1549301742-The_Theory_of_Difference_Schemes__Samarskii

392 Stability Theory of Difference Schemes

To prove this assertion, it suffices to verify that estimate (21) implies

estimate ( 19):

II Tn,j II= II Sn-1 Sn-2 · · · Sj+l Sj II

< 11 Sn - 1 11. 11 Sn - 2 11 ... 11 S'j + 1 11. 11 S'j 11 < ( 1 + Co Tr-j

Quite often, the reader can encounter the statement that "stability

with respect to the initial data implies stability with respect to the right-

hand side". How is one to understand the nature of this assertion?

We say that scheme ( 4) is uniformly stable with respect to the

initial data if the Cauchy problem is stable:

(22) Yn+l = SnYn, n = j, j + 1, ... , Yj , j = 0, 1, ... , n given,

for any j = 0, 1, ... ,n 0 -1, that is,

where M 1 is a constant independent of T and h both.
If the condition of unifonn stability is satisfied, then estimate (19)
holds true for the resolving operator Tn,j. Therefore, Theorem 1 asserts
that estimate (20) is valid for a solution of problem (4). This type of
situation is covered by the following results.

Theorem 3 If scheme ( 4) is uniformly stable with respect to the initial
data, then it is also stable with respect to the right-hand side under the
condition of the norm concordance

(24)

Moreover, condition (24) assures us of the validity of a priori estimate (20).
It is worth noting here that condition (21) is sufficient for uniform
stability with respect to the initial data.
The object of investigation is the two-layer sche1ne with constant op-
erators A and B, not depending on tn = nr:

Yn+1=Sy 71 +rf 11 , f 71 =B-^1 <p 71 , n=O,l, ... ,
(25)
S=E-rB-^1 A.