Resolve the multitude of microscale interactions to model stochastic partial differential equations

Abstract

Constructing numerical models of noisy partial differential equations is very delicate. Our long term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a second step we here consider a small domain, representing a finite element, and apply stochastic centre manifold theory to derive a one degree of freedom model for the dynamics in the element. The approach automatically parametrises the microscale structures induced by spatially varying stochastic noise within the element. The crucial aspect of this work is that we explore how many noise processes may interact in nonlinear dynamics. We see that noise processes with coarse structure across a finite element are the significant noises for the modelling. Further, the nonlinear dynamics abstracts effectively new noise sources over the macroscopic time scales resolved by the model.