Hon1ogeneous difference sche1nes for the heat conduction 483By analogy with the case of discontinuous coefficients we find thatSince an+I = k(O + (0.5 - ti) h k'(~) + O(h^2 ), we might have
\,Yith the aid of the relationswe establish(65) 17n+I = BQ + O(h) = 0(1)
by observing further that(66) 1/J* n = O(h).The accurate account of the error z can be clone as in Section 4,
0
leading to the same rate of convergence. No progress is achieved for a = a
in line with approved rules, because the choice of the coefficient should not
cause the emergence of a higher-order accuracy. From the forn1ula 77n+I =
g Q+O(h) it is easily seen that 1)n+l = O(h) and, hence, II z II= O(h^2 +rm~)
if g = 0, n1eaning that the heat source is located at one of the nodal points.
This guides a proper choice of the nonequidistant grid wh(Q) so that
the heat source will appear at one of the nodal points. When this is the
case, schen1e (60) converges uniforn1ly, on the sa1ne grounds as before, with
the rate O(lhl^2 +rm~). But a special choice of the coefficients ai given
by the formulas of the truncated scheme with second-order accuracy (see
Chapter 3, Section 7) improves our chances of constructing the difference
schen1e of accuracy-O(lhl^2 +rm~) for any g E [O, 1], that is, disregarding to
the possible locations of the heat source.
- A concentrated heat capacity. We now consider the boundary-value
problem for the heat conduction equation with some unusual condition
placing the concentrated heat capacity Co on the boundary, say at a single
point x = 0. The traditional way of covering this is to impose at the point
x = 0 an unusual boundary condition such as
(67) (,' ~ 0 01-l ""t = k^811 •) ' '
u. OJ. x = 0'
Co = const > 0 ,