中国物理B ›› 2016, Vol. 25 ›› Issue (1): 10203-010203.doi: 10.1088/1674-1056/25/1/010203

• GENERAL • 上一篇    下一篇

Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

Xin-Lei Kong(孔新雷), Hui-Bin Wu(吴惠彬), Feng-Xiang Mei(梅凤翔)   

  1. 1. College of Science, North China University of Technology, Beijing 100144, China;
    2. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
    3. School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
  • 收稿日期:2015-07-26 修回日期:2015-08-31 出版日期:2016-01-05 发布日期:2016-01-05
  • 通讯作者: Xin-Lei Kong E-mail:kongxinlei@ncut.edu.cn
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No. 11272050), the Excellent Young Teachers Program of North China University of Technology (Grant No. XN132), and the Construction Plan for Innovative Research Team of North China University of Technology (Grant No. XN129).

Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

Xin-Lei Kong(孔新雷)1, Hui-Bin Wu(吴惠彬)2, Feng-Xiang Mei(梅凤翔)3   

  1. 1. College of Science, North China University of Technology, Beijing 100144, China;
    2. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
    3. School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
  • Received:2015-07-26 Revised:2015-08-31 Online:2016-01-05 Published:2016-01-05
  • Contact: Xin-Lei Kong E-mail:kongxinlei@ncut.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No. 11272050), the Excellent Young Teachers Program of North China University of Technology (Grant No. XN132), and the Construction Plan for Innovative Research Team of North China University of Technology (Grant No. XN129).

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摘要： In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation, applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoffian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.

关键词: Birkhoffian equations, Hamiltonian equations, symplectic algorithm

Abstract: In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation, applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoffian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.

Key words: Birkhoffian equations, Hamiltonian equations, symplectic algorithm

中图分类号:  (Geometric mechanics)

  • 02.40.Yy
02.60.Cb (Numerical simulation; solution of equations) 45.20.Jj (Lagrangian and Hamiltonian mechanics)