1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schemes^401

Remark 1 Condition (14) is sufficient for stability in the sense of (15) of

scheme (1) if B = B(t) > 0 is a non-self-adjoint variable operator.

Remark 2 The fact that condition (14) is, in a certain sense, natural can

be clarified a little bit by considering the simplest example relating to the

difference scheme

b Yn+l - Yn + ayn = , O

T

n = 0, 1, ... , Yo = Uo,

where a and b are positive numbers corresponding to the differential equa-

tion

du

b-+au=O,

dt

The difference equation yields

Yn+l = (1- Tba)Yn,

The stability requirement

t > 0, u(O) = u 0 ,

is obviously satisfied if ll - ra/bl < 1 or -1 < 1 - ra/b < 1, that is, for

b > !ra. The similarity with the operator equation B > !ra is clear at

the first glance.

Exa1nple We now consider the weighted scheme

Yt + A (of; + ( 1 - a) y) = 0 ,

whose use permits us to illustrate the effectiveness of the stability condition
(14). For this, we ,write clown this scheme in canonical form accepted in
Section 1:

( 18) B=E+arA.

If A= A* > 0 is independent oft and a> -1/( rll A II), then the weighted
scheme belongs to the primary family (see Section 2). The necessary and
sufficient stability condition (14) is of the form

B - 0.5 TA= E +(a - 0.5) TA> 0.