A general method to remove the stiffness of PDEs

TL;DR

This paper introduces a novel numerical method that reduces PDE stiffness by adding damping and anti-damping terms, enhancing stability and convergence across various complex equations.

Contribution

The paper presents a general, stable, and convergent scheme for PDEs by combining implicit damping and explicit anti-damping terms, applicable to multiple complex equations.

Findings

Achieved absolute stability with the new scheme

Successfully applied to mean curvature flow and Kuramoto-Sivashinsky equations

Demonstrated effectiveness in Rayleigh-Taylor instability with surface tension

Abstract

A new method to remove the stiffness of partial differential equations is presented. Two terms are added to the right-hand-side of the PDE : the first is a damping term and is treated implicitly, the second is of the opposite sign and 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.