Overview

When time-dependent PDEs are solved numerically by spectral methods, the pattern is usually the same: spectral differentiation in space, finite differences in time. For example, one might carry out the time-stepping by an Euler, leap frog, Adams, or Runge{Kutta formula [But87, HaWa96, Lam91]. In principle, one sacrifices spectral accuracy in doing so, but in practice, small time steps with formulas of order 2 or higher often leave the global accuracy quite satisfactory. Small time steps are much more affordable than small space steps, for they affect the computation time, but not the storage, and then only linearly. By contrast, halving the space step typically multiplies the storage by 2 ^d in d space dimensions, and it may multiply the computation time for each time step by anywhere from 2 ^d to 2 ^{3 d}, depending on the linear algebra involved.

So far in this book we have solved three time-dependent PDEs, in each case by a leap frog discretization in t. The equations and the time steps we used were as follows:

Now it is time to explain where these choices of ? t came from.

Figure 10.1 shows the output from Program 6 (p. 26) when the time step is increased to ? t = 1.9 N ^?1, and Figure 10.2 shows the output from Program 20 (p. 83) with ? t = 6.6 N ^?2 Catastrophes! Both computations are numerically unstable in the sense...