1549301742-The_Theory_of_Difference_Schemes__Samarskii

344 Difference Schen1es with Constant Coefficients

provided condition (7) holds. With this relation in vrew, we derive the

inequality

(8)

where IIμ lie~ := maxxEl'h I μ(x, t) I, which expresses the stability of the

explicit scheme (3) with respect to boundary conditions and initial data for

(9) To= 2 1( L p h2 1)-l

a=l °'

By analogy with the one-dimensional case in the estimation of y (see Section

1.7), let us recast the difference equation as yj+i = Fj, where

and II Fj lie< II Y lie+ T 11 'P lie for T < T 0 , thereby justifying that

II JJ'+i lie= II Fj' lie< II~, lie+ T 11 'Pj' lie

for the same values of T < T 0. Summation over j' = 0, 1, ... , j yields the

inequality

(10)

j , I

II Yj+i lie < L T II 'P^1 lie,

j'=O

characterizing the stability of the explicit scheme with respect to the right-
hand side. For T < To relations (8) and (10) together imply the estimate

(11)

. ~ ·I j , I
II Y^1 +^1 lie < II Uo lie+ max II μJ lie + L T II 'P^1 lie·
O~J'~J+l ~ j'=O

With the notation h = min he,, the stability condition (9) takes the form

h2

T < ~ -2p,

thus demonstrating that the adn1issible step T of the explicit scheme de-
creases with increasing the di111ension p.