408 Stability Theory of Difference SchemesLet us show that inequalities (14) and (30) are equivalent. To this
end, we operate with the expression
N
By - 0.5 T Ay = L ck(t) (B ~k - 0.5 T A~k)
k=l
N
= L ck(t) (1 - 0.5 T ,\k) B~k,
k=l
by means of which it is plain to calculate the functional
N
(By,y)-0.5r(Ay,y) = ~2 L_, ck(t)(l-0.5r>-
1 J.
k=l
Because of this form, the equivalence of (14) and (30) is obvious.
We thus have shown that under conditions (26) inequality (14) is
sufficient for the stability of scheme (la) in the space HA, that is, relation
(29) occurs. Let us stress that the requirement of self-adjointness of the
operator B is necessary here, while the energy inethod demands only the
positivity of B and no more.
One can prove in a similar way the stability of scheme (la) in the
space H B, provided condition (30) holds.
- The p-stability condition. A more general definition of stability with
respect to the initial data became rather urgent and extremely important.
Let D = D* > 0 be a constant operator. Scheme (1) is said to be
p-stable with respect to the initial data if for a solution of problem (la) the
inequality holds for any Yo E H:
( 31) II Yn lln < p" II Yo lln '
where p =exp {c 0 r}, c 0 is a constant independent of h, T and irrelevant
to the choice of y 0 .- If sche1ne (la) is p-stable in the space HD, then it is
stable in the same space HD:
II Yn lln < NI1 II Yo lln ,
with constants M1 =exp {c 0 t 0 } for c 0 > 0 and M1 = 1 for c 0 < 0.
The two-layer scheme (la) with constant operators A and B can be
reduced to the explicit scheme
x n+l -x n + C Xn = 0
(32) or
T