1549301742-The_Theory_of_Difference_Schemes__Samarskii

420 Stability Theory of Difference Schemes

Joint use of Lemma 6 and inequality (60) permits us to state the

following.

Theorem 11 Let A= A(t) be a positive operator and condition (55) hold.

Then for scheme ( 46) with <7 > <7c the a priori estimate

( 61)

1 t

lly(t+r)ll < lly(O)ll+-c L rll<tJi^1 ll

t'=O

is true. If the following conditions

1 1

<7a=2-r6.

are simultaneously satisfied, then estimate (61) is attained for c = 1.

To prove this theore1n, let us write down scheme ( 46) in the form

where S = (E + <7rA)-^1 (E - (1 - <7)rA) and B = E + <7TA. Using the

triangular inequality and estimates (57)-(59) behind, we obtain the relation

T

II Yn+I II< II Yn II+ -c II <f1n II,

from which estimate (61) immediately follows.

11. A priori estimates in the case of a variable operator A. So far we have
    established stability in HA under the agreement that operator A is constant,
    that is, independent oft. In the case when A(t) = A*(t) > 0 depends on
    t, this obstacle necessitates imposing the Lipschitz continuity of the
    operator A( t) in the variable t

(62) I ( (A£ t) - A( t - T)) x, xi < T c 3 (A( t - T) x, x)

for all x EH, 0 < t < n 0 r, where c 3 is a positive constant independent of
hand T both.
A primary family of describing schemes is specified by the following
restrictions:

A(t) = A*(t) > 0 for all t E wT,

(63) A(t) is Lipschitz continuous in t,

B(t) > 0 for all t E wT.