1549301742-The_Theory_of_Difference_Schemes__Samarskii

452 Stability Theory of Difference Schernes

Since D > 0 and the space H is finite-din1ensional, D > liE and the

inverse operator n-^1 exists (Ii > 0). As far as B > 0 and D = D* > liE,

we have for a solution of the equation (0.5 TB + D) w = <p the estimate

llwlln < ll'Plln-1, so that

ll(g1)s,slln(t 8 ) < ll'Pslln-^1 (t 8 ) ·

By virtue of estimate (95) we are led to

lVIaking use of (97) and the triangle inequality, we derive the estin1ate for

a solution of problem (84b)

(98)

vVe summarize all the results obtained in the following assertion.

Theorem 9 Let conditions (85)-(87) be satisfied. Then scheme (84) is

stable with respect to the initial data and right-hand side and a solution of

problem (84) satisfies the a priori estimate

(99) llYn+1 llA(tn)

< M1 ~ ( lly(O)llA( 7 ) + llY1(0)11n( 7 ) + ~ T 11'Ps11n-^1 (ts)) ·

Corollary If D = E + T^2 R > E and n-^1 = E, then ll'Pslln-1 < 11 'P II and
for a solution of problem (84b) the following estimate is valid:

(100)

Finer estin1ates that are similar to estimates established for the string
vibration equation (for more detail see Chapter 5) hold true in a more
narrower class of schemes