1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schemes on non-equidistant grids 171

where 1/;(x) = (aux)x - du+ <p(x) is the error of approximation.

On account of the balance equation (1) the error of approximation 1/;;

reduces to

where

(7)

1/;i = 1li:, 1 + 1/J; J 17i = (aux)i - (k u')i-1/2,

'''· * = (1n. - <p.)^0 - d 1l· + -^1

'+'i ri 1 i i Ji.

'

q(x)u(x) dx.

Xi-1/2

In what follows the functions k, q and f will be smooth for xi _ 1 <

x < xi and x; < x < xi+ 1 and possess discontinuities of the first kind at a

single node xi. Combination of the expansions

for s < 0 and f(xi + s hi) = f;+o + s h;+ 1 I:+o + O(hf+ 1 ) for s > 0 with

0

formula (3) for <pi gives

&; = h; f;-o; ~i+1 f;+o + ( h7 f~-112) _. + O(nf).

x) z

Likewise,

1

n

' Xi-1/2

With the relations ai = ki 112 and ux,i = u~ 112 +0(hf) in view, we deduce
from the foregoing that for the scheme with coefficients (5)

(8)

whose ingredients behave as follows:

1fJ7* = O(nf),

(9)

_ hf (qu-J);_1/2 2

77;=

8

=O(h;),

7); = ai ux,i - (ku')i-1/2 = O(h7).