1549301742-The_Theory_of_Difference_Schemes__Samarskii

394 Stability Theory of Difference Schemes

Summarizing, we draw some conclusions which will be needed in the

sequel.

1. When the transition operator happens to be constant, the stability
   analysis with respect to the initial data is mostly based on estimates of the
   norms of the transition operator.


2. The condition of concordance between the norms of the right-hand
   side and the solution such as

is a severe constraint. If II B-^1 II< c 1 , where c 1 > 0 is a constant indepen-

dent of hand T, then II <p 11( 2 ) < c 1 11<p11(1) and instead of (20) we obtain the

estimate

t

(30) lly(t + r)ll(l) < M1 lly(O)ll(l) + M2 ~ T ll<tJ(t')ll(l), M2 = M1 c 1.

t'=O

In Section 2 a priori estimates will be obtained for which condition (24) of
the norm concordance is not required.

3. Scheme (4) is stable if II sj II< 1 + CaT for all j = 0, 1, ... ) no - 1.
   In practical applications of this sufficient stability criterion one needs to
   reveal some properties of the operators A and B ensuring con di ti on (21).
   Such conditions are established in Section 2 of the present chapter. They
   asquire the form of linear operator inequalities for the operators A and B
   acting in the Hilbert space Hn = Bh.


4. Approximation and convergence. The notions of approximation, conver-
   gence and accuracy for operator-difference schemes are introduced by anal-
   ogy with the corresponding notions for the operator schemes AnYn = <tJn
   arising earlier in Chapter 2, Section 4. Only a few editorial changes will
   appear in this matter. For instance, together with the norms II · ll(lh) and

II · ll(h) it will be sensible to introduce the additional norms

llYhy(tn)ll(lhr) = O<t~i:)'~<t - - n llY1ir(tn')ll(h)'

which complement further stability analysis.
Thus, let Bh be a linear space with norms II · ll(h) and 11 · ll( 2 h) on it.

We denote by B~^1 ) and B~^2 ), respectively, the resulting normed spaces and
assume in the sequel that

Yhr(tn) E B~l)) <f1hr(tn) E B~

2

)