II. FINITE DIFFERENCE METHOD 1 Difference formulae

u
x
Figure 8: The implicit method (18) applied to the the diffusion mixed problem
(13). Here D = 1, L = 80, T = 160, J = 40, N = 40. Athough Dk/h 2 > 1/2, the
implicit method is convergent.
numerical solution is obvious. However the convergence of ū to the analytic
solution u as h, k ! 0 is not always true. Indeed, in Figures II.6 and II.7,
with slightly different choice of parameters, we see different behaviours of the
numerical solution. Such a numerical example shows the importance of the
quantity (15). Actually one can prove that the condition
r = Dk/h 2 · 1
2
t
(16)
is a necessary condition for stability of the explicit numerical method. In
practice, it ensures a sufficiently high velocity of propagation of the discrete
signal. As for convergence, it is sufficient to remark that the explicit scheme
is consistent. More precisely, one can prove that, for a sufficiently regular
function, the local discretization error behaves like O(k + h 2 ). This implies
that the the scheme (14) is convergent (since it is stable and consistent) under
the condition (16). It is an example of conditionally convergent scheme.
Unconditionally convergent methods can be obtained using implicit schemes
where a linear system has to be solved at each time level tn. Such a scheme
can be obtained by using a backward difference, instead of a forward difference,
to approximate the derivatie ut. This way the differental equation (13)
20