1549301742-The_Theory_of_Difference_Schemes__Samarskii

422 Stability Theory of Difference Schemes

provided condition (64) holds. The energy identity for t = 0 admits the

form

2r ((B(O) - 0.5 r A(O)) y(t), y(t)) + £( r) = lly(O)ll;l(o) + 2r (<tJ(O), Yi(O)).

With conditions ( 64) and <p = 0 in view, we deduce from here that

(68) E(r) < lly(O)ll~(o).

Collecting (67) and (68) with regard to problen1 (1) with <p = 0 we establish

Such a reasoning discovers actually only one essential difference be-

tween the cases of variable and constant operators.

We now summarize the above results in some aspects as the analogs

of Theorems 5 and 7 that furnish the justification for what we wish to do.

Theorem 12 Let operators A = A(t) and B = B(t) be dependent on t

and conditions (63) and (64) be satisfied. Then for scheme (1) we have the

estimate

(69) llYn+1 llAn < M1

X {llYollAo + ll<tJollA~' + ll<tJnllA~' + k~l T ll(A-l<tJh, kllAk}'

where Mi =exp { ~c 3 t 0 } and A 11 = A(tn).

If c 3 = 0, then the operator A is independent of the variable t and

(69) becomes (37) on account of the equality

Theore1n 13 Let operators A = A(t) and B = B(t) be dependent on t
and conditions (63) and (65) be satisfied. Then for scheme (1) we have the
estimate

n

(70) llYn+1ll~n < M1
2
{llYoll~ 0 + 2

1

€ ~ rll<f1kll

2

},

k=O

Estimates (37) and (45) are obtained for the case of constant operators
A by merely setting M1 = 1 or c 3 = 0.