1549301742-The_Theory_of_Difference_Schemes__Samarskii

400 Stability Theory of Difference Schemes

Theorem 1 The condition

( 14) B >!..A - 2

is necessary and sufficient for the stability in HA of scheme (1) from the

primary family with respect to the initial data with constant M 1 = 1, that

is, condition (14) is necessary and sufficient for the validity of the estimate

(15) 11 Yn 11 A < 11 Yo 11 A > n = 1, 2, ... ,

where Yn is a solution of problem (la).

Sufficiency. Granted condition (14), the energy identity for problem

(la) (with <p = 0)

(16) 2r((B - 0.5r A)yt, Yt) + (Ay, y) = (Ay, y)

implies the inequality (Ay,f;) < (Ay,y) or ll:iill~ < llYll~, yielding

Necessity. Suppose that scheme (la) is stable and estimate (15) is

satisfied. We are going to show that this leads to the operator inequality

(14), that is,

( 17) (Bv, v) > 0.5r (Av, v) for any v EH.

We begin by placing identity (16) on the first layer (n = 0):

By virtue of ( 15) this identity can be satisfied only for

meanmg

If y 0 E H is an arbitrary element, then so is the element v = Yt(O) =
-B-^1 Ay 0 E H. Indeed, taking any ele1nent v = Yt(O) E H we find Yo =
-A-^1 Bv E H for A-^1 does exist. Thus, the inequality is met for any
v = Yt(O) E H, that is, the operator inequality (14) takes place, thereby
completing the proof.