A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics (Q1001669)

The goal of the present paper is to study an approximation scheme for a reaction-diffusion equation with finite delay. This model is used in order to describe the evolution of a population with density distribution in such a way that the resulting finite dimensional ordinary differential system contains the same asymptotic dynamics as the reaction-diffusion equation. The difficulty in proposing an appropriate scheme lies in preserving the asymptotic dynamics. The system has two competing infinite dimensional behaviours, the first comming from diffusion and the other one from the delay effects. The authors propose an approximate scheme based on an operator splitting approach. First, they do a time delay discretization and show that the discrete solution converges to the continuous one. Afterwards, they perform the approximation using spectral projections and do the spatial discretization. Finally, the convergence of the numerical scheme is shown.