensional hyperbolic equations separately.

We

Translated

by

Feinstein

R.

REFERENCES 1.

Douglas.

2.

Douglas, 439,

J.

and Gunn, J.E.,

J.

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81.

Math.

71-80,

82,

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421-

1956.

3.

Yanenko.

N. N.,

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4.

Yanenko,

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IZV.

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Mat.,

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1961. 5.

D’Yakonov.

Ye.G.,

6.

D’ Yakonov, Ye. G., Reshenie nekotorykh cheskoi fiziki pri pomoshchi metoda

Zh.

multidimensional the

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7.

Samarskii,

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Godunov,

A.A., S.K.

Difference

9.

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matics,

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Mat.,

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Solution

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kraevykh

Boundary zadach).

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1960.

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1962.

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A.A.

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Tikhonov,

A.A., A.N.

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Zh.

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431-466. Mat.,

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Yanenko,

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Douglas,

J.,

V.A. Suchkov. No. 5. 903-905. Numer.

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