The Mechanism by Which Many Partial Difference Equations Destabilize

Abstract

The motivation for this work was to attempt to understand the reason that certain classes of explicit numerical algorithms for computing solutions to partial differential equations develop intense local patches of instability after long times, even when all the linear stability criteria are satisfied. Now, numerical analysts are the academic world’s greatest plumbers, and so practitioners of the art of numerical computation have invented many ingenious schemes to circumvent these problems (see, for example [1]). However, beyond declaring that the instabilities are nonlinear in character and beyond a few careful analyses of the problem (for example [2], [3]), the numerical analyst has not investigated in any great detail the nature of the breakdown.