General tooth boundary conditions for equation free modelling
Technical report
Title | General tooth boundary conditions for equation free modelling |
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Report Type | Technical report |
Authors | Roberts, A. J. (Author) and Kevrekidis, I. G. (Author) |
Institution of Origin | University of Southern Queensland |
Number of Pages | 22 |
Year | 2006 |
Publisher | University of Southern Queensland |
Place of Publication | Toowoomba, Australia |
Web Address (URL) | http://arxiv.org/abs/math.DS/0603433 |
Abstract | [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 'macroscopic' length scales of interest. The plan is to use the microscopic rules restricted to small 'patches' of the domain, the 'teeth', using interpolation to bridge the 'gaps'. Here we explore general boundary conditions coupling the widely separated 'teeth' of the microscopic simulation that achieve high order accuracy over the macroscale. We present the simplest case when the microscopic simulator is the quintessential example of a partial differential equation. We argue that classic high-order interpolation of the macroscopic field provides the correct forcing in whatever boundary condition is required by the microsimulator. Such interpolation leads to Tooth Boundary Conditions which achieve arbitrarily high-order consistency. The high-order consistency is demonstrated on a class of linear partial differential equations in two ways: firstly through the eigenvalues of the scheme for selected numerical problems; and secondly using the dynamical systems approach of holistic discretisation on a general class of linear PDEs. Analytic modelling shows that, for a wide class of microscopic systems, the subgrid fields and the effective macroscopic model are largely independent of the tooth size and the particular tooth boundary conditions. When applied to patches of microscopic simulations these tooth boundary conditions promise efficient macroscale simulation. We expect the same approach will also accurately couple patch simulations in higher spatial dimensions. |
Keywords | multiscale computation, time scales, gap-tooth methodology, coupling boundary conditions, high order consistency |
ANZSRC Field of Research 2020 | 490399. Numerical and computational mathematics not elsewhere classified |
460207. Modelling and simulation | |
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/9xv23/general-tooth-boundary-conditions-for-equation-free-modelling
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