1549301742-The_Theory_of_Difference_Schemes__Samarskii

478 Homogeneous Difference Sche1nes for Tiine-Dependent Equations

with i1^1 = EPu/ at ax and f' = [} f I ax incorporated, we arrive at

It seen1s clear that representation (53) for the residual 1/; is an i1nme-

cliate i1nplication of the balance equation ( 52) by observing that

(ku'). = (ku'). - 0.5hi(ku')'. + ~ h^2 (ku')

11

z-1/2 · z z 8 z z + O(h^3 z )

and, hence,

(ku')'. = [(k11

1

)i-l/ 2 J,,,i- 8 ~ [h~+ 1 (ku')'.

1

-h~(ku

1

);

1

J +0(127).

z

By virtue of the relations

vi = vi-1/2 + O(h;),

(ku^1 )

1

=u-f, (k u^1 )

11

= u' - !'

the desired result will be substantiated if we succeed in showing that

~. (h;+l V; - h; v,) = (h; V;-1;2)x,i + O(lin.

z

An alternative form of the residual

is best suited for our purposes in the applications of formula ( 54), thereby
justifying the final results

17 = 0.5 a (u;; + u,,) - k u' + r (o- - 0.5) a tl,r:t + O(h^2 ) = O(h^2 +rm~).

The next question we have raised above is the accuracy of sche1ne ( 49),
the accurate account of which can be clone using a priori estimates of the
problem (50) solution with further reference to the special structure (50) of
the right-hand side of 1/;. Following established practice, we introduce the