On symmetry breaking of Allen-Cahn

TL;DR

This paper investigates the unexpected convergence issues in numerical solutions of the Allen-Cahn equation, introduces a symmetry-preserving filter to correct this, and provides a rigorous theoretical framework for the improved convergence.

Contribution

It presents a novel symmetry-preserving filter technique and a new theoretical framework to ensure correct steady state convergence in numerical Allen-Cahn solutions.

Findings

High-precision discretizations can lead to incorrect steady states.

The proposed filter restores correct symmetry and convergence.

Theoretical proof of convergence for filtered solutions.

Abstract

We consider numerical solutions for the Allen-Cahn equation with standard double well potential and periodic boundary conditions. Surprisingly it is found that using standard numerical discretizations with high precision computational solutions may converge to completely incorrect steady states. This happens for very smooth initial data and state-of-the-art algorithms. We analyze this phenomenon and showcase the resolution of this problem by a new symmetry-preserving filter technique. We develop a new theoretical framework and rigorously prove the convergence to steady states for the filtered solutions.