1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schemes for the heat conduction 493

11. Cylindrically symmetric and spherically symmetric heat conduction
    problems. In explorations of many physical processes such as diffusion
    or heat conduction it may happen that the shape of available bodies is
    cylindrical. In this view, it seems reasonable to introduce a cylindrical
    systen1 of coordinates ( r, i.f!, z) and write clown the heat. conduction equation
    with respect to these variables (here a;= r):

(81) 8t ou =;:ox^1 o ( k(x,t)x ox ou) +f(x,t),

In the physical language, this is a way of saying that the temperature

is independent of i.f! and z.

In the case of a spherical symmetry the heat conduction equation

acquires the fonn

( 81 ') 8t ou = x2^1 ox a ( k(x, t) x2 , ou) ox + f(x, t).

Homogeneous difference schemes for stationary equations in spherical and

cylindrical coordinate systems have been designed in Chapter 3,

Still using its framework, the starting point correcting that situation

is to impose at the point x = l the usual condition of the first or third kind

(82)

and then require the boundedness of a solution at the point x = 0:

. OU
llm k x - = 0 for (81),
x-o ox

Inn. k x^2 - OU = 0 for (81 I ).

x-o ox

The cylindrically symmetric heat conduction problem is reproduced by

(83)

OU

8t=Lu+f(x,t),

t > 0'

1 o ( ou)

Lu=;: oa: xk(x,t) ox ,

O<x<l,

U ( X, 0) = U 0 ( X) , 0 < X < 1 ,

xk-aul =0,

OX x=ll

tl( 1, t) = J.l2 ( t) , t > 0.

In line with the usual practice we introduce on the segment 0 < x < 1 an
equidistant grid

wh={x;=ih, i=O,l, ... ,N, hN=l}