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

摘要

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 bytau partial derivative(tt)u +(1- tau f'(u))partial derivative(t)u - mu partial derivative(xx)u = f(u),where tau, mu 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 experiments(1) 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 - tau f'. Moreover, important evidence of the validity of those results beyond the formal hypotheses g(u) 0 for any u is numerically established. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.