# Higher order accuracy in the gap-tooth scheme for large-scale dynamics using microscopic simulators

Paper

*CTAC 2004: 12th Biennial Computational Techniques and Applications Conference.*Melbourne, Australia Sep 2004 Cambridge, United Kingdom .

Paper/Presentation Title | Higher order accuracy in the gap-tooth scheme for large-scale dynamics using microscopic simulators |
---|---|

Presentation Type | Paper |

Authors | Roberts, A. J. (Author) and Kevrekidis, I. G. (Author) |

Editors | May, Rob and Roberts, A, J. |

Journal or Proceedings Title | ANZIAM Journal (Australian & New Zealand Industrial and Applied Mathematics Journal) |

Australia and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal | |

Journal Citation | 46 (5), pp. C637-C657 |

Number of Pages | 21 |

Year | 2005 |

Place of Publication | Cambridge, United Kingdom |

ISSN | 1446-1811 |

1446-8735 | |

Web Address (URL) of Paper | http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/981/847 |

Conference/Event | CTAC 2004: 12th Biennial Computational Techniques and Applications Conference |

Event Details | CTAC 2004: 12th Biennial Computational Techniques and Applications Conference Event Date Sep 2004 Event Location Melbourne, Australia |

Abstract | We are developing a framework for multiscale computation which enables models at a 'microscopic' level of description, for example Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators, to perform modelling tasks at the 'macroscopic' length scales of interest. The plan is to use the microscopic rules restricted to small patches of the domain, the 'teeth', followed by interpolation to estimate macroscopic fields in the 'gaps'. The challenge begun here is to find general boundary conditions for the patches of microscopic simulators that appropriately connect the widely separated 'teeth' to achieve high order accuracy over the macroscale. Here we start exploring the issues in the simplest case when the microscopic simulator is the quintessential example of a partial differential equation. In this case analytic solutions provide comparisons. We argue that classic high-order interpolation provides patch boundary conditions which achieve arbitrarily high-order consistency in the gap-tooth scheme, and with care are numerically stable. The high-order consistency is demonstrated on a class of linear partial differential equations in two ways: firstly, using the dynamical systems approach of holistic discretisation; and secondly, through the eigenvalues of selected numerical problems. When applied to patches of microscopic simulations these patch boundary conditions should achieve efficient macroscale simulation. |

Keywords | gap-tooth scheme; microscopic simulators; macroscopic; eigenvalues |

ANZSRC Field of Research 2020 | 490303. Numerical solution of differential and integral equations |

490399. Numerical and computational mathematics not elsewhere classified | |

490105. Dynamical systems in applications | |

Public Notes | File reproduced in accordance with the copyright policy of the publisher/author. |

Byline Affiliations | Department of Mathematics and Computing |

Princeton University, United States |

https://research.usq.edu.au/item/9x886/higher-order-accuracy-in-the-gap-tooth-scheme-for-large-scale-dynamics-using-microscopic-simulators

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