1549301742-The_Theory_of_Difference_Schemes__Samarskii

406 Stability Theory of Difference Schemes

Thus, the functional J B [y] is transformed into the same form (23) as was

done before for J A [y]:

] B [y] = 2 T ( ( B v, v) - ~ (A v, v)) ,

where

v = B-1/2 c1/2 B-1/2 Ay, y = A-1 B1/2 c-1/2 B1/2 v.

This implies the equivalence of conditions (25), which assures us of the

validity of Theorem 2.

Therefore, the method of estimation of the transition operator norm

permits us to prove that condition (14) is necessary and sufficient for the

stability of scheme (1) with respect to the initial data in the space HA (for

B # B*) and in the space Hs (for B = B* > 0) with constant M 1 =1.

7. The method of separation of variables. If the operators A and B are
   self-adjoint

(26) A=A>O, B = B > 0,

then the stability of scheme (la) in the spaces HA and H 8 for

B > - 2:.A 2

can be proved by means of the inethod of separation of variables following
established practice (for more detail see Chapter 5).
Let N be the dimension of a finite-dimensional space H, >.k be eigen-
values and ~k be orthonormal eigenfunctions of the problem (see Chapter
1, Section 1 and Chapter 2, Section 1)

(27) k=l,2, ... ,N,

where, in addition, (B~k,~m) = /jkm (bu= 1 and /jkm = 0 fork# m). All
the eigenvalues >.k of problem (27) are positive, because A > 0.
In such a setting a solution of problem (la) is sought in the form
N
(28) y(t) = z ck(t) ~k.
k=l

By virtue of the relations

N N

Ay = L ck A ~k = L >.k ck B ~k

k=l k=l