1549301742-The_Theory_of_Difference_Schemes__Samarskii

Hom.ogeneous difference schemes on non-equidistant grids 175

this approach a little better, we consider the simplest case, provided that

the asymptotic expansion

(14)

holds, where a 1 ( x) does not depend on h. In such a setting it is required

to find a grid function Yh ( x) satisfying the relation

(15)

The traditional way of covering this is to introduce two grids w h, and w h 2

with steps h 1 and h 2 and cornmon nodes, the set of which is denoted by

w h, and to forn1 the grid function

(16)

while c 1 and c 2 remain as yet unknown. Substitution of expansions (14) for

Y1i, and yh 2 into (16) gives

whence it follows that fih - uh= O(hk^2 ) if

This is certainly so with

(17) c 2 = 1 - c 1 ,

hk,

c - - 2

1- 7k1_hk' 21 2

In particular, keep.ing h 1 = h and h 2 = ~ h, we find wh = wh and c 1 =
-1/(2k^1 - 1).
Thus, the improved accuracy of a grid solution on some set of nodes
wh is connected with solving problem (12) twice (first on the grid wh, and
then on the grid whJ and drawing up the linear combination (16) with
coefficients (17). The grids w h, and w h 2 are chosen so that their intersection
coincides with w h. For instance, by applying a scheme of second-order
accuracy satisfying the relation