Stable integrated RBF calculation using preconditioning and high-order compact approximation for second-order PDEs

Tien, C. M. T. and Mai-Duy, N. and Ngo-Cong, D. and Tran, C. D. and Tran-Cong, T. (2015) Stable integrated RBF calculation using preconditioning and high-order compact approximation for second-order PDEs. In: Eleventh International Conference on Computational Fluid Dynamics in the Minerals and Process Industries 2015 , 7-9 Dec 2015 , Melbourne, Victoria.

Abstract

It is well known that the accuracy of several radial basis function (RBF) methods, including those based on multiquadric (MQ) RBFs, depends on a free shape parameter. For smooth solutions, it is theoretically claimed that without round-off error the highest accuracy for a given number of nodal points is regularly achieved with extreme values of the shape parameter, where the RBFs become increasingly flat. However, the limit values of the shape parameter (increasingly flat) often leads to very ill-conditioned problems. To alleviate this difficulty, we present a RBF method for solving second-order PDEs, in which (i) the RBF approximations are constructed using the integral approach where the starting points are fourth-order derivatives; and (ii) a simple but effective preconditioning technique is employed in the process of converting the RBF coefficient space into the physical space. In this paper, we first numerically study the effect of the shape parameter on the solution accuracy of the present RBF method through Poisson equation; and, then apply the method employed with extreme values of the shape parameter to simulate several fluid flow problems where highly accurate and stable solutions are produced.


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Item Type: Conference or Workshop Item (Commonwealth Reporting Category E) (Paper)
Refereed: Yes
Item Status: Live Archive
Additional Information: Author assigns all right, title and interest throughout the world in and to the copyright in the print and electronic forms of the Work to the Publisher. Numerical methods for solving PDEs using integrated radial basis function (author's note)
Faculty / Department / School: Current - Faculty of Health, Engineering and Sciences - School of Mechanical and Electrical Engineering
Date Deposited: 10 Jun 2016 00:16
Last Modified: 09 Mar 2017 03:44
Uncontrolled Keywords: RBF; integrated RBF; shape parameter; ill-conditioning; ODE; PDE
Fields of Research : 09 Engineering > 0913 Mechanical Engineering > 091307 Numerical Modelling and Mechanical Characterisation
Socio-Economic Objective: E Expanding Knowledge > 97 Expanding Knowledge > 970109 Expanding Knowledge in Engineering
URI: http://eprints.usq.edu.au/id/eprint/28294

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