Volume 27, Issue 3
Construction of the Local Structure-Preserving Algorithms for the General Multi-Symplectic Hamiltonian System

Jialing Wang, Yushun Wang & Dong Liang

Commun. Comput. Phys., 27 (2020), pp. 828-860.

Published online: 2020-02

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  • Abstract

Many partial differential equations can be written as a multi-symplecticHamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we systematically give a unified framework to construct the local structure-preserving algorithms for general conservative partial differential equations starting from the multi-symplectic formulation and using the concatenating method. We construct four multi-symplectic algorithms, two local energy-preserving algorithms and two local momentum-preserving algorithms, which are independent of the boundary conditions and can be used to integrate any partial differential equations written in multi-symplectic Hamiltonian form. Among these algorithms, some have been discussed and widely used before while most are novel schemes. These algorithms are illustrated by the nonlinear Schrödinger equation and the Klein-Gordon-Schrödinger equation. Numerical experiments are conducted to show the good performance of the proposed methods.

  • Keywords

Multi-symplectic formulation, multi-symplectic algorithm, energy-preserving algorithm, momentum-preserving algorithm, concatenating method, average vector field method.

  • AMS Subject Headings

65L12, 65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wjl19900724@126.com (Jialing Wang)

wangyushun@njnu.edu.cn (Yushun Wang)

dliang@mathstat.yorku.ca (Dong Liang)

  • BibTex
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@Article{CiCP-27-828, author = {Wang , Jialing and Wang , Yushun and Liang , Dong }, title = {Construction of the Local Structure-Preserving Algorithms for the General Multi-Symplectic Hamiltonian System}, journal = {Communications in Computational Physics}, year = {2020}, volume = {27}, number = {3}, pages = {828--860}, abstract = {

Many partial differential equations can be written as a multi-symplecticHamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we systematically give a unified framework to construct the local structure-preserving algorithms for general conservative partial differential equations starting from the multi-symplectic formulation and using the concatenating method. We construct four multi-symplectic algorithms, two local energy-preserving algorithms and two local momentum-preserving algorithms, which are independent of the boundary conditions and can be used to integrate any partial differential equations written in multi-symplectic Hamiltonian form. Among these algorithms, some have been discussed and widely used before while most are novel schemes. These algorithms are illustrated by the nonlinear Schrödinger equation and the Klein-Gordon-Schrödinger equation. Numerical experiments are conducted to show the good performance of the proposed methods.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0160}, url = {http://global-sci.org/intro/article_detail/cicp/13923.html} }
TY - JOUR T1 - Construction of the Local Structure-Preserving Algorithms for the General Multi-Symplectic Hamiltonian System AU - Wang , Jialing AU - Wang , Yushun AU - Liang , Dong JO - Communications in Computational Physics VL - 3 SP - 828 EP - 860 PY - 2020 DA - 2020/02 SN - 27 DO - http://dor.org/10.4208/cicp.OA-2018-0160 UR - https://global-sci.org/intro/cicp/13923.html KW - Multi-symplectic formulation, multi-symplectic algorithm, energy-preserving algorithm, momentum-preserving algorithm, concatenating method, average vector field method. AB -

Many partial differential equations can be written as a multi-symplecticHamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we systematically give a unified framework to construct the local structure-preserving algorithms for general conservative partial differential equations starting from the multi-symplectic formulation and using the concatenating method. We construct four multi-symplectic algorithms, two local energy-preserving algorithms and two local momentum-preserving algorithms, which are independent of the boundary conditions and can be used to integrate any partial differential equations written in multi-symplectic Hamiltonian form. Among these algorithms, some have been discussed and widely used before while most are novel schemes. These algorithms are illustrated by the nonlinear Schrödinger equation and the Klein-Gordon-Schrödinger equation. Numerical experiments are conducted to show the good performance of the proposed methods.

Jialing Wang, Yushun Wang & Dong Liang. (2020). Construction of the Local Structure-Preserving Algorithms for the General Multi-Symplectic Hamiltonian System. Communications in Computational Physics. 27 (3). 828-860. doi:10.4208/cicp.OA-2018-0160
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