1549301742-The_Theory_of_Difference_Schemes__Samarskii

38 Preliminaries

was quite applicable in determination of the periodic solution Yi+N =Yi of

problem (61): aiYi-1-CiYi +biYi+l = -fi, provided that the conditions

of periodicity

and the additional conditions

(77) ai > 0, bi> 0 J Ci > ai + b;

hold.

We give below without proving the algorithm of the cyclic elimination

method which will be used in the sequel:

(78)

(79)

(80)

Cl'i+l = ----bi Ji + ai f3i

f3i+l = ----

Ci - ai Cl';

i=2,3, ... ,N;

i = N - 2, N - 3, ... , 1 ;

Yi= Pi+ YN qi J i = 1, 2, ... JN -1.

This algorithm is stable because the solutions of (79) are found by
the right elimination n1ethod being stable under conditions (77) with the
denominator 1-Cl'N+i q 1 -IN+i > 0. Indeed, it follows from (77)-(78) that
Cl'i < 1, /i > 0 and Cl' 2 + / 2 < 1. Assuming Cl'i + /i < 1 we get

(81)

Combination of (79) with (81) gives qN-l < 1 and qi< 1, thereby justifying
the relation 1 - Cl'N+i q 1 -1N+i > 0.