A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions

Roberts, A. J. and MacKenzie, T. and Bunder, J. E. (2014) A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions. Journal of Engineering Mathematics, 86 (1). pp. 175-207. ISSN 0022-0833

Abstract

Developments in dynamical systems theory provide new support for the macroscale modelling of pdes and other microscale systems such as lattice Boltzmann, Monte Carlo or molecular dynamics simulators. By systematically resolving subgrid microscale dynamics the dynamical systems approach constructs accurate closures of macroscale discretisations of the microscale system. Here we specifically explore reaction–diffusion problems in two spatial dimensions as a prototype of generic systems in multiple dimensions. Our approach unifies into one the discrete modelling of systems governed by known pdes and the ‘equation-free’ macroscale modelling of microscale simulators efficiently executing only on small patches of the spatial domain. Centre manifold theory ensures that a closed model exists on the macroscale grid, is emergent, and is systematically approximated. Dividing space into either overlapping finite elements or spatially separated small patches, the specially crafted inter-element/patch coupling also ensures that the constructed discretisations are consistent with the microscale system/pde to as high an order as desired. Computer algebra handles the considerable algebraic details, as seen in the specific application to the Ginzburg–Landau pde. However, higher-order models in multiple dimensions require a mixed numerical and algebraic approach that is also developed. The modelling here may be straightforwardly adapted to a wide class of reaction–diffusion pdes and lattice equations in multiple space dimensions. When applied to patches of microscopic simulations our coupling conditions promise efficient macroscale simulation.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: © 2013 Springer Science+Business Media. Permanent restricted access to published version due to publisher copyright policy.
Faculty / Department / School: Current - Faculty of Health, Engineering and Sciences - School of Agricultural, Computational and Environmental Sciences
Date Deposited: 27 Mar 2017 04:23
Last Modified: 27 Mar 2017 04:38
Uncontrolled Keywords: centre manifolds, computer algebra, discrete modelling closure, gap tooth method, multiscale computation, multiple dimensions, reaction–diffusion equations
Fields of Research : 01 Mathematical Sciences > 0102 Applied Mathematics > 010204 Dynamical Systems in Applications
Socio-Economic Objective: E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Identification Number or DOI: 10.1007/s10665-013-9653-6
URI: http://eprints.usq.edu.au/id/eprint/27228

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