1549301742-The_Theory_of_Difference_Schemes__Samarskii

220 Homogeneous Difference Schemes

Let ~ = xn + ()h, 0 < () < 1, that is, xn < ~ < xn+I' n > 0. The

balance equation (3) on the segment xn < x < xn+i is of the form

Wn+I - Wn + [w] = xJ+i (qu - J) dx

Xn

or

(15)

On the remaining segments [xi , X;+iL i f n, identity (3) holds true. As a

final result, instead of (12), we get the scheme

where

tp; = f; ' i In, if n + 1,

(16)

Q

l.fn = fn + 2 h '

In the physical language, this is a way of saying that the source is spread
over two intervals.
Rewriting the same identity on the segn1ent [xi_ 112 , X;+ 1 ; 2 ] reveals
another scheme

(17)

for 0 < () < ~,

for ~ < () < 1.

Q

l.fn = fn + -n

Let us stress that the integro-interpolational method is a rather flex-
ible and general tool in designing difference sche111es relating to stationary
and nonstationary problems with one or several spatial variables.