# Accurate macroscale modelling of spatial dynamics in multiple dimensions

Technical report

*Accurate macroscale modelling of spatial dynamics in multiple dimensions.*Adelaide, Australia. University of Adelaide.

Title | Accurate macroscale modelling of spatial dynamics in multiple dimensions |
---|---|

Report Type | Technical report |

Authors | Roberts, A. J. (Author), MacKenzie, Tony (Author) and Bunder, J. E. (Author) |

Institution of Origin | University of Adelaide |

Number of Pages | 58 |

Year | 2012 |

Publisher | University of Adelaide |

Place of Publication | Adelaide, Australia |

Web Address (URL) | http://arxiv.org/abs/1103.1187 |

Abstract | Developments in dynamical systems theory provides 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 modelling of systems by a type of finite elements, 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 exist on the macroscale grid, is emergent, and is systematically approximated. Dividing space either into overlapping finite elements or into 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 | dynamical systems theory; mathematical modelling |

ANZSRC Field of Research 2020 | 490199. Applied mathematics not elsewhere classified |

490302. Numerical analysis | |

490409. Ordinary differential equations, difference equations and dynamical systems | |

Public Notes | Files associated with this item cannot be displayed due to copyright restrictions. |

Byline Affiliations | University of Adelaide |

Department of Mathematics and Computing |

https://research.usq.edu.au/item/q1639/accurate-macroscale-modelling-of-spatial-dynamics-in-multiple-dimensions

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