A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions
Article
Roberts, A. J., 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. https://doi.org/10.1007/s10665-013-9653-6
| Article Title | A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions |
|---|---|
| ERA Journal ID | 239 |
| Article Category | Article |
| Authors | Roberts, A. J. (Author), MacKenzie, T. (Author) and Bunder, J. E. (Author) |
| Journal Title | Journal of Engineering Mathematics |
| Journal Citation | 86 (1), pp. 175-207 |
| Number of Pages | 33 |
| Year | 2014 |
| Place of Publication | Dordrecht, Netherlands |
| ISSN | 0022-0833 |
| 1573-2703 | |
| Digital Object Identifier (DOI) | https://doi.org/10.1007/s10665-013-9653-6 |
| Web Address (URL) | http://link.springer.com/article/10.1007%2Fs10665-013-9653-6 |
| 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. |
| Keywords | centre manifolds, computer algebra, discrete modelling closure, gap tooth method, multiscale computation, multiple dimensions, reaction–diffusion equations |
| ANZSRC Field of Research 2020 | 490105. Dynamical systems in applications |
| Public Notes | Files associated with this item cannot be displayed due to copyright restrictions. |
| Byline Affiliations | University of Adelaide |
| Department of Mathematics and Computing | |
| Institution of Origin | University of Southern Queensland |
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