Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

doi:10.1088/1674-1056/aca9c8

中国物理B ›› 2023, Vol. 32 ›› Issue (2): 20204-020204.doi: 10.1088/1674-1056/aca9c8

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

Beibei Zhu(朱贝贝)¹, Lun Ji(纪伦)^2,3, Aiqing Zhu(祝爱卿)^2,3, and Yifa Tang(唐贻发)^2,3,†

1 School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China;
2 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
3 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

收稿日期:2022-10-08 修回日期:2022-11-27 接受日期:2022-12-08 出版日期:2023-01-10 发布日期:2023-01-31
通讯作者: Yifa Tang E-mail:tyf@lsec.cc.ac.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11901564 and 12171466).

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

Beibei Zhu(朱贝贝)¹, Lun Ji(纪伦)^2,3, Aiqing Zhu(祝爱卿)^2,3, and Yifa Tang(唐贻发)^2,3,†

1 School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China;
2 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
3 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received:2022-10-08 Revised:2022-11-27 Accepted:2022-12-08 Online:2023-01-10 Published:2023-01-31
Contact: Yifa Tang E-mail:tyf@lsec.cc.ac.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11901564 and 12171466).

摘要/Abstract

摘要： We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit, K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for two non-canonical Hamiltonian systems. Numerical tests show that the proposed methods exhibit good numerical performance in preserving the phase orbit and the energy of the system over long time, whereas higher order Runge-Kutta methods do not preserve these properties. Numerical tests also show that the K-symplectic methods exhibit better efficiency than that of the same order implicit symplectic, explicit and implicit symplectic methods for the original nonseparable non-canonical systems. On the other hand, the fourth order K-symplectic method is more efficient than the fourth order Yoshida's method, the optimized partitioned Runge-Kutta and Runge-Kutta-Nyström explicit K-symplectic methods for the extended phase space Hamiltonians, but less efficient than the the optimized partitioned Runge-Kutta and Runge-Kutta-Nyström extended phase space symplectic-like methods with the midpoint permutation.

关键词: non-canonical Hamiltonian systems, nonseparable, explicit K-symplectic methods, splitting method

Abstract: We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit, K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for two non-canonical Hamiltonian systems. Numerical tests show that the proposed methods exhibit good numerical performance in preserving the phase orbit and the energy of the system over long time, whereas higher order Runge-Kutta methods do not preserve these properties. Numerical tests also show that the K-symplectic methods exhibit better efficiency than that of the same order implicit symplectic, explicit and implicit symplectic methods for the original nonseparable non-canonical systems. On the other hand, the fourth order K-symplectic method is more efficient than the fourth order Yoshida's method, the optimized partitioned Runge-Kutta and Runge-Kutta-Nyström explicit K-symplectic methods for the extended phase space Hamiltonians, but less efficient than the the optimized partitioned Runge-Kutta and Runge-Kutta-Nyström extended phase space symplectic-like methods with the midpoint permutation.

Key words: non-canonical Hamiltonian systems, nonseparable, explicit K-symplectic methods, splitting method

中图分类号: (Numerical differentiation and integration)

02.60.Jh

02.60.Cb (Numerical simulation; solution of equations) 45.20.Jj (Lagrangian and Hamiltonian mechanics) 45.20.dh (Energy conservation)

Beibei Zhu(朱贝贝), Lun Ji(纪伦), Aiqing Zhu(祝爱卿), and Yifa Tang(唐贻发). Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems[J]. 中国物理B, 2023, 32(2): 20204-020204.

Beibei Zhu(朱贝贝), Lun Ji(纪伦), Aiqing Zhu(祝爱卿), and Yifa Tang(唐贻发). Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems[J]. Chin. Phys. B, 2023, 32(2): 20204-020204.

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Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

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