Higher-accuracy schemes^213One trick we have encountered is to take only a finite number of the mem-
bers in series ( 19):m m
oJm)(s, h) = L cxk(s) h^2 k, f3(m)(s, h) = L {3k(s) h2k,
k=O k=Omaking it possible to determine the coefficients a(m), d(m) and tp(m) by
formulae (18), where the polynomials cx(m) and f3(m) stand for ex and {3. As
a final result we get a scheme of accuracy O(h^2 m+^2 ) in the class of piecewise
continuous functions k(x), q(x), f(x) E Q(^0 l[O, l]. Any such scheme is called
a truncated scheme of rank m.
Form= 0 it refers to schemes of zero rank and accuracy O(h^2 ) with
regard to k, q, f E Q(o). When providing current manipulations, the
expressions for d and tp involved in this scheme0
a(o) = K^1 = ;· p( x + sh) ds,
-1d(O) = d + (hd.).r,
0 s
d - ~ j ds ;· q(x +th) di,
* - k(x+sh)
-1 -1/20 s
o ;· ds ;·
tp* =a k(x+sh) f(x +th) di- 1 - 1 /2
are different from those being used in the best scheme (14)-(15) of Section 2.
Truncated schemes"provide a possibility to attain any order of accuracy for
arbitrary piecewise continuous functions k(x), q(x) and f(x) and appear to
be useful in many aspects.
An exact scheme and truncated schemes can be designed on an arbi-
trary non-equidistant grid wh by the same methods as we employed before.
In practice the use of truncated schemes in the case of equation (1)
with variable coefficients necessitates carrying out calculations of multiple
integrals on each interval of the grid. Replacing those integrals by finite
sums we are able to create more simpler schemes of accuracy 0( h^4 ) and
O(h^6 ), whose coefficients can be expressed through the values of k, q and f