1549301742-The_Theory_of_Difference_Schemes__Samarskii

Operator-difference schemes 391

where

Tn+l, o =Sn Sn-1 ... S1 So, Tn+l,n+l = E.

The operator T,,+ 1 , j is called the transition operator from the layer j to

the layer n+l, while the operator Tn+l, o refers to the resolving operator.

The triangular inequality yields

n

(18) II Yn+l 11(1) < II Tn+1,o II · II Yo ll(i) +LT II Tn+1,j+1 II · II fj 11(1),

J =Cl

where II · 11(1) is any suitable norm on the space Bh, 111aking it possible to

arrive at the following assertion.

Theorem 1 For the stability of scheme (15) it is sufficient that for any

0 < j < n < n 0

( 19)

Moreover, having stipulated this condition, the solution of the difference

problem (3) satisfies the a priori estimate

for all 0 < n < n 0.

Note that estimate (20) implies (12) with constant M 2 = M 1 t 0 and

II <fj llc 2 J = II Bj-l <p~ 11(1) incorporated.

Theore1n 2 For the stability of scheme (3) it is sufficient that for the norm
of its transition operator sj the estimate

(21)

is valid for all j = 0, 1, ... , n 0 - 1, where c 0 > 0 is a constant independent
of T and h. Moreover, under condition (21) a priori estimate (20) holds
with constant M1 =exp {c 0 t 0 }.