1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schemes for the heat conduction 463

The next step is to reduce the resulting scheme to more convenient

forn1s in trying to avoid cumbersome calculations. The case CJ = 0, relating

to the explicit scheme, is simple to follow:

permitting us to find a solution on every new layer.

The case CJ f:- 0, relating to the implicit scheme, is connected with the

equation related to the unknown y = yj +^1 :

CJTAy-y=-F, F = y + T (l - CJ) A y + T <p,

which can be written in the augmented form

(8) A;'fJ;_ 1 - C;f}; + Ai+1~Vi+i = -F;, i = 1, 2, ... , N - 1,

Ai= CJra;/h^2 , Ci= A;+ Ai+!+ l,

with the supplementary boundary conditions for i = 0 and i = N

The difference bou\ldary-value problem associated with the difference equa-
tion (7) of second order can be solved by the standard elimination method,
whose computational algorithm is stable, since the conditions Ai f:- 0,
ICil > IA;I + IA;+1I are certainly true for CJ> 0.
The recurrence forn1ula

. 1.
FJ = -~
' (! z

is aimed at finding the right-hand side F;
smaller volun1e of computations.

F/ of equation (8) with a