Several compact local stencils based on integrated RBFs for fourth-order ODEs and PDEs

Hoang-Trieu, T.-T. and Mai-Duy, N. and Tran-Cong, T. (2012) Several compact local stencils based on integrated RBFs for fourth-order ODEs and PDEs. CMES: Computer Modeling in Engineering and Sciences, 84 (2). pp. 171-203. ISSN 1526-1492

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Abstract

In this paper, new compact local stencils based on integrated radial basis functions (IRBFs) for solving fourth-order ordinary differential equations (ODEs) and partial differential equations (PDEs) are presented. Five types of compact stencils - 3-node and 5-node for 1D problems and 5×5-node, 13-node and 3×3 -node for 2D problems - are implemented. In the case of 3-node stencil and 3×3-node stencil, nodal values of the first derivative(s) of the field variable are treated as additional unknowns (i.e. 2 unknowns per node for 3-node stencil and 3 unknowns per node for 3×3-node stencil). The integration constants arising from the construction of IRBFs are exploited to incorporate into the local IRBF approximations (i) values of the governing equation (GE) at selected nodes for the case of 5-, 5×5- and 13-node stencils, and (ii) not only nodal values of the governing equation but also nodal values of the first derivative(s) for the case of 3-node stencil and 3×3-node stencil. There are no special treatments required for grid nodes near the boundary for 3-node stencil and 3×3-node stencil. The proposed stencils, which lead to sparse system matrices, are numerically verified through the solution of several test problems.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Copyright © 2012 Tech Science Press.
Depositing User: epEditor USQ
Faculty / Department / School: Historic - Faculty of Engineering and Surveying - Department of Mechanical and Mechatronic Engineering
Date Deposited: 17 Jul 2012 07:22
Last Modified: 21 Jul 2014 05:08
Uncontrolled Keywords: compact local approximations; high-order ODEs; high-order PDEs; integrated radial basis functions
Fields of Research (FOR2008): 01 Mathematical Sciences > 0101 Pure Mathematics > 010109 Ordinary Differential Equations, Difference Equations and Dynamical Systems
01 Mathematical Sciences > 0102 Applied Mathematics > 010201 Approximation Theory and Asymptotic Methods
09 Engineering > 0913 Mechanical Engineering > 091307 Numerical Modelling and Mechanical Characterisation
Socio-Economic Objective (SEO2008): E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Identification Number or DOI: doi: 10.3970/cmes.2012.084.171
URI: http://eprints.usq.edu.au/id/eprint/21553

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