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Hamilton系统的对称辛广义加性Runge-Kutta方法

贾旻茜, 张宇欣, 游雄   

  1. 南京农业大学理学院, 南京 210095
  • 收稿日期:2021-03-02 出版日期:2022-07-14 发布日期:2022-08-03

贾旻茜, 张宇欣, 游雄. Hamilton系统的对称辛广义加性Runge-Kutta方法[J]. 计算数学, 2022, 44(3): 379-395.

Jia Minqian, Zhang Yuxin, You Xiong. SYMMETRIC AND SYMPLECTIC GENERALIZED ADDITIVE RUNGE-KUTTA FOR HAMILTONIAN SYSTEMS[J]. Mathematica Numerica Sinica, 2022, 44(3): 379-395.

SYMMETRIC AND SYMPLECTIC GENERALIZED ADDITIVE RUNGE-KUTTA FOR HAMILTONIAN SYSTEMS

Jia Minqian, Zhang Yuxin, You Xiong   

  1. College of Sciences, Nanjing Agricultural University, Nanjing 210095, China
  • Received:2021-03-02 Online:2022-07-14 Published:2022-08-03
Sandu和Günther[SIAM J.Numer.Anal.53(2015)17--42]对形如$\dot{y}=\sum\limits_{k=1}^{N}f^{[k]}(y)$的微分方程提出广义加性Runge-Kutta (GARK)方法.本文利用双色有根树导出GARK方法的阶条件,给出辛条件和对称性条件,并构造了三个二阶对称辛GARK (SSGARK)方法和两个四阶SSGARK方法.对三个经典测试问题的数值实验结果显示,与文献中几个非对称或非辛的ARK/GARK方法相比,新的SSGARK方法能更有效地保持Hamilton量.
Sandu and Günther[SIAM J. Numer. Anal., 53(2015) 17——42] has established Generalized Additive Runge-Kutta (GARK) methods for differential equations of the form $\dot{y}=\sum\limits_{k=1}^{N}f^{[k]}(y)$. This paper derives the order conditions for GARK methods via bicolored rooted trees. Symplecticity and symmetry conditions are also presented. Three symmetric and symplectic GRAR (SSGARK) methods of order two and two SSGARK methods of order four are constructed. When applied to three classical test problems, the new SSGARK methods are shown to be more effective in preserving the Hamiltonian energy, compared to several non-symmetric or non-symplectic ARK/GARK methods.

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