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

Roberts, A. J. (2006) Resolving the multitude of microscale interactions accurately models stochastic partial differential equations. LMS Journal of Computation and Mathematics, 9. pp. 193-221.

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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 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 time scales resolved by the model.

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Item Type:	Article (Commonwealth Reporting Category C)
Refereed:	Yes
Item Status:	Live Archive
Faculty / Department / School:	Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited:	11 Oct 2007 00:37
Last Modified:	02 Jul 2013 22:36
Uncontrolled Keywords:	differential equations; numerical modelling; manifold theory
Fields of Research :	01 Mathematical Sciences > 0101 Pure Mathematics > 010109 Ordinary Differential Equations, Difference Equations and Dynamical Systems 01 Mathematical Sciences > 0103 Numerical and Computational Mathematics > 010301 Numerical Analysis 01 Mathematical Sciences > 0104 Statistics > 010406 Stochastic Analysis and Modelling
Socio-Economic Objective:	E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Identification Number or DOI:	10.1112/S146115700000125X
URI:	http://eprints.usq.edu.au/id/eprint/1241

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