On the convergence of the fractional step method for heat conductivity equations

ON THE CONVERGENCE OF THE FRACTIONAl.STEP METHOD FOR HEAT CONMJCTIVITY EQUATIONS* A.

A.

SAMARSKII

(MOSCOW) (Received

1. The splitting for

method

the numerical

equation

with

conductivity

heat

constant

down for the case proof

of

with

the

shall the

the

Prove

the

ence

these

schemes parabolic

take

the

of

functionals

[sl . tour

of

r.

In the

[21

into

We use

proved

for

C be an arbitrary 6=G+IJ,

cylinder

x= ijT =zX

In this

of

the for

In contrast

+

to

net. the

local not

multi-dimensional an arbitrary

to

[31,

differ-

region

at each

characterised the

methods

arbitrary

by the developed

non-uniform

two-dimensional

(XI, z?)

a point

[O
with we look

region the for

given in

and

stage

we to for

pattern

[3!,

and

[d

nets. bounded

coordinates the

in

Ye

we shall

[II,

we

be used

of

O(T)

“(T*)

note

can also

difference

o(h*)

and a

speed

which enables us, in particular, As usual we give the argument account.

schemes

is

speed

for

schemes

of

[31-[51.

given.

method

[l!

written

region,

with

in contrast

splitting

equations.

homogeneous

Convergence

2. Let

the

scheme is

scheme

is

step

makes them suitable

homogeneous

scheme

a family

this

in

conductivity

difference

and a non-uniform

mean with

suggested

coefficients

= const.

we examine;

with

and this

six-point

T/h*

in the

is

heat

and a rectangular

mean) of

fractional

region

schemes

quasi-linear use

condition modified

convergence

equation

the

the

constant

variables

(in

method)

[zI

In

with

two space

the

step

multi-dimensional

coefficients.

an arbitrary

one-dimensional connect

of

additional of

the

equation

convergence

show that case

of

1962)

June

(fractional

solution

the

9

solution

by the

con-

xl

x2.

and

of

the

problem

t.2, t) E QT = Gx (0 < t < T), (1) CC=1

l

Zh.

vych.

mat.

2. NO. 6.

111’7-1121. 1347

1962.

.$.A.

1348

u

II‘

Ul

=

(2, 0,

u = U(X,

assume:

cl

In the

A.

Lipschitz

with

the

3.

fir)

the

zi=

of’“)

firlf x1

@)

(a$), =

rc-x

0,

and

straigbt

all

nodes

is

+I

_

oh={xi

boundary

net

parallel

the

net

oh lying

Ed = xjr)E 7

and

rn=x~)

xa increases

on going

We put

chains

introduce = 0,

the 1,

integral ,i’=O,

+,

space-time

the

nodes

into

of

I,...,

the

net

it

the

of

of

steps,

(uniform steps

with

i. e.

length

respect

the 4,.

set

of

set

net

nodes

nodes

xr.

The segnent

with

step

to = 0,

T) and we

tj+,, , =.tj + r/2 = (i + f) [O<

time

contains

t < T]},

(Xi* tj*)EQT

ljl) E QT)

the

is

x &,

set

of

!A = oh x o,,

2,

both

where

K’. The point

a=&

where

the of

K-+,

that

of

Let q,,

SZ=((x+

clear

set

T by the points

to

0, = (tj.E

we draw

the chain the

and

region,

chain

ya

parts

zp)

ox,;

r is the , o~~)=o~+~~,~.

direction,

0,

I is

and

hp) = rIf” -

a+Eq,

axis

points

TJK

net

point

be boundary

(fractional)

is

internal

I

xp)

coordinates

the

%=

R and

the

coordinate

and ya

interany

?${x~E??),

the net

called

xr,

net

of

is

A’ equal

,...,

net 5;

the

the

drawn through

coordinates

steps

Through

on g,,

1. The net

f,i+$

with

of

~r’=~~fT~,n,

and fractional

of

to

8,,

()xo consists of a finite we give our argument the contour I- twice.

be the

region.

XI to

intermediate

.... K -

net

k,,

au!&,

(: exI:licitly:

line

The

E G}

t,=Kr=T,

jz ,...,

2.

Pu,@x~,

rectangular

the

a*k,iax; satisfy

2).

region

functions

J$, in a given

T is *divided

t, . .. . . tj=

ET

fron

TO=:ro,n+yo,n,

t <

of

are

,$k-l)

s,,

all

the

straight

. . . , a = 1,

line

x1 of

t (a = I,

non-uniform

the node

Let

the

of

‘pd, a a@&,,

for

to (see

It2.

fl.

hfd =

I’)

the

j

(3)

~~u~~~,

and

of

the

arbitrary

respectively.

T=(qE

? of

functions

the coordinate axis [31 ). For simplicity any strarght line %‘;p, intersects

that

.

