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Meshless Conservative Scheme for Multivariate Nonlinear Hamiltonian PDEs

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Abstract

For multivariate nonlinear Hamiltonian equations, we propose a meshless conservative method by using radial basis approximation. Based on the method of lines, we first discretize the Hamiltonian functional using radial basis function interpolation, and then obtain a finite-dimensional semi-discrete Hamiltonian system. Moreover, we define a discrete symplectic form and verify that it is an approximation to the continuous one and is conserved with respect to time. For time discretization, two conservative methods (symplectic method and energy-conserving method) are employed to derive the full-discretized system. Approximation errors together with conservation properties including symplecticity and energy are discussed in detail. Finally, we present several numerical examples to illustrate that our method is accurate and effective when processing nonlinear Hamiltonian equations with scattered nodes. Besides, the numerical results also confirm the excellent conservation properties of the proposed method.

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Authors and Affiliations

  1. Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China

    Zhengjie Sun & Zongmin Wu

  2. Shanghai Center for Mathematical Sciences, Shanghai, People’s Republic of China

    Zongmin Wu

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  1. Zhengjie Sun
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  2. Zongmin Wu
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Correspondence to Zongmin Wu.

Additional information

This work is supported by NSFC (11631015, 91330201), Joint Research Fund by National Natural Science Foundation of China and Research Grants Council of Hong Kong (11461161006).

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Sun, Z., Wu, Z. Meshless Conservative Scheme for Multivariate Nonlinear Hamiltonian PDEs. J Sci Comput 76, 1168–1187 (2018). https://doi.org/10.1007/s10915-018-0658-1

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  • DOI: https://doi.org/10.1007/s10915-018-0658-1

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