An efficient indirect RBFN-based method for numerical solution of PDEs

Mai-Duy, Nam and Tran-Cong, Thanh (2005) An efficient indirect RBFN-based method for numerical solution of PDEs. Numerical Methods for Partial Differential Equations, 21 (4). pp. 770-790. ISSN 0749-159X

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Abstract

This article presents an efficient indirect radial basis function network (RBFN) method for numerical solution of partial differential equations (PDEs). Previous findings showed that the RBFN method based on an integration process (IRBFN) is superior to the one based on a differentiation process (DRBFN) in terms of solution accuracy and convergence rate (Mai-Duy and Tran-Cong, Neural Networks 14(2) 2001, 185). However, when the problem dimensionality N is greater than 1, the size of the system of equations obtained in the former is about N times as big as that in the latter. In this article, prior conversions of the multiple spaces of network weights into the single space of function values are introduced in the IRBFN approach, thereby keeping the system matrix size small and comparable to that associated with the DRBFN approach. Furthermore, the nonlinear systems of equations obtained are solved with the use of trust region methods. The present approach yields very good results using relatively low numbers of data points. For example, in the simulation of driven cavity viscous flows, a high Reynolds number of 3200 is achieved using only 51 × 51 data points.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Author's Orignal deposited in accordance with the copyright policy of the publisher.
Depositing User: epEditor USQ
Faculty / Department / School: Historic - Faculty of Engineering and Surveying - Department of Mechanical and Mechatronic Engineering
Date Deposited: 11 Oct 2007 01:18
Last Modified: 02 Jul 2013 22:48
Uncontrolled Keywords: radial basis function; trust region methods; Poisson equation; Navier-Stokes equation; driven cavity viscous flow
Fields of Research (FOR2008): 09 Engineering > 0913 Mechanical Engineering > 091399 Mechanical Engineering not elsewhere classified
09 Engineering > 0915 Interdisciplinary Engineering > 091508 Turbulent Flows
01 Mathematical Sciences > 0101 Pure Mathematics > 010110 Partial Differential Equations
Socio-Economic Objective (SEO2008): E Expanding Knowledge > 97 Expanding Knowledge > 970109 Expanding Knowledge in Engineering
Identification Number or DOI: doi: 10.1002/num.20062
URI: http://eprints.usq.edu.au/id/eprint/2981

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