The Explicit-Implicit-Null method: Removing the numerical instability of PDEs

Authors:
Laurent Duchemin;Jens Eggers
Affiliations:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, F-13384 Marseille, France;Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom
Venue:
Journal of Computational Physics
Year:
2014

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Abstract

A general method to remove the numerical instability of partial differential equations is presented. Two equal terms are added to and subtracted from the right-hand side of the PDE: the first is a damping term and is treated implicitly, the second is treated explicitly. A criterion for absolute stability is found and the scheme is shown to be convergent. The method is applied with success to the mean curvature flow equation, the Kuramoto-Sivashinsky equation, and to the Rayleigh-Taylor instability in a Hele-Shaw cell, including the effect of surface tension.