1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schemes on non-equidistant grids 177

When u(x), f(x) and the coefficients of equation (11) are smooth enough,

the expression for the error of approximation ij;h ( x) is sin1ple to follow:

n-1

(19) 1/;h(x) =I:: f3s(x) hs + f3n(x, h) hn,

s=l

where f3s(x), 1 < s :::; n - 1, are independent of h and lf3n(x, h)I < M,

Ji/[ = const > 0 is also independent of h. From such reasoning it seems

clear that any sufficiently smooth function 01 s ( x) admits the representation

n-1

(20) Lh C1 3 (x) = L01 3 (x) +uh+ I:: / 8 (x) h^8 + /n(x, h) hn.

s=l

In this view, it is not unreasonable to attempt the difference Yh - uh

in the form (18). Applying the operator Lh to identity (18) yields

n-1

Lh (Yh - uh)= I:: Lh C1 5 (x) hs + L01n(x, h) hn.

s=l

Using the relation Lh(Yh - uh)= 1/;h and fonnula (20) for Lh01 8 , we arrive

at

n-1 s-1

Lh(Yh - 1th)= I:: ( L C1 3 +I:: /m hm) h^8 + O(hn),

s=l m=l

g1vmg

n-1 s-1

(21) 1/Jh = I:: ( L C1 3 + I:: /m hm) h^8 + O(hn).

s=l m=l

Comparison of formulae (19) and (20) justifies the validity of expansion

( 18) if functions 01 s ( x) are solutions to the equations

s-1

L01 3 = f3 3 (x) - I:: lm(x) for s=l,2, ... ,n-1.

m=l

Observe that expansion (19) is not obliged to contain all of the powers of
h, some of them may be omitted. In this case the appropriate coefficient
equals f3s = 0.
Adaptive grids rnay be of assistance in raising the order of accuracy
without increasing the total number of nodes. If the subsidiary inforn1ation