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, band-limited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger 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 = {}, 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, band-limited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger 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} }
TY - JOUR T1 - Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs JO - Communications in Computational Physics VL - 5 SP - 1375 EP - 1396 PY - 2018 DA - 2018/04 SN - 19 DO - http://dor.org/10.4208/cicp.scpde14.32s UR - https://global-sci.org/intro/article_detail/cicp/11134.html KW - AB -

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, band-limited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger 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.

Wenjun Cai, Huai Zhang & Yushun Wang. (2020). Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs. Communications in Computational Physics. 19 (5). 1375-1396. doi:10.4208/cicp.scpde14.32s
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