Ho1noge11eous difference schemes for the heat conduction 481This fact follows in1111ediately from a priori esti1nate ( 42):Remark 2 Unifonn convergence with the rate O(lhl^2 + r) of the forward
difference scheme with CJ = 1 can be establi8hed by 1neans of the maximum
principle and the reader is invited to carry out the necessary 1nanipulations
on his/her own.- An one-point heat source. Of special interest is the nonstationary heat
conduction problem in the situation when a heat source is located only at
a single point x = ~ under the agreement that at this point the solution of
problem (1 )-(3) satisfies the condition of conjugation
( 58) [u]=O, [k ~~] =-Q for x=~,
where Q = Q(t) is a power of the heat source.
The discontinuity condition of the heat flowis to be understood as the discontinuity property of the first derivatives
k ~~. That is to say, the solution u. = u(x, t) has a week discontinuity on
the straight line x = ~ by relating at the same time the coefficient k( x) and
the function f(x, t) to be smooth enough.
By means of the integro-interpolation method it is possible to con-
struct a homogeneous difference scheme, whose design reproduces the avail-
ability of the heat source Q of this sort at the point x = ~. This can be
clone using an equidistant grid wh and accepting~ = xn + fJh, 0 < fJ < 0.5.
U ncler such an approach the difference equation takes the standard form
at all the nodes X; f:. xn ( i f:. n). In this line we write clown the balance
equation on the segment a: 11 _ 112 < x < xn+i; 2 for fixed t = t = ij+o 5.
With the aid of the relations
"'n+0.5 ~ :rn+0.5
j (k u')' dx = j (k u')' cLr + j (k u.')' dx(^1) 'n-0.5 X71-IJ 5
= k ·u 'l.rn-05. · + [k cl '] = Wn+l/2 - wn-1/2 - Q ,
Xn+0.5
w = ku' ,