General tooth boundary conditions for equation free modelling

Roberts, A. J. and Kevrekidis, I. G. (2006) General tooth boundary conditions for equation free modelling. Technical Report. University of Southern Queensland, Faculty of Sciences, Department of Maths and Computing, Toowoomba, Australia. (Unpublished)


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[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.

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Item Type: Report (Technical Report)
Item Status: Live Archive
Additional Information: USQ publication.
Depositing User: epEditor USQ
Faculty / Department / School: Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited: 11 Oct 2007 00:30
Last Modified: 02 Jul 2013 22:35
Uncontrolled Keywords: multiscale computation, time scales, gap-tooth methodology, coupling boundary conditions, high order consistency
Fields of Research (FOR2008): 01 Mathematical Sciences > 0103 Numerical and Computational Mathematics > 010399 Numerical and Computational Mathematics not elsewhere classified
08 Information and Computing Sciences > 0801 Artificial Intelligence and Image Processing > 080110 Simulation and Modelling

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