’

x,,

one property

region

Consider

where

3 r the

conditions

point x E G parallel number of intervals assuming

for

Lipschitz

Below we use only section

region

conditions

the

ak,fax,--

0 <

1, 2.

l

Contli t ions

T

a=

(2)

the

fied.

the

(cl = const),

>

XEG,

0) =uo(x),

has a unique solution d oblem (l)-(3) - r in .‘?Tt (2) the following conditions are satis-

(1)

continuous

t),

u(x,

t E IO, Tj;

k, (a, tj > We shall

Saaarskii

is

tj. = j*z,

a node

internal where

Fractional

step

nethod

for

heat

ckmductivity

equations

1349

% = (tj, E (0 < f < T)). Te shall function

use

given

L= (<w 1

the notation on 9.

) .p),

‘0% need

the

We shall

.41-l/2)=

c&x) -y),h,,

Y&=(Y

given

in

[31. y kt~,)

write

$ @.fI) + q--l)),

h,=hf;l’lf,

following

h,*=hp+‘),

yg=

Let

y(s,

tj*) = yj* be a net

_ - y(z (**a) , ,, ,,, (y-y

t-la))ih,,

+(1z,+h,+)j2,

yr,=

dtmi!

=

yxe=(Y(+laJ-

YM,,,

{yj+a!2-yj+(=-1)/2)/~.

sums and norms:

(y, v)’

=x yiqp

tip’,

Oh (Y, VI;, =

y=

2

“'h

yiV&za) hj$!$‘)

(f

for

a = 1,

-

for

IIYUZ,

7/U,

a = 2),

oj;t=) IIY II= V(Y* Y)‘*

lIYI!r

=

V(Y,

Yh

II Y /isa = 11% /la,’ where

qz=O 4.

The operator

Lau is

for

replaced

(Qa

=

Yala,

= Y,

xEr,,b. by the

homogerIeous

uniform

net

scheme AaY = (“,Y; Lavlng that

second

order

1;s

on a uniform

net

for

of

pattern

ao = katmx)

lem (l)-(3)

functionals

we obtain

in correspondence y;= = A= y@f for

(Z,

ao is

the with

tj_ta:*)

Y (2, tj+a,2) = si (X, tj+a,*l

t4?

in

[41:

and (51)

so

ELI,

for

described

Y@) = SY + (1 -

5) &

a= 1, 2; i = 0, 1, 2,. . . , K - 1;

2 E 7,;

0 B 3 d 1,

y (2, 0) =

Y = Y (x, ++,,s), t’ E [tj, tj+*]

shall

t’ = tj+,~,_ It is clear from (4)-(5) that I the first boundary problem for all chains

we solve direction

x0(.

for

2 E oh;

(Q (5) (6)

?; = Y (2, rj+(,_+,,),

and for

hi%

take

uo (zf

TI,

yi Cl= (Y -

J&Y = (a, (2, t*1 Y;,‘&’

in particu-

scheme used in (21. Ye put the probthe local one-dimensional scheme:

(a, 6% 1) ,, Cl > 0, (G q E OF”’ x [O < 6 where

(see

A.,u - L,u = 0 (hi).

The class lar,

approximation

a

definiteness

at each 4,

moment tj++

in the

given

we

A.A.

1350

5.

f&et us nom go on to

scheme of

(4)-(6).

problem

Let

study

u be the

(4)-(6).

For

“&

=

the convergence

solution

z = y -

Aad”) +

2 I? tj+&

= 0,

conditions

A are

We put ;jir in the

the ibe

local’

errors

basic

integral

and sum with

Putting respect

the

>

the solution

(7)

)

(I mEoh,

(3)

O-

By analogy

of

the the

identity

respect

following

ljlll : 0 in (15) to the initial

solution

inequalities

I* z,I usin,:

(2 + i,

we obtain

tj+a/*)>C1

of

y the

conditions

2(2,0)=0,

x E +fa;

satisfied.

the

(l)-(3),

Aad@- y

lpa =

-q&,

and accuracy

problem

(9)

with

(7)

we find

form

6. ‘Titlt the estimate method of integral

the

of

u we obtain

*a ($, ,?uppose

Snnrnrskii

of

the problem

and a special

(7)-(g)

method

relation

lvith

$ + yi = 0. Let o = fc. !.et us multiply

to Oh. Using

Green’s

of

we use summing

us rvrite down (7) by (z+zjfi&

formula

A&p)) * = f II If< bra + 2;) 1122, = I,7

(13)

identities:

we s6.e that data

(for

the

~~chcne (4)-(6)

;Iny !jql and 7):

is

stable

*it!!

Fractional

7.

Lemma 1.

then have

for the

step

If

tnethorl

for

tYn can be put

the solution estimate

of

heat

in

problem

the

conductivity

1351

equations

forn

(7)-(q)

for

o = !L and any $,

and -r we

where

We put

(17)

in

where

co is later.

l/tj

Using initial

(15)

and make the

an arbitrary

positive

As a result

we obtain

I‘ emmu 2. I,

constant the

transformations:

which

2,

If

p( t,)

. . .)

then

following and a(tj)

the

simple are

inequality

we shall

put

equal

to

inequality

the inequality % I (V, z)‘J ~0.5//~~~+0.52~ condition z(x, 0) = 0 we find

L.et us now use the

J =

following

lemma,

non-negative

+0.5r21~~~]~~, and also

the proof functions

of

which (t,,

the

we omit:

= j?,

1352

A.A.

