A holistic finite difference approach models linear dynamics consistently

Roberts, A. J. (2003) A holistic finite difference approach models linear dynamics consistently. Mathematics of Computation, 72 (241). pp. 247-262. ISSN 0025-5718


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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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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Awaiting Author's version. Deposited in accordance with the copyright policy of the publisher.
Faculty / Department / School: Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited: 11 Oct 2007 01:18
Last Modified: 02 Jul 2013 22:48
Uncontrolled Keywords: centre manifold approach; finite difference models; linear dynamics
Fields of Research : 01 Mathematical Sciences > 0101 Pure Mathematics > 010109 Ordinary Differential Equations, Difference Equations and Dynamical Systems
Identification Number or DOI: 10.1090/S0025-5718-02-01448-5
URI: http://eprints.usq.edu.au/id/eprint/2980

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