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

Article

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. https://doi.org/10.1112/S146115700000125X
Article Title

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

ERA Journal ID32385
Article CategoryArticle
Authors
AuthorRoberts, A. J.
Journal TitleLMS Journal of Computation and Mathematics
Journal Citation9, pp. 193-221
Number of Pages29
Year2006
Place of PublicationLondon, United Kingdom
ISSN1461-1570
Digital Object Identifier (DOI)https://doi.org/10.1112/S146115700000125X
Web Address (URL)http://www.lms.ac.uk/jcm/9/lms2005-032/sub/lms2005-032.pdf
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.

Keywordsdifferential equations; numerical modelling; manifold theory
ANZSRC Field of Research 2020490409. Ordinary differential equations, difference equations and dynamical systems
490302. Numerical analysis
490510. Stochastic analysis and modelling
Byline AffiliationsDepartment of Mathematics and Computing
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