Conservative schemes 157convenience in analysis, it is supposed that F[,f(s)] is a linear nonnegative
functional such that
(1) F [c 1 f 1 (s) + c 2 f 2 (s)] = c 1 F [f 1 (s)] + c 2 F [f 2 (s)], -1/2 < s < 1/2,
where c 1 and c 2 are arbitrary constants;
(2) F [f(s)] > 0 for ,f(s) > 0.
In spite of the fa.ct that A [k( s)] is usually a nonlinear functional (see scheme
( 151 )), we may assun1e for the sake of simplicity that A [ k ( s)] is a linear
nonnegative functional and consider, in addition to schemes (16)-(17), those
with the coefficient a( x) still subject to the relationship (cf. ( 151 ))( l 61 ) _1 a(x) -A[ - k(x+sh)^1 l ·
Conditions for the second-order approximation (18) imply some restrictions
on the pattern functionals A[k(s)] and F[.f(s)]. In preparation for this,
plain calculations give<p(x)=F[f(x+sh)] =F[f(x)+shf'(x)+O(h^2 )]
= f(x) F[l] + h J'(x) F[s] + O(h^2 ).
Since J(x) is taken arbitrarily, it follows from the foregoing and (18) that
(20) F[l]=l, F[s]=O.
By the sarne token,a(x) =A [ k(x) +sh k'(x) +! s^2 h^2 k"(x) + O(h^3 )]
= k(x) A[l] + h k'(x) A[s] + ~ h^2 k"(x) A[s^2 ] + O(h^3 ),
a(x + h) ,= k(x) A[l] + h k'(x) A[l + s]
+! h^2 k^11 (x) A[(l + s)^2 ] + O(h^3 ),
a(x + h) - a(x) = (A[l + s] - A[sl)k'(x)
h+! h (A [ (1 + s )^2 ] - A[ s^2 ]) k'' ( x) + 0( h^2 ),
a(x + h~ + a(x) = A[l] k(x) +! h (A[l + s] + A[sl) k'(x) + O(h^2 ).