1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations

general case:

(--+) B;

CY i+l = C'; - CY; A; '

A; {3; + F;

C'; - CY; A; '

(~)

Y i = CYi+1 Yi+1 + f3;+1 ,

11

i = 1, 2, ... , N - 1,

i=l,2, ... ,N-1,

i = N - 1, N - 2, ... , 1, 0.

Here the sy1nbols (-+) and (,____)indicate the directions of index count: either

frotn i to i + 1 or from i + 1 to i.

In connection with the preceding algorithm, it is natural to raise the

question of correctness and stability providing a possibility of applying the

method and obtaining a solution with a prescribed accuracy. Special inves-

tigations give definite answers to to these questions.

6. Stability of the elinunation method. Let us stress that the conditions
   C'; - CY; A; -::f. 0 and 1-CYN x 2 -::f. 0 cannot be excluded or relaxed during the
   course of the right elimination method. Just for this reason the restrictions
   on coefficients for well-posed and stability conditions needs investigation.
   Common practice involves sufficient conditions

i=l,2, ... ,N-1,

( 16)

yielding ICY;I < l for all i = 1,2, ... ,N.

The proof is carried out by induction. Assuming ICY;I < 1 we will show

that ICY;+ 1 I < l. Since lo: 1 I = lx 1 I < 1, the sa111e propertry will cover all

i = 2, 3, ... , N on account the chain of the inequalities:

from which it follows that IC'i - o:; A;I > 0, since B;-::/- 0.
Granted In; I < 1, observe that