A sweep for an infinite thermal conductivity

A SWEEP FOR AN INFINITE THERMAL CONDUCTIVITY+ N .N. KALITKIN Moscow (~ccaiuad

WE consider

10 May 1967)

the divergent difference

equation defined on the net 0 < i < N:

kiui_.i - (ki + ki+i + ai)ni + ki+iui+l = --bi. We write the first and last equations

of this chain of equations

(1) in the following

form: -_(ko + ki + co) no + kui = -bo, A

kNUr?-i

-

(kN + kN+t + aN)uN

=

-bs.

(ia)

number of problems reduce to equation (l), for example, the numerical

solution

of the onedimeneional

parabolic equati on

au C(U,

with three-point

-5

t’at

a =

&d

z

[

implicit difference

k(U,

5,

+

f’z

f(U,

5, t)

I

schemes

(then (la) corresponds

to the

boundary conditions). Equation (1) ie solved

by the sweep method (see, for example, [l] ):

fo = qo = 0,

Ei+i =

ki+i ki+i + (1-

k
=

ki+l+(l

ui = fi+lUi+i + qi+l,

Calculation examples

-gi)ki

i=

by formulas (2a) is impossible it is precisely

(2a)

i = 0, 1, 2, . . . , N.

+y’

N,N--l,

ii)ki + ai ’

. . . . 1,O

(UN+,

=

0).

(26)

if k = m. However, in two important

this case which occurs.

Firstly,

the equation of the

diffusion of a magnetic field within the limits of the magnetohydrodynamic approximation. Here k * 0-l , where o ia the conductivity of the medium, which may be zero. l

Zh. +hid.

Secondly, the non-linear heat conduction Mat. mat. Fir.

8, 3, 684-686, 1968.

262

equation: the thermal

263

conductivity

may not be formally infinite,

the maximum number representable IO magnetic eupercooductors

field equations

k does not vanish (if the exotic case of

is not considered):

oot exceed the conductivity

but in practice it may be greater than

oo tbe computer.

in practice the conductivity

of a plasma does

of pure copper at room temperature.

In this case the

flow variant of a sweep is suitable

[2], permitting k = 00 (but it is nneitable

if

k - 0). For the heat conduction the preceding cases encountered

equation a method is reguind

k = 0 and k I 00 to be calculated

more often).

Therefore,

boundaries

or out&de

both

(the first case ie

we take (2) as the initial formulas.

Within d.c domain (called isothermal of the net, let the coefficients

which enables

below) bounded by the nodes p end q

ki be large, being much greater than on the

the domain,

Weintroduce

the notation

i-1

Ai=(*-Zp)k,+C

i-i

aj,

and expand (2a) in terms of the small eeveral terms of the expansion,

Bi=kpqp+C

bj

parameter Ai/ki

0 > PI

in this domain,

Retaining

we obtain

(3b)

q-1 uq m [B, -

The expreesion

2 (-4$j/k,) j==p+l

y (Aj’/kj) j-P+1

I-‘.

(3b) has an error of the secood order with respect to the

parameter of the expansion, term in (3a) is retained only numerical

][ kq + Aq -

calculation

and (3b) hae an error of the third order. SO

as to obtain second-order

we can omit it and put

accuracy

The last

in (3b); in a

N. N. Kditkin

fi

1 -(Aj/kj)

X

5%

If in the ieothermal

domain

?i z

’

1 +(ntlk,)

tBilki),

(4) in (2b) we verify

the came at all points,

We will

the coefficients

thermal

conductivity possible

the transition calculation

condition loss

< 10~“” even

proposed

formulas

itself

(3).

(l-t$)

of accuracy

(3) or (4.

domain

and is hence

(4) is more accurate even

Firstly,

analogue

figures

than (2a).

associated (in the majority than (2a).

Therefore

the

conductivity.

the heat flows

of the expression

It is necessary

[ = 1;

accuracy

domain

It is

We determine

errors of order

for not very large thermal

in the isothermal

is great.

(2a) where the

(2b).

< 10 -n/#, they give greater

are preferable

the temperature

(3b), (3~) have

out to__n signifiount

be found from the difference

here the loss

to formulae by formula

In the isothermal

Formulas

)

formula

domain

by formulas

to be determined

is carried

We make two remarks,

balance

we pass

n = 11) and (Ai/ki

If (Ai/ki)

cannot

otherwise

of accuracy.

if the calculation

of computers

of the sweep

the function

by formula

by (2a) requires

with considerable (Ai/ki)%

is small;

to calculate

w

that in the isothermal

is approximately

alwaye

&-,k,l(k,+A,),

f-6 I (k, + -4~).

fly -+

calculate

(3c)

ki + ,_

Q+l<‘i
Substituting

p+l=Gi
w = .-kall/az

to write (I) es an energy

equation Lcj

and consider

it as a recurrence

at an adjacent satisfies

-

node.

-

relation

Beginning

one of the boundary

Wi+I

UjUj

=

--bi

expressing

the calculation conditions

the flow at a node in terms

with this recurrence

(the stability

relation

of the calculation

is

obvious). Secondly, left boundary putting

we consider

of the difference

p = 0.

isothermal

the case net.

But if k~ is large

domain.

Ir canthen

where

the isothermal

If & is small,

domain

formulas

that

the

(3) may be applied,

the point i- 0 is an “interior” be shown

adjoins

point of the

;

Infinite

235

thermal conductivity

It is easy to see that :I (I ,< i < q) in this case are small quantities l/ki

and eq is also a small quantity of order

isothermal

domain the quantities

The case where the isothermal is always

described

by formulas

of order

kq/kq_ I. Therefore within the

Ui will be of the same order of smallness. domain adjoins

the right boundary of the net

(3).

Ttadated

by J. Berry

REFERENCES

1.

GODUNOV, S.K. and RYABEN’KII, V.S. Introduction to the Theory of Difference Schemes (Vvedenie v teoriyo raznoetnykh &hem). MOSCOW, Fizmatgiz, 1962.

2.

DEGTYAREV, L.V. and FAVORSKII, A.P. Zh. vphirl. Mot mat. Fit. 8, 3, 679-684,

A flow variant of the sweep method. 1968.