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Type: Journal article
Title: Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain
Author: Wang, W.
Roberts, A.
Citation: Discrete and Continuous Dynamical Systems Series A, 2011; 31(1):253-273
Publisher: Southwest Missouri State Univ
Issue Date: 2011
ISSN: 1078-0947
Department: Faculty of Engineering, Computer & Mathematical Sciences
Statement of
Wei Wang, Anthony Roberts
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.
Keywords: Stochastic reaction-diffusion equation
overlapping finite elements
interelement coupling
Rights: Copyright status unknown
DOI: 10.3934/dcds.2011.31.253
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