1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schemes 421

As before we assume the existence of an inverse operator B-^1 (t), which

assures us ofsolvability of problem (1) for any input data y 0 and ip(t). This

family obviously contains the primary family introduced in Section 2.2.

The method of energy inequalities shows in such a setting that the

conditions

(64)

(65)

B(t) > 0.5 T A(t)

B(t) > sE+0.5rA(t)

for all t E wT ,

for all t E wT , 0 < c < 1 ,

turn out to be sufficient for the stability of scheme (1) with variable oper-

ators A(t) and B(t). The norms 11 · llA' 11 · llA-' the1nselves happen to be

dependent on the variable t:

Therefore, it makes sense to speak about stability in the space HA(t) (in-

stead of HA) and HB(t)·

The energy identity (13) with A= A(t) is the starting point in special

investigations. To obtain a recurrence inequality, we should modify the

express10n

(Ay, y) = (A(t) y(t), y(t)) = (A(t - r) y(t), y(t))

• ((A(t) - A(t - r)) y(t), y(t))

and estimate the second sum111and on the right-hand side with the aid of
inequality (62):

(A(t)y(t),y(t)) < (1 + rc 3 )(A(t - r)y(t),y(t)).

Upon substituting this esti1nate into (13) we obtain the energy inequality

'
(66) 2r ((B(t) - 0.5 T A(t)) y(t), y(t)) + E(t + r)

< (1+TC 3 )£(t)+2r (ip(t),y(t)) 1

where
E(t + r) = (A(t) y(t + r), y(t + r)) = llY(t + r)ll~(t).

Inequality ( 66) with the member ip = 0 i1nplies that