172 Homogeneous Difference Schemes- The order of accuracy on non-equidistant grids. As before, the inner
products are defined by
N-1 N
(y,v).= L Yiv;n;,
i=l(y, v]. = L Yi V; h;.
i=lFor problem (6) with the right-hand side (8) the estimatei-1
is valid with μi = L lik 1/JZ for i = 2, 3, ... , N and μ 1 = 0.
k=lThe derivation of this estimate coincides with that of inequality (12)
carried out in Section 3 by inserting a;/ h; in place of a;/ h.
In light of the asymptotic relations (9) estirnate (10) implies that if
k, q, f E QC^2 l[O, 1], then scheme ( 4)-(5) is of second-order accuracy on the
sequence of non-equidistant grids w h ( J{):
where h = J(l, h^2 ] is the mean square step.
When only one coefficient k( x) E QC^2 ) is discontinuous, while other
coefficients q, f E cC^2 ) are continuous, any conservative scheme ( 4) gen-
erating a second-order approximation is of second-order accuracy on the
sequence of non-equidistant grids w h ( J{). This fact is an immediate impli-
cation of the expansions 7); = a;1ti»i - (ku')i-i; 2 = O(h7), valid for the
aforementioned schemes, and 1/J7 = O(n?).
We are now .interested in the question concerning the accuracy of
scheme ( 4) with second-order approximation on an arbitrary non-equidis-
tant grid. A discontinuity point x = ~ is free to be chosen for the relevant
coefficients:
0<8<1.
Being concerned with 1/J7 specified by (7) and involved in the representations
7); =(aux); - (ku')i-1/2,