1549301742-The_Theory_of_Difference_Schemes__Samarskii

440 Stability Theory of Difference Schemes

Identity (53) yields

(57) 2T (By~) yt) + J(t + T) < ( 1 + (

2

+cc) c^3 T) J(t) + 2T (ip, yt).

Having at our disposal the energy inequality (57), we can derive the a priori

estimates in just the same way as was done for constant operators A and

R. For exan1ple, for B > 0 inequality (57) for proble1n (la) implies the

estimate

(58) if R > - l + 4 c A )

where !Vfi depends only on c, c 3 and t 0.

We bring together the basic facts in the following assertion.

Theore1n 6 Let variable operators A= A(t) = A*(t) > 0 and R = R(t) =

R*(t) > 0 be Lipschitz continuous int and let

(59)

l+c

R(t) >

4

A(t) for all 0 < t = n T < t 0 ,

where c = const > 0 is independent of T and h. Then for scheme (1) the

following estimates hold:

(60) llY(t + T)ll(t) < !Vfi llY(T)ll(r)

• M2 r<t' max <;,_t [llip(t')llA-1(t') + ll'Pr(t')llA-'(t')]

for B(t) > 0, 0 < t = nT < t 0 ,

( 61) llY(t + T)ll(t) < !Vfi llY(T)ll( ) + M2 max llip(t')ll

T r<t'<;,_t

for B(t) > €E, where€= const > 0, M 1 > 0 and !Vh > 0 are independent
of T and h both.

To avoid needless repetitions, we omit here the proof of the theorem.

Remark Some requirements of Theorem 6 can be relaxed. For instance,
the condition B > 0 can be replaced by the following one:

(62) B > -C4 T^2 A'

where c 4 = const > 0 is independent of T and h both. Under condition
(62) estimate (60) holds true for T < T 0 , T 0 = l/(4c 4 ).