346 Difference Schen1es with Constant Coefficientswhere y = yJ, y = yj-i, r = pr/h^2 and 6. = 4ph-^2 Also, we point out
here without proving that scheme (13) is stable for any T and h. This fact
follows from the general stability theory developed in Chapter 6. Also, this
scheme generates conditional approximation, since its residual behaves for
T = O(h^2 ) like
p T2
1/J=Au+VJ-u 1 - h 2 urt=O(h^2 )if we accept 'P = f and h 1 = h 2 = · · · = hp = h.
- Sche1nes with weights. VVhen discretizing equation (2) int, the scheme
with weights arises natnrally in one or another form:
or
xEwh, i=jr>O,
(14)
y(x, 0) = '1l 0 (x),In preparation for this, we agree to consider 'P = J = f(x, tj+i; 2 ). As
before, we suppose once again that G is a parallelepiped and A is specified
by formula ( 4). We investigate the order of approximation by appeal to the
express10n
O" U + ( 1 - O") U = u+u ( 1)
2- O" - 2 T Ut
for O" u + (1 - O") 1l and, after this, touch upon the residual
= A u ; '1l + ( ()" - ~ ) T A '1lt + 'P - ll t
= Lu+ (.()" - ~)TL u + J - ii.+ ( 'P - f) + 0( I h 12 + r^2 )
where Lu = 6. u, u = u(x, tj+i; 2 ), 'P = f and u = 8u/8t. It is therefore
cone! uded that