1549301742-The_Theory_of_Difference_Schemes__Samarskii

Operator-difference schemes 393

where y 0 is given. If a scheme with a constant transition operator Sis stable

with respect to the initial data, then it is uniformly stable with respect to

the initial data, since

(26) rr -1n,j - 'T' in-1JO. - JJ cn-j ·

Theore1n 4 The stability wilh respect to the initial data of scheme (25)

with constant operators is necessary and sufficient for the stability with

respect to the right-hand side, provided condition (24) of the norm of con-

cordance holds. Moreover, in that case a priori estimate (20) is valid.

Sufficiency. The stability with respect to the initial data means the

boundedness of the resolving operator

(27)

In accordance with what has been said above, this implies condition (19)

and so it remains to use Theorem 1.

Necessity. Let scheme (25) be stable with respect to the right-hand

side, meaning the validity of the inequality for the solution of problem ( 4b)

n

(28) llYn+1ll(l) < M1 ~ T llB-

1

<tJjll(l), n = 0, 1, · · · ·

j=O

This estimate holds true for any right-hand side Jj = B-^1 <pj. This provides
enough reason to conclude from (17) that

(29)

n

Yn+l = ~ TTn+l,j+l .fj ·

j=O

The assertion will be proved if we succeed in showing that property
(27) is true. Choosing T fj = bj,o we deduce from (29) that

and

II Yn+l 11(1) < II Tn,0 II · II f 11(1) ·

On the other hand, (28) yields II Yn+l 11(1) < M 1 11 f llcir Comparison of
these inequalities gives (27), from which the stability with respect to the
initial data immediately follows.