Higher-accuracy schemes 211We now introduce a local coordinate system at a point x = X; by merely
setting x = x;+sh, s = (x-x;)/h, in which the segment [x;_ 1 , X;+ 1 ] carries
into the segment (pattern) -1 < s < 1 with the node x = x; corresponding
to the points= 0. Then we concentrate primarily on v;(x) = v~(x;+sh) =
hoJ(s,h) and v;(x) = v;(x; +sh)= hf3i(s,h), -1::; s::; 1, vi(x;) =ha;
and, because of (10), v~(x;) = hai+ 1. Being the pattern functions, CYi(s, h)
and {3i(s, h) are subject to the conditions(16) L CY= -d ( -_^1 - -dCY) - h^2 q(s) _ CY - 0,
ds p(s) ds
-l<s<l,CY(-l,h) = 0, CY^1 ( -1, h) = p( -1) ,
L{3=0, -l<s<l, {3(1,h)=O, {3^1 (l,h)=-p(l),
where p(s) = p(xi +sh) and q(s) = q(xi +sh) depend only on the values
q( s) and f>( s) (p( x) and q( x)) on the segment -1 ::; s ::; 1 (on the segment
xi_ 1 ::; x ::; x;+ 1 ). Omitting subscript i in formula (14) we obtain the
homogeneous conservative scheme for y(x) = u(x), x E wh:(17)where(18)
x E wh, y(O) = u 1 , y(l) = u 2 ,
a(x) = CY(O, h) =A [p(x +sh), q(x +sh)] ,
0
d(x) = a/x) J CY(s, h) q(x +sh) ds
-1
1
+ a(x ~ h) J f3(s, h) q(x +sh) ds,
0
0
lfJ(x) = -)^1 ;· CY(s, h) f(x +sh) ds
a(x
-1
1
+ a(x~h) J f3(s,h)f(x+sh) ds.
0