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