1549301742-The_Theory_of_Difference_Schemes__Samarskii

466 Homogeneous Difference Sche1nes for Ti1ne-Dependent Equations

holds. Under this condition the relation

takes place. Su111ming up the preceding other j = 0, 1, 2, ... , we obtain the

inequality

j

(13) 11yi+^1 11 c < 11Y^011 c + I: T 111c/ 11 c ,

j'=O

which expresses the stability of scheme (10) in the space C if

(14)

This condition is only sufficient for the indicated property. A necessary and

sufficient condition for the stability of the explicit scheme with respect to

the initial data in the space HA is

( 15)

2

T <-- ~) where

But it may happen that the coefficient k( x) varies very fastly. In that
case the estimate~< 4c 2 /h^2 is too rough and condition (14) gives a severe
restriction. In mastering the difficulties involved, we are forced to apply
the maximum principle to the weighted scheme (9) written in the canonical
form for any O":

The boundary conditions y(O) = y(l) = 0 together with Theorem 3 in
Chapter 4, Section 2 for equation (16) give us the estin1ate