1549301742-The_Theory_of_Difference_Schemes__Samarskii

674 Methods for Solving Grid Equations

collection in a certain sense of t.he paran1eters T 1 , T 2 , ... , T 11 specified by

formula ( 43). In what follows the set so formed is denoted by

Scheme (14) with new parameters

k = 1, 2, ... , n,

possesses the property of computational stability. Assuming that the effect

of rounding errors is equivalent to possible perturbations of input data: the

initial approximation, the right-hand side and the operator A involved in

the iteration scheme (14). Under such an approach a numerical solution 'f.h

of proble111 (14) can be treated as t.he exact solution of the problen1

·ih+1 - Yk - _ ,, 1 _

----+Ayk=ik+1+--wk+11 k=0,1,2, ... ,n.

Tk+l Tk+l

In this context, it should be noted that the rnember C\+ 1 covers the error

1

of --Yk·

Tk+l

When possible perturbations of the operator A are neglected for one

reason or other, the intern1ediate solutions Yrn turn out to be bounded in

norm:

II Ym _ II < c^1 II Yo _ II+ ( 1 + n cm ) ( -^1 in.ax II f; - II+ c^1 max II W; _ II ) ,

c, c, fl l:<:;z:<:;m c, l:<:;z:<:;m

where cm = 1 if m f:. 2P; cm = 0 if in= 2P and the error z 11 = Yn - u of n
iterations satisfies the estimate

II Yn - u II< qn II Yo - u II+

1

• qn max II Ji - f II+

4

/C 1nax II W; 11.

fl l:<:;z:<:;n 3 v~ l<z<n

The derivation of these estimates espressing computational stability of iter-
ative i11ethods with optimal sets of Chebyshev's pararneters { r;} is omitted
in the present book. In the sequel we involve only the collection { r;}, allow-
ing a simpler writing of the ensuing formulas without concern of symbols
"*"
The results of calculations for the exa1nple from Section 5 through
the use of the explicit scheme ( 14) with ordered sets of parameters { r;} are
presented in the Table 5.