390 Stability Theory of Difference SchemesThus, the three-layer scheme (8) is said to be stable if for any initial
data y 0 , y 1 and for any right-hand side ip(t) its solution satisfies the estimate(14)where M 1 and M 2 are positive constants independent of h and T and dis-
regarding to the choice of Yo, y 1 and ip( t).
The main problem under consideration amounts to the following one.
Suppose that equation ( 4) is uniquely solvable with respect to Yn+l for
any Yn and ip(t). What are conditions on the operators A and B for a
scheme to be stable in the sense of the above definition? In other words, we
wish to establish sufficient stability conditions for sche1ne ( 4) and obtain
a priori estimates of the fonn (12). Moreover, sufficient conditions should
be convenient for practical verifications in the case of concrete difference
schemes associated with equations of mathematical physics.
In what follows stability of differential schemes will be given special
investigation irrelevant to approximation and convergence.
- Sufficient stability conditions for two-layer schemes in linear normed
spaces. We now raise the question concerning sufficient stability condi-
tions for two-layer schemes in linear normed spaces. In full details these
investigations will be carried out in Section 2 for the case when B1i = H1i
is a real Hilbert space.
In what follows the Cauchy problem ( 1) is supposed to be solvable,
that is, the inverse operator B-^1 exists. Therefore, scheme ( 4) can be
writ ten in the fonn
(15) Yn+I =Sn Yn + T fn, j~ = B;;,^1 'Pn' n = 0, 1, ... ' Yo E B1i,
where
( 16)
is the transition operator. The operator Sn depends on tn = n r, hand r,
however, neither for Sn nor for Bn, An, Yn, 'Pn and Yo the dependence on
h and T is explicitly indicated.
(17)
By virtue of the recurrence relation (15) we find thatn
Tn+l,O Yo+ I.= T Tn+l,j+l fj,
j=O