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