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.
Zadeh, H. R., Mohammadi, M., Babolian, E. (2015). Solving a Class of PDEs by a Local Reproducing Kernel Method with An Adaptive Residual Subsampling Technique. CMES-Computer Modeling in Engineering & Sciences, 108(6), 375–396. https://doi.org/10.3970/cmes.2015.108.375
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