Volume 19, Issue 5
Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs

Wenjun Cai, Huai Zhang & Yushun Wang

Commun. Comput. Phys., 19 (2016), pp. 1375-1396.

Published online: 2018-04

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  • Abstract

This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schr ¨odinger equation and the coupled Schr ¨odinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.

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@Article{CiCP-19-1375, author = {Wenjun Cai, Huai Zhang and Yushun Wang}, title = {Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs}, journal = {Communications in Computational Physics}, year = {2018}, volume = {19}, number = {5}, pages = {1375--1396}, abstract = {

This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schr ¨odinger equation and the coupled Schr ¨odinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.scpde14.32s}, url = {http://global-sci.org/intro/article_detail/cicp/11134.html} }
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