1549301742-The_Theory_of_Difference_Schemes__Samarskii

218 Homogeneous Difference Schemes

The function u( x) is linearly interpolated on each of the segments [x;_ 1 , x;]

and [ x; , X;+ 1 ], so that

u·+(x-x·)u z z x )z ·

u(x) ~ {

for

u; + (x - xi) ux,i for

which simplifies the huge job done with the integrals

"'i+1 "'i+1 ~·i+1

;· wdx ;· ku' dx ~ llx i

J

. ' k(x) dx,
x· 1 xz Xi

Xi Xi

J

wdx ~ Ux i ;· k(x)dx,

'

Xi-I Xi-I

x· Xi

J

1

qu(i - x;_ 1 ) di~ -u; j q(i) (i - xi_ 1 ) di

:Ci-I

Xi

+ ux,i j q(i) (x; - i) (t - x;_ 1 ) di,

"'i+1

j q(i) (x;+ 1 - i) di

x;+1

+ ux,i j q(i) (x;+ 1 - i) (i - x;) di.

x·

'

Substituting the resulting expressions into identities (10)-(11) and sub-
tracting the second identity from the first one, we arrive at the difference