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A modified finite volume method incorporating radial basis functions for simulating diffusion

Moroney, Timothy J. & Turner, Ian W. (2005) A modified finite volume method incorporating radial basis functions for simulating diffusion. The ANZIAM Journal, 46(E), pp. 458-471.

Abstract

The finite volume method is the favoured numerical technique for solving (possibly coupled, nonlinear, anisotropic) diffusion equations. The method transforms differential equations into a system of nonlinear, algebraic equations through the process of discretisation. The accuracy of this discretisation determines to a large extent the accuracy of the final solution. We present a new method of discretisation which is designed to achieve high accuracy without imposing excessive computational requirements. The method employs radial basis functions as a means of local gradient interpolation. When combined with high order Gaussian quadrature integration methods, the interpolation based on radial basis functions produces an efficient and accurate discretisation. The resulting nonlinear, algebraic system is solved efficiently using a Jacobian-free Newton--Krylov method. Information obtained from the Newton--Krylov iterations is used to construct an effective preconditioner in order to reduce the number of nonlinear iterations required to achieve an accurate solution. Results to date have been promising, with the method giving accuracy several orders of magnitude better than simpler methods based on shape functions.

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ID Code: 7525
Item Type: Journal Article
Additional Information: The contents of this journal can be freely accessed online via the journal’s web page (see link).
Additional URLs:
ISSN: 1446-1811
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Analysis (010301)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Copyright Owner: Copyright 2005 Australian Mathematical Society
Deposited On: 10 May 2007
Last Modified: 29 Feb 2012 23:12

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