Splitting schemes for FitzHugh--Nagumo stochastic partial differential equations

TL;DR

This paper develops and analyzes splitting integrators for the stochastic FitzHugh--Nagumo PDE system, demonstrating their convergence and effectiveness in simulating nerve signal propagation under stochastic influences.

Contribution

The paper introduces novel splitting schemes for the stochastic FitzHugh--Nagumo system and proves their strong convergence rate of 1/4, supported by numerical experiments.

Findings

Numerical solutions have finite moments.

Splitting schemes achieve strong convergence rate of 1/4.

Numerical experiments confirm scheme performance.

Abstract

We design and study splitting integrators for the temporal discretization of the stochastic FitzHugh--Nagumo system. This system is a model for signal propagation in nerve cells where the voltage variable is solution of a one-dimensional parabolic PDE with a cubic nonlinearity driven by additive space-time white noise. We first show that the numerical solutions have finite moments. We then prove that the splitting schemes have, at least, the strong rate of convergence $1/4$. Finally, numerical experiments illustrating the performance of the splitting schemes are provided.