1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schemes 403

The operator A - = E = A* - > 0 is independent of t and the operator B - is

positive for <7 > 0. Stability condition (14) in the space HA = JI takes for

now the form B - ~TA= A-^1 + (<7 - ~)rE > 0, valid with <7 > ~- The

condition <7 > ~ is sufficient for the occurrence of the relation

11 Yn 11 < 11 Yo 11 forr7>0.5, A:f=A*, A>O.

5. Stability with respect to the initial data in HB. Let us write down the
   second energy identity for scheme (la) assuming B also to be a self-adjoint
   operator: B = B* > 0. At the first stage we take the inner product of (la)
   and 2ry:

( 19)

Relying on the formulae

y = 2 1 (A y + y ) - T 2 Yt

and making use of Lemma 1, we find that

2r(Byt,Y) = (B(y-y),y+y) +r^2 (Byt,Yt)

= II fJ II~ - II Y II~+ r

2

II Yt II~,

Substitution of these expressions into (13) leads to

Theorem 2 Let the operators A and B involved in scheme ( 1) be inde-
pendent oft, A = A > 0 and B = B > 0. Condition (14) is sufficient
for scheme ( 1) to be stable with respect to the initial data in the space H B
with constant M 1 = 1.
Indeed, let B 2 ~TA, then

llYt II~ -0.5rl1Ytll~ = ((B-0.5rA)yt,Yt) > 0