A collocation method based on one-dimensional RBF interpolation scheme for solving PDEs

Mai-Duy, N. and Tanner, R. I. (2007) A collocation method based on one-dimensional RBF interpolation scheme for solving PDEs. International Journal of Numerical Methods for Heat and Fluid Flow, 17 (2). pp. 165-186. ISSN 0961-5539

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Abstract

Purpose -- To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains. Design/methodology/approach -- The proposed method is based on a Cartesian grid and a one-dimensional integrated-radial-basis-function (1D-IRBF) scheme. The employment of integration to construct the RBF approximations representing the field variables facilitates a fast convergence rate, while the use of a 1D interpolation scheme leads to considerable economy in forming the system matrix and improvement in the condition number of RBF matrices over a 2D interpolation scheme. Findings -- The proposed method is verified by considering several test problems governed by second- and fourth-order PDEs; very accurate solutions are achieved using relatively coarse grids. Research limitations/implications -- Only 1D and 2D formulations are presented, but we believe that extension to 3D problems can be carried out straightforwardly. Further development is needed for the case of non-rectangular domains. Originality/value -- The contribution of this paper is a new effective collocation formulation based on RBFs for solving PDEs.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Deposited in accordance with the copyright policy of the publisher.
Depositing User: Dr Nam Mai-Duy
Faculty / Department / School: Historic - Faculty of Engineering and Surveying - Department of Mechanical and Mechatronic Engineering
Date Deposited: 10 Feb 2008 23:31
Last Modified: 02 Jul 2013 22:54
Uncontrolled Keywords: integrated radial basis function; collocation method; Cartesian grid; multiple boundary conditions
Fields of Research (FOR2008): 01 Mathematical Sciences > 0103 Numerical and Computational Mathematics > 010302 Numerical Solution of Differential and Integral Equations
09 Engineering > 0915 Interdisciplinary Engineering > 091501 Computational Fluid Dynamics
09 Engineering > 0913 Mechanical Engineering > 091307 Numerical Modelling and Mechanical Characterisation
Socio-Economic Objective (SEO2008): E Expanding Knowledge > 97 Expanding Knowledge > 970109 Expanding Knowledge in Engineering
Identification Number or DOI: doi: 10.1108/09615530710723948
URI: http://eprints.usq.edu.au/id/eprint/3568

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