P (‘j+l)

a

COT

Sanarskii

i=i,z,...,

P ftf) + o ttj+i),

2

(co > 0)

j’s1

gives 3

P ttj+i) d c

ttj+l)

+

Wco’j

C

TS

(20)

(tjP)*

j’=J

It

follows

from the

inequality

(lg),

from l,emma 3,

that jti

This

proves

lemma 1.

8. Theorem 1. If

conditions

A are

satisfied,

then

scheme (5)-(7) converges in the mean with speed o = 0.5 and as /I,, T tend independently to zero

the

difference

0((Ih2(/~)+O(t)

for

on any non-uniform

space

ret:

IIY txt tj+x/*) - u fz, tj+a,-J II6 *%f(II*” 113 + t)*

a=2,2,

j=O,

i, . . . . K-i,(21)

a=l, where AI is

a positive

constant

which

does

not

2,

depend

on the

choice

step)

follows

from

of

net. For the since, jz(r,

estimate

from

(10)

‘j+,,I)II

(14) far order of

order

of

requirements

step

If

the

converges

in

2.

y.

the

than

are

scheme conditions

following tbe

(1)

0 < y d 1,

(18)

h2 [j?J_t 0 (t”). To estimate we have Q j’ = 0 (11

on a fractional

The conditions of

o = 3 (on a whole

a = 1. Thus on fractional accuracy as on integral

Theorem

(4)-(6)

for

and (12)

The convergence weaker

(31)

mean

we use

(18)

steps

the

(5)-(7)

au/ax,,

scheme

the

(5)-(7)

has

Same

can be proved

with

considerably

4.

(5 = 0.5

C%,/&r,, a2u/&,* (3)

identity

steps.

conditions for

wit;1 o = 2 and the

are

:;atisfy

&L/&~,

satisfied

and for

au/&,

in ?r k,,

then

any h, and

the T

the Hiilcier

ilk,/&,

satisfy

scheme

so

that

condit’ions in ?ii

Frtictionnl

the Hijlder

conditions

TO prove

O(T~), of

heat

given the

them here.

is

uniformly

is

true

to

the

all

the

is

7 < 7

this

ease

If

we suppose

[3!

we can prove

for

9 f

o Q 1 and for

is

the

It ence

the

boundary

we have

of

for

given

the

If

on c,

only

simPleat

erluatrue

that

able

for

the

not method

estimate

rof , max! 7o!.

to

then the

on the

using 131).

here This

using

leads

to different

in this

case also

of

for

For the

2 of

boundary

method

h2 = ii+@,

o,,

formulation

that

the

estimate

(5)-(7)

where

net

always

the

(flak

any h, and T (Theorem

to

obtain

scheme

0($)+0(x),

conditions with respn=t

then

and max!

of

(ia ) ha

we are

contrast

[31 and we shall one’-dimensional

qnfO,

been

speed

step

for

in

We have proved

is attained

the

1 and 2 remain

local

r,*O,

convergence

the

holds

(in

.

Lemma 1.

z/hi < [2(i -o)maxu,]-1,

must be stressed

~~(P,I~3,=O(I~h~$a)+

of local errors q~, used here, which leads cannot be extended to the case p > 2. In

any p > 1 with

estimate

approximation

form nets.

that

u = 1.

uniform

maximum value

u = 1 this

out

(21)

that

that

Theorems

I\oy are

there

summation estimate (18). of

of

I?,53

t.o t.

reasoning

0 where T@ depends

instead

resnect fact

However,

for

nroved

IO. The method of u priori

equations

form

expressions It

the

preceding (I).

convergent

for

to use

and make use

general

The corresllonding

conr!uctivity

heat

y > 0 with

2 we have

conductivity,

of

for

order

j~pa$a=O((/k~+y&J

equations

give

nethmf

of

Theorem

‘Be have

9. tion

step

$, case

[3!).

conditions of

schemes

with-

the differon non-uni-

second order

accuracy

to space.

Translated

by R. Feinstein

RRFERENCES

1.

Yanenko,

N. N.,

Dokl.

Akad.

Nauk

SSR,

125, NO. 6.

1207-1210,

1959.

1354

A.A.

2.

Yanenko,

N.N.,

3.

Samarskii.

4.

Tikhonov,

Xh.

A.A., A.N.

vych.

Zh.

Samarskii

2. No.

mat.,

vych.

mat.,

and Samarskii,

5,

2. NO.

A.A.,

Zh.

933-937.

1962.

5, 787-811.

1962.

uych.

1, No.

mat.,

1. 5-63.

1961. 5.

Samarskii,

A. 4.)

Xh.

uych.

mat.,

2, No.

4,

603-634,

1962.

6.

Samarskii,

A.A.,

.Th.

uych.

mat.,

1, No.

3.

441-460.

1961.

7.

Tikhonov, 832,

A.N. 1962.

and Satmarskii.

A.A.,

Zh.

vych.

mat.,

2,

No.

5,

812-