Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces

TL;DR

This paper analyzes a convex-concave splitting scheme for the Allen-Cahn equation, showing that the effective time step scales with the square of the interface width parameter, leading to slow interface movement and stability.

Contribution

It provides a comprehensive analysis of the scheme's stability and slow motion behavior across various potential types, establishing a new connection to mean curvature flow approximation.

Findings

Effective time step scales with ^2, causing slow interface movement.

Stability achieved by 'freezing' interfaces, not geometric constraints.

New link established between Allen-Cahn equations and thresholding mean curvature flow.

Abstract

The convex-concave splitting discretization of the Allen-Cahn is easy to implement and guaranteed to be energy decreasing even for large time-steps. We analyze the time-stepping scheme for a large class of potentials which includes the standard potential as well as two extreme settings: Potentials with quadratic convex part (uniform positive curvature), and potentials which are concave between the potential wells and either linear or infinite outside (highly concentrated curvature). In all three scenarios, the 'effective time step size' of the scheme scales with the square of the small parameter $\varepsilon$ governing the width of transition layers. A weaker 'slow motion' result is proved under much more general assumptions. Thus, stability is achieved by effectively 'freezing' the interfaces in place. The time step limitation is not geometric in origin, but depends on the phase-field parameter $\varepsilon$. Along the way, we establish a new link between an Allen-Cahn type equation and a thresholding approximation of mean curvature flow.