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Meshless Symplectic and Multi-symplectic Local RBF Collocation Methods for Hamiltonian PDEs

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Abstract

This study presents a novel meshless symplectic and multi-symplectic local radial basis function (RBF) collocation method (LRBFCM) for Hamiltonian partial differential equations. Specifically, the nonlinear wave equation and nonlinear Schrödinger equation are considered. The discretization in space is based on LRBFCM and then in time by symplectic integrator. The conservativeness of the method is explored and the accuracy is assessed. The LRBFCM is simple and efficient, since it can avoid the ill-conditioned problem and shape-parameter-sensitivity of global RBF method. Numerical experiments with uniform knots and random knots are designed to illustrate the effectiveness of the method.

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Data Availability Statement

The data used to support the findings of this study are available from the corresponding author upon request.

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Acknowledgements

We would like to express our deep appreciation to the anonymous reviewers for their valuable comments and suggestions.

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  1. College of Economics and Management, Nanjing Forestry University, Nanjing, China

    Shengliang Zhang

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  1. Shengliang Zhang
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Correspondence to Shengliang Zhang.

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Zhang, S. Meshless Symplectic and Multi-symplectic Local RBF Collocation Methods for Hamiltonian PDEs. J Sci Comput 88, 90 (2021). https://doi.org/10.1007/s10915-021-01605-w

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  • DOI: https://doi.org/10.1007/s10915-021-01605-w

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