1549301742-The_Theory_of_Difference_Schemes__Samarskii

Methods for designing difference schen1es 217

Making use of the approximation

h

J ( qu - .f) dx ~ ( qu - f)o h,

0

we impose the boundary condition for Yi with the approxirnation error

O(h^2 ) at the point x = 0:

(8)

The boundary condition (2) of the third kind can be established at the
point x = 1 in a similar way.
So far we have studied some versions of the IIM on the basis of the bal-
a.nce equation (the balance method). We now consider the second rnethod
for the design of homogeneous difference schen1es by means of the IIM, in
the framework of which equation (1) has to be integrated twice: first, we
integrate equation (1) from xi to x:
x
(9) w(x) - w(xi) = ;· (q1l - f) dx.
Xi

Second, we integrate the preceding over x from xi to xi+i and frorn xi-i
to X(

( 10)

The interchange of the integration order leads to

x· z

Xi

;· (t - xi_ 1 ) (qu - J) di.