1549301742-The_Theory_of_Difference_Schemes__Samarskii

168 Homogeneous Difference Schemes

3.4 HOMOGENEOUS DIFFERENCE SCHEMES ON

NON-EQUIDISTANT GRIDS

1. Schemes on non-equidistant grids. Quite often, in practical irnplementa-
   tions difference schemes on non-equidistant grids are in common usage for
   solving differential equations. In Chapter 2, Section 1 we have produced for
   the simplest equation u^11 = -f a difference scheme on the non-equidistant
   grid

and have determined for this scheme the error of approximation.

In order to develop a homogeneous conservative scheme on the non-

equidistant grid wh, we write down the balance equation

xi+l/2 xi+l/2

(1) wi_ 112 - wi+ 1 ; 2 - j q(x)u(x)dx = - j f(x)dx, w -- -k u' ,

Xi-1/2

on the interval (x; 112 , xi+i; 2 ) with the ends xi 112 = xi - 0.5 hi and
X;+ 1 ; 2 = xi + 0.5 hi+i · By analogy with Section 2.2 the forthcoming sub-
stitutions

·"i+l/2

J

Xi-1/2

xi+1/2

(^0 0 1)
J
q u dx ~Ii z d 'l u i l d z q(x)
Ii z
·"i-1/2
0
----h = - ai Ux , i '
z
(
o 1 x; dx )-1
ai= hi J k(x)
.r:i-1
clx,