1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer sche1nes 399

err A. Since A< 11A11 E and E >A/II A 11, we have B > (1/11 A ll+err)A > 0

under the restriction er > -1/(rll A II). The operator A = A* > 0 is

independent oft, so that conditions (1)-(3) are satisfied and the scheme

concerned belongs to the pri1nary family for er > -h^2 / ( 4c 2 T), where c 2 =

max a(x).

xEwh

3. The energy identity. We study the problem of stability of scheme ( 1) by
   the method of the energy inequalities involving as the necessary manipula-
   tions the inner product of both sides of equation (1) with 2ryt = 2('[; - y):

(9)

Using the formula

( 10)

we rewrite (9) in the form

(11) 2r((B - 0.5rA)yt, Yt) + (A(y + y), f;-y) = 2r(ip, Yt).

Lemma 1 Let A be a. self-a.djoint opera.tor, then

(12) (A(y + y), y - y) = (Ay, y) - (Ay, y).

Indeed, the chain of the relations

(Ay, y) = (y, Ay) = (Ay, y)

occurs due to the fact that A is self-adjoint operator and, therefore,

(A(y + y), y - y) = (Ay, f;) + (Ay, y) - (Ay, y) - (Ay, y)

= (Ay, y) - (Ay, y).

Substituting (12) into (11) we obtain the energy identity for scheme (1):

(13) 2r((B - 0.5r A)yt, Yt) + (Ay, y) = (Ay, y) + 2r(ip, Yt).

4. Stability with respect to the initial data in HA. Stability of sche1ne ( 1)
   with respect to the initial data is investigated with further estimation of a
   solution of problem (la).