Generic Numerical Analysis of Stochastic Reaction Diffusion Model with applications in excitable media

TL;DR

This paper develops a unified numerical framework for stochastic reaction-diffusion models, proving convergence and analyzing noise effects on wave dynamics in excitable media.

Contribution

It introduces the gradient discretisation method (GDM) for stochastic models and applies a hybrid mixed mimetic approach to study noise impact on wave propagation.

Findings

Convergence of numerical schemes under natural data assumptions

High noise can cause wave backfire or propagation failure

Application to excitable media demonstrates noise effects on wave dynamics

Abstract

The stochastic reaction-diffusion model driven by a multiplicative noise is examined. We construct the gradient discretisation method (GDM), an abstract framework combining several numerical method families. The paper provides the discretisation and proves the convergence of the approximate schemes using a compactness argument that works under natural assumptions on data. We also investigate, using a finite volume method, known as the hybrid mixed mimetic (HMM) approach, the effects of multiplicative noise on the dynamics of the travelling waves in the excitable media displayed by the model. Particularly, we consider how sufficiently high noise can cause waves to backfire or fail to propagate.