Sharp convergence to steady states of Allen-Cahn

TL;DR

This paper investigates the convergence behavior of solutions to the Allen-Cahn equation, introduces a Fourier filtering technique to improve numerical accuracy, and classifies steady states with explicit convergence rates.

Contribution

It provides a comprehensive classification of steady states, introduces a novel Fourier filter method, and establishes sharp convergence results with explicit profiles and rates.

Findings

Numerical solutions can converge to incorrect steady states without filtering.

A new Fourier filter technique prevents symmetry-breaking errors.

Explicit convergence rates and profiles for steady states are established.

Abstract

In our recent work we found a surprising breakdown of symmetry conservation: using standard numerical discretization with very high precision the computed numerical solutions corresponding to very nice initial data may converge to completely incorrect steady states due to the gradual accumulation of machine round-off error. We solved this issue by introducing a new Fourier filter technique for solutions with certain band gap properties. To further investigate the attracting basin of steady states we classify in this work all possible bounded nontrivial steady states for the Allen-Cahn equation. We characterize sharp dependence of nontrivial steady states on the diffusion coefficient and prove strict monotonicity of the associated energy. In particular, we establish a certain self-replicating property amongst the hierarchy of steady states and give a full classification of their energies and profiles. We develop a new modulation theory and prove sharp convergence to the steady state with explicit rates and profiles.