1549301742-The_Theory_of_Difference_Schemes__Samarskii

432 Stability Theory of Difference Schemes

By inserting in (16) v = fJ and z = y we transform (15) into

(17) (A(fJ + y), fJ - y) = 0.5 [ (A(fJ + y), )j + y) + (A(y - y), f; - y)]

• 0.5 [ (A(y + y), y + y) + (A(y - y), y - y)].

We now substitute (17) and (13) into (12) and take into account that

(A(y - y), fJ - y) = r^2 (Ay 1 , Yi), Yr= (y - ~Q)/r = Yt,

(A(y - y), y - y) = r^2 (Ayr, Yr),

making it possible to establish the basic energy identity for the three-layer

scheme (1 ):

( 18) 2r ( Byt , yt) + [ t (A(y + y), y + y) + r^2 ( ( R - t A) Yt, Yt) l

= [ t (A(y + y), y + y) + r^2 ( (R - t A) Yr, Yr) l + 2r (<tJ, Yt) ·

In giving it we preassumecl only property (9) concerning the self-adjointness

of the operators A and R and no 1nore.

3. Stability with respect to the initial data. Recall the definition of stability
   with respect to the initial data and the right-hand side. Scheme (1) is said
   to be stable with respect to the initial data if for problem (la) the a priori
   estimate holds:

(19) llY(t + r)ll(l) < M1 llY(r)ll(lo).

Scheme ( 1) is said to be stable with respect to the right-hand side if for
problem ( 1 b) the estimate

(20) 'llY(t + r)ll(i) < M2 max ll<tJ(t')l( 2 )

O<t'<::t

or the estimate

holds.
Making use of the triangle inequality and collecting (19) and (20) or
(21), we obtain estimate (5) or (6).