1549301742-The_Theory_of_Difference_Schemes__Samarskii

550 Economical Difference Schemes for Multidimensional Problems

Let a value y = y" be given. Starting from F, we tnove further

along the rows ·i 2 = 1, 2, ... , N 2 - l to solve problen1 (16) by the standard

elin1ination n1ethod, whose use pennits us to cletern1ine the values fJ at all

the nodes of the grid wh. After that, we calculate F and move along the

colun1ns i 1 = 1, 2, ... , N 1 - 1 in an atte1npt to solve problem (17) and find

the values y = yn+l. In passing from the (n+l)th layer to the (n+2)th layer

the same procedure is workable, thus causing the alternating directions.

Since the eli1nination method requires several operations at one node,

the total trnn1ber of which is independent of the grid step, Lhe algorithm just

established will be economical if one succeeds in showing that scheme (9)-

( 14) is absolutely stable. The following sections place a special emphasis

on stability and convergence of the scheme concerned.

3. Stability. Subsequent considerations of stability of scheme (9)-(14) are
   conducted with a priori elimination of the intermediate value y. This can
   be clone by subtracting equation (10) fron1 equation (9) and re-ordering of
   the relevant one as

(18)

Substituting (18) into (9) yields

( Hl)

With the relation y = y + TYt in view, the intention is to use (19) in the

canonical form

(20)

Under such an approach formula ( 18) should also be valid for x 1 = 0 and

x 1 = l 1 , since otherwise (A 1 f/); 1 refers to the undetermined values for i 1 = 1

and i 1 = N 1 - l. Knowing y = p and i; = fl for x 1 = 0 and x 1 = l 1 , we

deduce fron1 ( 18) that

by observing that these coincide with the boundary conditions ( 13)-( 14).
Thus, we have proved that a solution of problem (9)-(14) satisfies equation
( 20) subject to the supplementary conditions

( 21)