1549301742-The_Theory_of_Difference_Schemes__Samarskii

166 Ho1nogeneous Difference Sche1nes

As far as Ai < AN, the preceding decomposition implies that

The difference t;; = v; - w;, solving the problem

t;(O) = t;(l) = 0,

with a > c 1 > 0 and d > 0, is aimed at the further estimation of vi in

complete agreement with the len1n1a from Chapter 1, Section 2.8. The

results of this section provide support for the view that II t; lie -:; II w lie

and, therefore,

Surnmarizing, we anive at

(14)

The function 1/;* is representable by

(15) 1/;*=μ~, ' .c

where μi = L~~\ ·h1/;~ for i = 2, 3, ... , N and μ 1 = 0. Because of this
forrn, it is possible to write down 1/; = (TJ + μ)x and involve estimate (14)
in subsequent reasonings:

2 2

llzlle < -(l,l17+μIJ-:;-((l,l11IJ+(l,lμll).

c1 c1

4. On convergence and accuracy. The results obtained in the preceding two
   sections may be of help in establishing the rate of convergence for scheme
   (2).