Why large time-stepping methods for the Cahn-Hilliard equation is stable

TL;DR

This paper develops a new stability theory for large time-stepping methods applied to the Cahn-Hilliard equation, explaining observed stability phenomena in semi-implicit schemes with stabilization.

Contribution

It introduces a novel stability analysis that accounts for the stability of semi-implicit methods with stabilization in the small dissipation regime.

Findings

Semi-implicit methods exhibit remarkable stability with large time steps.

Stability persists even with a stabilization parameter of order one.

The new theory explains the empirical stability observed in practice.

Abstract

We consider the Cahn-Hilliard equation with standard double-well potential. We employ a prototypical class of first order in time semi-implicit methods with implicit treatment of the linear dissipation term and explicit extrapolation of the nonlinear term. When the dissipation coefficient is held small, a conventional wisdom is to add a judiciously chosen stabilization term in order to afford relatively large time stepping and speed up the simulation. In practical numerical implementations it has been long observed that the resulting system exhibits remarkable stability properties in the regime where the stabilization parameter is $\mathcal O(1)$, the dissipation coefficient is vanishingly small and the size of the time step is moderately large. In this work we develop a new stability theory to address this perplexing phenomenon.