Kinetic schemes for assessing stability of traveling fronts for the Allen-Cahn equation with relaxation

TL;DR

This paper develops finite volume numerical schemes for reaction-diffusion systems with relaxation, specifically analyzing the stability of traveling fronts in the hyperbolic Allen-Cahn equation through kinetic interpretation and numerical experiments.

Contribution

It introduces discretizations based on kinetic interpretation for hyperbolic Allen-Cahn equations and validates stability results of traveling waves numerically.

Findings

Numerical schemes effectively approximate the hyperbolic Allen-Cahn equation.

Numerical experiments confirm theoretical stability results.

Results extend validity beyond initial hypotheses.

Abstract

This paper deals with the numerical (finite volume) approximation of reaction-diffusion systems with relaxation, among which the hyperbolic extension of the Allen--Cahn equation represents a notable prototype. Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump process. Numerical experiments are provided for exemplifying the theoretical analysis (previously developed by the same authors) concerning the stability of traveling waves, and important evidence of the validity of those results beyond the formal hypotheses is numerically established.