Finite Amplitude Instabilities of Partial Difference Equations

A theory is developed to account for the effect of finite amplitude perturbations on the stability of partial difference equations. A simple criterion is derived for determining whether finite amplitude perturbations can cause instabilities in the difference equation for values of the time step parameter below that predicted by linear theory. We find a fundamental difference between the nature of the instability for difference equations in which the time derivative is modeled as (a) a forward difference (diffusion-like) and (b) a central difference (wave-like). In the former, the critical amplitude threshold is achieved by external influences (initial conditions, forcing, etc.). In the latter, the instability is achieved in two stages. First, the critical amplitude threshold is attained dynamically, by a gradual focusing mechanism in which a continuous band of neutrally stable modes in the neighborhood of the most critical plays a central role. In this mechanism, the initial amplitude and the grid length determine the length of time for the instability to materialize. Once the critical amplitude is realized, the instability explodes rapidly in an exponential manner. The theoretical predictions are found to be in close agreement with the results of numerical experiments. Finally, the notion of global stability and integrability are discussed.