II. FINITE DIFFERENCE METHOD 1 Difference formulae

above consistence and stability, and using the fact that the nonlinear term
f(x, y) is monotonic.
Ex.II.3E (The difference method applied to a partial differential equation
) We already showed the solution of the heat equation on the line ¡1 <
x < +1 by using the convolution properties of the Fourier transform. Let
us show how to solve numerically the heat equation for x in some bouded
interval: the mixed problem (initial value and boundary value problem)
⎧
⎪⎨
⎪⎩
ut ¡ D uxx = 0
u(x, 0) = f(x)
u(0, t) = g1(t), u(0, t) = g2(t), 0 < t < T
(13)
where D is the diffusion coefficient, and f(x), g1(t), g2(t) are given functions.
Let us consider a discretization with nodes (xj, tn) , with xj = jh, tn = nk
and h = L/J, k = T/N. Let us denote by ūj,n the numeric solution in the
node (xj, tn). By the above initial condition
while the boundary conditions imply
ūj,0 = f(xj), j = 0, 1, ..., J
ū0,n = g1(tn), ūJ,n = g2(tn).
For 0 < j < J, n > 0 a possible discretization of (13) in the point (xj, tn) is
got by means of a forward difference for the derivative ut and by a central
difference for the second derivative uxx:
ut » ūj,n+1 ¡ ūj,n
, uxx »
k
ūj+1,n ¡ 2ūj,n + ūj−1,n
h2 By substituting in (13), we obtain the following relation for j = 1, ..., J ¡ 1,
and n = 0, ..., N ¡ 1
where
ūj,n+1 = (1 ¡ 2r)ūj,n + r(ūj+1,n + ūj−1,n
(14)
r = Dk/h 2 . (15)
The method is explicit: in xj the numerical solution ū at time tn+1 is
obtained from ū in the three points xj−1, xj, xj+1 at time tn, by an explicit
formula and not by solving a system. Hence in this case the existence of the
18