Open Access

ARTICLE

Solving a Class of PDEs by a Local Reproducing Kernel Method with An Adaptive Residual Subsampling Technique

H. Rafieayan Zadeh¹, M. Mohammadi^1,2, E. Babolian¹

1 Faculty of Mathematical Sciences and Computer, Kharazmi University, 50 Taleghani Ave., Tehran 1561836314, Iran
2 Corresponding author

Computer Modeling in Engineering & Sciences 2015, 108(6), 375-396. https://doi.org/10.3970/cmes.2015.108.375

Abstract

A local reproducing kernel method based on spatial trial space spanned by the Newton basis functions in the native Hilbert space of the reproducing kernel is proposed. It is a truly meshless approach which uses the local sub clusters of domain nodes for approximation of the arbitrary field. It leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs). An adaptive algorithm, so-called adaptive residual subsampling, is used to adjust nodes in order to remove oscillations which are caused by a sharp gradient. The method is applied for solving the Allen-Cahn and Burgers’ equations. The numerical results show that the proposed method is efficient, accurate and be able to remove oscillations caused by sharp gradient.

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Keywords

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