1549301742-The_Theory_of_Difference_Schemes__Samarskii

436 Stability Theory of Difference Schemes

Theorem 3 Let the conditions of Theorem 1 be satisfied. Then scheme ( 1)

is stable with respect to the right-hand side and for it the estimate holds

fort > r:

(35) II Y(t + r) II< II Y(r) II+ Nh max (ll<tJ(t')llA-1 + ll<tJr(t')llA-1),

r<t'5,t

where M2 = const > 0 depends only on t 0 •

Theorem 4 Let A = A* > 0 and R = R* > 0 be constant nonnegative

operators and B = B( t) be a variable non-self-adjoint positive definite

operator

(36) B > sE, c = const > 0,

where a number c is independent of hand r, and let the condition

(37)

1

R > - -A 4

be satisfied. Then a solution of problem ( 1 b) admits a priori estimate

(38)

2 Vt [ t ] 1/2

II y(t + r) II < ----;--- t~ T II <p(t') 11

2

Look at identity (31). It follows from (36) and (32) for s 0 = c that

(39) r c II Yr 112 +II Y(t + r) 112 < II Y(t) 112 +=II <p(t) 112.

c

With the relations II Y(r) 11=0 and 11 Y(t + r) 112 > 0 in view, summing up

overt= r, 2r, ... , nr yields

( 40)

or

( 41)

t 1 t

L rllYr(t')ll^2 < - (^2) c L rll<tJ(t')ll2.
t'=r t'=r
Further development of estimate (38) from ( 41) is based on the fol-
lowing assertion of auxiliary character.