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A holistic finite difference approach models linear dynamics consistently

Author: A. J. Roberts
Journal: Math. Comp. 72 (2003), 247-262
MSC (2000): Primary 37L65, 65M20, 37L10, 65P40, 37M99
Published electronically: June 4, 2002
MathSciNet review: 1933820
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Abstract: I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of finite difference models under nonlinear and other perturbations on grids with finite spacing. For example, the linear advection-diffusion equation is found to be stably modelled for all advection speeds and all grid spacings. The theorems establish an extremely good form for the artificial internal boundary conditions that need to be introduced to apply centre manifold theory. When numerically solving nonlinear partial differential equations, this approach can be used to systematically derive finite difference models which automatically have excellent characteristics. Their good performance for finite grid spacing implies that fewer grid points may be used and consequently there will be less difficulties with stiff rapidly decaying modes in continuum problems.

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Additional Information

A. J. Roberts
Affiliation: Department of Mathematics and Computing, University of Southern Queensland, Toowoomba, Queensland 4352, Australia

Received by editor(s): April 6, 2000
Received by editor(s) in revised form: November 14, 2000
Published electronically: June 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society