Resolving the multitude of microscale interactions accurately models stochastic partial differential equations

Abstract: Constructing numerical models of noisy partial differential equations is a very delicate task. 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 consider here a small domain, representing a finite element, and derive a one-degree-of-freedom model for the dynamics in the element; stochastic centre manifold theory supports the model. 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 a multitude of microscale noise processes may interact in nonlinear dynamical systems. The analysis finds 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 macroscale timescales resolved by the model.