154 Homogeneous Difference Schemes- Homogeneous conservative schemes. In the preceding section we have
designed the conservative scheme (14) by means of the integro-interpo-
lational method. In the general case the coefficients a, d and <p of scherne
(14) are some functionals of the coefficients k(x), q(x) and f(x) involved in
the differential equation
(16)a(x) = A[k(x +sh)] ,
d(x) = F[q(x +sh)],
'P(:r) = F[f(a: +sh)].
The domain of the pattern functional A[k(s)] is Q(^0 l[-1, 1] (function k(s) E
QC^0 l[-l, 1]), while the domain of F[f(s)] is QC^0 l[-1/2, 1/2]. In other words,
the functional A[k(s)] (or F[f(s)]) is defined for all piecewise continuous
functions k(s) (or f(s)) given on the segment -1 < s < 1 (or -1/2 < s <
1 /2). In trying to recover the coefficient a( x) the intention is to use formula
(16) with k(s) = k(x +sh). This is consistent with the passage from the
pattern -1 < s < 1, on which the function k(s) is defined, to the pattern
x - h < x' < x + h, so that the function k(x') should provide a possibility
of subsequent calculations of a( x).
At the next stage we consider the homogeneous conservative scheme
( 17)(ayx) x -cly=-<p(x),
y(O)=u 1 , y(l) = U 2 , a> c 1 > 0,
for which the coefficients are expressed by formulae (16). Comparison of
conservative schemes (17) or (14) with the three-point scheme of general
form ( 4) from Sect ion 1 shows that scheme ( 17) is governed and constructed
in accordance with the rule b; = Cl;+i. We will not pursue analysis of this:
the ideas needed tci do so have been covered.
The requirement that scheme ( 4) of Section 1 should be conservative
("divergent") is equivalent to being self-adjoint of the appropriate difference
operator. To make sure of it, we refer to a second-order operator
0
in the space <;h of all grid functions y = {y;} defined on the grid wh and
0
vanishing on the boundary: y 0 = YN = 0. The space [h is equipped with