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

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 – given byτ∂ttu+(1−τf′(u))∂tu−μ∂xxu=f(u), where τ,μ>0 and f is a cubic-like function with three zeros – represents a notable prototype.Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump process, given by the substitution of the Fourier's law with the Maxwell–Cattaneo's law. A corresponding pseudo-kinetic scheme is also proposed for the Guyer–Krumhansl's law.For the Maxwell–Cattaneo case, numerical experiments1 are provided for exemplifying the theoretical analysis (previously developed by the same authors) concerning the stability of traveling waves under a sign condition on the damping term g:=1−τf′. Moreover, important evidence of the validity of those results beyond the formal hypotheses g(u)>0 for any u is numerically established.