Stochastic simulations of nonlinear reaction-diffusion equations using an exponential integrator

Abstract

Stochastic simulations can be generated from deterministic reaction-diffusion equations by discretising in space and time and interpreting coefficients in the resulting system of discretised equations as probabilities governing movement and reaction events. In this paper, we present a novel variant of this approach for nonlinear reaction-diffusion equations that employs an exponential integrator when discretising in time. The proposed method yields valid probabilities, defined by the entries of appropriate matrix functions, without the strict conditions on the time step required by a commonly-employed time discretisation scheme. Simulation results presented for one and two dimensional Porous-Fisher type models demonstrate the veracity of the method across several test problems.