Leapfrog equations

The leapfrog method of finite differencing, it will be recalled, requires expressing the time derivative over two time steps. This keeps the centers of the differencing operators in the same place. For the heat-flow equation Fourier-transformed over space,  

$${ q_{t+1} \ -\ q_{t-1} \over 2 \, \Delta t } \eq - \ { \sigma \over c }\ k^2 \ q_t$$ (78)

It is a bit of a nuisance to analyze this equation because it covers times t-1, t, and t+1 and requires slightly more difficult analytical techniques. Therefore, it seems worthwhile to state the results first. The result for the heat-flow equation is that the solution always diverges. The result for the wave-extrapolation equation is much more useful: there is stability provided certain mesh-size restrictions are satisfied, namely, $\Delta z$ must be less than some factor times $\Delta x^2$.This result is not exciting in one space dimension (where implicit methods seem ideal), but in higher-dimensional space, such as in the so-called 3-D prospecting surveys, we may be thankful to have the leapfrog method.

The best way to analyze equations like (78) which range over three or more time levels is to use Z-transform filter analysis. Converted to a Z-transform filter problem, the question posed by (78) becomes whether the filter has zeroes inside (or outside) the unit circle. Z-transform stability analysis is described elswhere Such analysis is necessary for all possible numerical values of k². Its result is that there is always trouble if k² ranges from zero to infinity. But with the wave-extrapolation equation, instability can be avoided with certain mesh-size restrictions, because $( \hat k \, \Delta x )^2$ lies between zero and four.