Abstract:
Explicit difference approximations of parabolic initial boundary value problems are usually stable only if a difference grid with a limited time-step is used. By considering the one-dimensional diffusion equation as an example, it is shown in the following work that simple smoothing formulas can be constructed which, when applied to solutions computed with unstable explicit difference equations, result in stable approximations of the solution of the differential equation. Such computational procedures can be expressed as explicit difference analogues of the problem considered. Conversely, explicit difference approximations, which need not be defined for all points of the difference grid but must be stable for the specific grid used, can be written as non-unique combinations of an explicit difference approximation, which need not be stable, and a smoothing formula. By appropriate choice of these explicit difference approximations and smoothing formulas this procedure will be defined for all grid points. This new technique thus has the advantage that explicit difference approximations with comparatively weak stability requirements and/or small truncation errors can be used in practice.

Reference:
Joubert, G. 1969. Explicit approximation methods for initial-value problems. University of Cape Town.