1549301742-The_Theory_of_Difference_Schemes__Samarskii

76 Basic Concepts of the Theory of Difference Schemes

Example 4. The first boundary-value problem for the heat con-

duction equation:

(34)

au 82 u

Lu= Ft - ax 2 = .f(x, t), 0 < x < 1, 0 < t < t 0 ,

u(O, t) = μ 1 (t), u(l, t) = ft 2 (t), u(x, 0) = u 0 (x).

Choosing the equidistant grid w 1 n = {(x; = ih, tj = jr), i = 0, 1, ... , N1,

j = 0, 1, ... , N 2 } and the simplest four-point pattern from Example 4 in

Section 1.2, we are now in a position to set up the difference problem

Yt = Yx:r + 'P '

which admits the index form

Y, /+1_yj z _ yj 1-1 -2yj+yj z z+1 j

-'----'- 2 + 'Pi '

T h

yj = μ 1 (tj), yJv 1 = μ 2 (tj), y~ = u 0 (x;).

Here the right-hand side 'P can be defined in a number of different ways:

etc.

The difference problem (34h) illustrates the implementation of the so-called
explicit scheme in which the values of the solution on the upper layer
y j +^1 are expressed through the values on the current layer by the explicit
formulae

Common practice involves also the implicit scheme

Yt = Yxx + 'P ' y( X, 0) = U 0 ( J;) ,

y(O, t) = μ 1 (t), y(l, t) = μ 2 (t),

which finds a wide range of applications. In order to have for later use the
values i) = y j +^1 on the (j + 1 )th layer, we rnust solve by the elimination
method the system of algebraic equations with a tridiagonal matrix