A contribution to the difference solution of the heat conduction equation in curvilinear coordinates

A CONTRIBUTION TO THE DIFFERENCE SOLUTION OF THE HEAT CONDUCTION EQUATION IN CURVILINEAR COORDINATES* (Moscow) (Received

Consider

parabolic

17

1962)

equation

=

i:

&

(I)

1 3 aij

are

with

problem (I,, 3.2) A method of the practical Below cessive

we give

for q

D(0

zi~D,

(Xl. x2),

another

three

be more convenient

than

use the

the

l

Zh.

uych.

mat.,

3,

contour

o
r~l‘.

longitudinal-transverse for

mixed r:

(21

in

A-2) A21

various

1069

sucprove

=

[II:

A 012 A

I (A2i2hlhz

‘786-788,

reasons,

[I].

introduced

4,

the

by

successive substitution suitable for this problem was described in [I].

in

2h,hz

No.

1p == f(l),

using

scheme

notation

r- 012

u

= azl)

bounded

<

may sometimes,

A,(Az+ Al2

of

schemes

which

at2 and xi

t=O;

longitudinal-transverse numerical solution

substitution

We shall

allaz2

region

1963.

A-2)

&a =

a22

&A-2 __-, h;

to

1070

I.D.

Scheme

Sofronov

I: v ET=

AllV,

Here v and w are The first stitution of

change

the

the

To examine

are

solved

directions, terms

order

of

the

Aaaw,

w (A12 +

An)

(4)

w.

functions.

two equations

all

=

Ez

auxiliary

in both

weight

z&IET=

w-v

uu

the

of

equation

operation

stability

of of

by one-dimensional

third (1)

the

the

equation in the

is

scheme

operators

All,

write

scheme,

successive

explicit. (4)

A2g,

sub-

The equal allows

us to

Al2 + A21.

equation

(4)

in the

form (E: Eliminating

the

auxiliary (E

Putting

(E -

z&r) v = U”,

ul=

_

functions (E

zAll)

~p~h~~+O&x,)

t&2) w = 21,

_

from the

last

equation

vAz2) u1 = [E’ + r (*12 + A21)l “’

from

,

u1 = [E’ + z (A,, + Azl)] w.

(5)

(5)

we find

1 - 22112 (1 -I- 2111)(I + zl.22) ’

h=

(6)

where

112= 4 -

a12

hb2

Let us show that

the

hand side

is

of

(6)

absolute less

I 11 +

2r

1/1ldl2

-

oh

cos 2

ozhz

cos 2

magnitude

than

the

oh1

sm --j-

of

the

Zlll

+

t&2

.

numerator

on the

right-

denominator:

2Th I < i + 2T I 112I < i +

(1 +

. o&a

sin 2

+

t%b22)

Ihl < 1 and so the scheme is Therefore that Scheme I approximates to equation

<

-

2Tlfh1~22. -

( -ml1

-

unconditionally (1)

we transform

1/&y

<

0.

stable. equation

To show (5)

as

follows: E (u’ -

a”) = z (An + ADZ)u’ + r (Al2 + An) u” - 22hAa+‘,

or I41 -

uo

E--y-=11

v

+

5 [(A,, + 1122) - (AIS $- An)] F

- ~~AnAasu’. (7)

The

Scheme

tlreduction*

Scheme I,

the

scheme , calculate advantage

over

layer.

property

S.K.

of

unconditionally

feature stable,

the

ul -=A Scheme

value

and A.V. in the

of

the

In the

u0

Au ‘/1*

=

t

case

in the

variables in the

of

In the

/\ and then,

u1

03)

base

according layer.

and non-linear

coefficients

in the

drew my attention

a hyperbolic

was noted

This

in

to

equation.

[2].

An-

Scheme II

is

(1 +

I -

initial

I22 +

242)

(1 +

M/2)

l22t/2)

(9)

’

equation

5 (An + &I) - $ AnA

v

.

(10)

III:

Z-W

time,

of

Zabrodin

reduction

7.41 + uo 2

uo

E-

E -

1071

equation

since

to

r

~

operator

r (111 +

approximates

2s ,

Scheme I that

Godunov

h =

and it

conduction

can be used here.

total

be introduced

tOreductiontl

advantageous

‘I.

A22u

z/2

can conveniently

auxiliary other

heat

2,

fi’----_=

AllV,

UK we calculate

has the

multipliers this

to

step “crest”

scheme

the

u’l* -

uo

v z/a=

In contrast auxiliary the

of

II: E

to

solution

difference

=

A122 +

212

enables

E-

u%’

and fourth

and this

240 =Auv,

A21w

2

212

third

v -

w-v

A21u’js +

z

E2/2=

equations them to

= Amw,

t/2

Al22

2

the mixed

be solved

ul ’

z

derivative

is

by one-dimensional

110 = Au’/‘.

spaced

(11)

over

successive

substitutions. The last E,-

Direct

I41-

scheme uo

approximates u1+ 28

h=l_

equation

z Ala+A,,

2

A-3-=-

calculation

to

shows that,

2 for

[

(1)

ul--u” --A 7

better: -1

+ 0 (r2).

(12)

Scheme III,

(l++ll2)e+

(32J

(l--+ha)Z+

(32l)z

r(111+E22+211*)

(1 +&I)

(1 +$Za2)’

where 121 = 4 hz

The scheme is clearly small steps. There is

o&l o&l sin 2 cos U$ sin 2

stable for sufficiently a region of values of

opha sin . 2

large T where

and sufficiently it is difficult

to

I.D.

1072

Sofronov

evaluate h accurately but any possible instability for be removed by increasing or reducing the value of -r.

these

steps

can

In the last scheme the asymmetry of the directions x1 and n2 can be compensated for by putting Al2 and AZ1 in one whole step equal to the corresponding right-hand sides of (3). and in the next, respectively, to A-a (AI + A-1) al-2 2h&z *

Aa (Al + L-1) ma 2hrha ’ The above

argument

can

be extended

to a greater

number

of dimensions.

Translated

by

Yu.Ya.,

Dokl.

R. Feinstein

REFERENCES 1.

Yanenko, SSSR,

2.

Godunov, 1982.

V. A. and Pogodin, N. N. , Suchkov, 128, No. 5, 903-905, 1959. S.K.

and Semendyaev,

K.A.,

Zh.

vych.

mat.,

2,

Akad.

No.

Nauk

1, 3-14,