Volume 33, Issue 4
High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems

Kai Liu & Xinyuan Wu

J. Comp. Math., 33 (2015), pp. 356-378.

Published online: 2015-08

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  • Abstract

The multi-frequency and multi-dimensional adapted Runge-Kutta-Nyström (ARKN) integrators, and multi-frequency and multi-dimensional extended Runge-Kutta-Nyström(ERKN) integrators have been developed to efficiently solve multi-frequency oscillatory Hamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators. We first consider the symplecticity conditions for the multi-frequency and multi-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators and ERKN integrators, respectively. On the basis of the theoretical analysis and by using the idea of composition methods, we derive and propose four new high-order symplectic and symmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency of the proposed high-order symplectic and symmetric methods.

  • Keywords

Symplectic and symmetric composition methods, Multi-frequency and multidimensional ERKN integrators, ARKN integrators, Multi-frequency oscillatory Hamiltonian systems.

  • AMS Subject Headings

65L05, 65L06, 65M20, 65P10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

752964253@qq.com (Kai Liu)

xywu@nju.edu.cn (Xinyuan Wu)

  • BibTex
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  • TXT
@Article{JCM-33-356, author = {Liu , Kai and Wu , Xinyuan }, title = {High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {4}, pages = {356--378}, abstract = {

The multi-frequency and multi-dimensional adapted Runge-Kutta-Nyström (ARKN) integrators, and multi-frequency and multi-dimensional extended Runge-Kutta-Nyström(ERKN) integrators have been developed to efficiently solve multi-frequency oscillatory Hamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators. We first consider the symplecticity conditions for the multi-frequency and multi-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators and ERKN integrators, respectively. On the basis of the theoretical analysis and by using the idea of composition methods, we derive and propose four new high-order symplectic and symmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency of the proposed high-order symplectic and symmetric methods.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1502-m2014-0082}, url = {http://global-sci.org/intro/article_detail/jcm/9848.html} }
TY - JOUR T1 - High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems AU - Liu , Kai AU - Wu , Xinyuan JO - Journal of Computational Mathematics VL - 4 SP - 356 EP - 378 PY - 2015 DA - 2015/08 SN - 33 DO - http://doi.org/10.4208/jcm.1502-m2014-0082 UR - https://global-sci.org/intro/article_detail/jcm/9848.html KW - Symplectic and symmetric composition methods, Multi-frequency and multidimensional ERKN integrators, ARKN integrators, Multi-frequency oscillatory Hamiltonian systems. AB -

The multi-frequency and multi-dimensional adapted Runge-Kutta-Nyström (ARKN) integrators, and multi-frequency and multi-dimensional extended Runge-Kutta-Nyström(ERKN) integrators have been developed to efficiently solve multi-frequency oscillatory Hamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators. We first consider the symplecticity conditions for the multi-frequency and multi-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators and ERKN integrators, respectively. On the basis of the theoretical analysis and by using the idea of composition methods, we derive and propose four new high-order symplectic and symmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency of the proposed high-order symplectic and symmetric methods.

Kai Liu & Xinyuan Wu. (2019). High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems. Journal of Computational Mathematics. 33 (4). 356-378. doi:10.4208/jcm.1502-m2014-0082
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