Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain

TL;DR

This paper develops a rigorous macroscopic discrete model for stochastic reaction-diffusion equations, capturing the interplay of noise and spatial diffusion through a novel coupling approach and averaging techniques.

Contribution

It introduces a new method for deriving semi-discrete stochastic models from reaction-diffusion PDEs using finite element coupling and asymptotic analysis.

Findings

The macroscopic model accurately captures the dynamics of the original PDEs.

Subgrid scale interactions between noise and diffusion are crucial for model accuracy.

The approach provides a rigorous framework for semi-discrete approximation of stochastic PDEs.

Abstract

Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptotic expansion of the slow modes show that the macroscopic discrete model will be a family of coupled stochastic ordinary differential equations which describe the evolution of the grid values. This modelling shows the importance of subgrid scale interaction between noise and spatial diffusion and provides a new rigourous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.