1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with several spatial variables 341

In a common setting, let G = G + r be a p-dimensional domain with

the boundary f and let X = ( X l > X 2 , ... , Xp) denote an arbitrary point in

it. It is required to find in the cylinder QT = G X [O < t < T] a continuous

function u(x, t), x E G, satisfying the governing equation

(1)

OU

at = 6. u + f(x, t), x EG, t > 0,

and the supplementary conditions

u(x, 0) = '1l 0 (x), x E G, u(x,t)=μ(x,t), xEf, t>O,

where

is the well-known Laplace operator.

For the purposes of the present section, let us introduce the grid wh =

{xi E G} in G and denote by ih the set of all nodes of wh belonging to

r and by wh the set of all inner nodes X; E G, so that wh = wh + fh·

The starting point in the further development of the difference scheme is

the approximation of the elliptic operator 6. u. We learn from Section 1 of

Chapter 4 that at all of the inner nodes 6. u ~ Au for x E wh.

By inserting in (1) the difference operator A in place of the Laplace

operator we are led to the system of differential-difference equations

(2)

av

at = Av+ 'P (x, t),

where v(x, t) is defined on the grid wh for every t > 0. The size of the

systen1 (2) is equal to the total number of the inner nodes N of the grid

wh. Here the function 'P(x,t) approximates f(x,t) on the grid wh.

At the next stage we introduce the grid in t

w 7 = { tj = J T, J = 0, 1, 2, ... , Jo, J 0 T = T}
with step T. In order to pass to the difference scheme for a new function
y(x,t) defined on the grid whr = wh X W 7 = {(x;, tj), x E wh, t E w 7 },
it is necessary to replace the system of differential equations (2) by some
difference scheme written in terms of t. Choosing, for instance, Euler's
scheme we obtain the explicit scheme

J=0,1,2, ... '

(3)

y^0 = y(x, 0) = '1l 0 (x), J = 0, 1, 2, ....