1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schemes 415

Furthermore, we substitute this estimate into ( 13). Having stipulated con-

dition (44), it follows from the foregoing that

11 ii 11 ~ < 11 Y 11 ~ + 2

7

c: 11 <tJ 112

or

l!Yj+1ll~ < llYj II~+ 2

7

c: ll<tJj 112

Sun1ming up then over j = 0, 1,,.,, n leads to (45). Thus, Theorem 7 is

completely proved.

Remark 1 Theoren1 7 remains valid in the case of a variable operator

B = B(t) and Theorem 2 continues to hold for a variable operator A= A(t).

The reader is invited to verify these facts on his/her own with the aid of

the proofs of the aforementioned theorems.

Remark 2 If A= A*, B = B* > 0 and conditions (36) are satisfied, then

scheme ( 1) is p-stable in the space H B with respect to the initial data, that

is, a solution of problem (la) satisfies the inequality

11 Yn 11 B < Pn 11 Yo 11 B ·

On the strength of Theorem 3 in Section 1 this implies the following

estimate for a solution of problem (1):

n

II Yn+l lls < Pn+l II Yo lls + L T Pn-k ll<tJklls-^1 ·

k=O

It is worth noting here that the positiveness of the operator A is unnecessary

in this matter. Apparently, this remark needs certain clarification. Let, for

instance, A> -c*E and c* > 0. When this is the case, the condition

or

T

A+p-lB>O

T

can be met only for p =exp {c 0 r} > 1, that is, for c 0 > 0. Under the

agreement B > c:E, c: > 0, we find that

' p - l

A+ B>A+c 0 B>(-c*+c 0 c:)E

T

and, therefore,

l-p

A> B

T

if we accept c 0 > c*/c:. To make sure of it, we may choose as B, for example,

the operator

B=E+rR, R=0.5A', A'=A+c*E>O.

Then c: = 1, c 0 = c and p = exp { c T}.