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

Wei Wang; Anthony Roberts

doi:10.3934/dcds.2011.31.253

Article Contents

2011, Volume 31, Issue 1: 253-273. Doi: 10.3934/dcds.2011.31.253

This issue Previous Article Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity Next Article On the ill-posedness result for the BBM equation

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

Wei Wang^1, and
Anthony Roberts^1,

1.
School of Mathematical Sciences, University of Adelaide, South Australia 5005

Received: March 2010

Revised: January 2011

Published: January 2011

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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 rigorous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.

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Mathematics Subject Classification: Primary: 60H15; Secondary: 65M55.

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