辛算法的纠飘研究

刘晓梅; 周钢; 王永泓; 孙薇荣

辛算法的纠飘研究

1.
上海交通大学机械与动力学院, 上海 200240
2. 上海交通大学数学系, 上海 200240

基金项目: 国家自然科学基金资助项目(50876066)

详细信息

中图分类号: O 241; O 302
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- 被引次数: 0
出版历程
- 收稿日期: 2011-10-21
- 网络出版日期: 2013-01-31

Rectifying drifts of symplectic algorithm

1.
School of Mechanical and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
2. Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

摘要

摘要: 辛算法较RK(Runge-Kutta)方法,保持辛结构不变或保持哈密顿系统规律性不变是突出的优点,但点态数值精度并不理想.推导出了三阶、四阶辛算法的漂移量计算公式,通过补偿漂移量就能提高点态数值精度,既保辛结构又保证点态数值高精度,适合于工程应用.建立了分数步对称辛算法(即FSJS算法)的纠漂公式,制定了漂移的约束标准.相关算例的数值结果表明:三阶FSJS算法漂移量最小,点态数值精度更高.
- 辛算法 /
- Runge-Kutta法 /
- 相位漂移 /
- 哈密顿函数
Abstract: Symplectic algorithm preserves the symplectic structure and laws for Hamiltonian systems compared with Runge-Kutta(RK) methods, but the point-wise numerical precision is worse for elliptic Hamiltonian systems. In order to improve it, the average statistic drift formulae of the third-order symplectic method and the fourth-order scheme were deduced. The precision was improved through compensating the drifts and step segmentation. A standard was built to find a better symplectic scheme in phase drift. The results of examples show that the third-order fractional step and symmetric symplectic algorithm(FSJS3 algorithm) is higher than the fourth-order one in phase accuracy, which is recommended for engineering application.
- symplectic /
- Runge-Kutta methods /
- phase shift /
- Hamiltonian functions

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参考文献(9)

[1]	李庆扬,王能超,易大义.数值分析[M].北京:清华大学出版社,2008 Li Qingyang,Wang Nengchao,Yi Dayi.Numerical analysis[M].Beijing:Tsinghua University Press,2008(in Chinese)
[2]	冯康,秦孟兆.哈密顿系统的辛几何算法[M].杭州:浙江科学技术出版社,2003 Feng Kang,Qin Mengzhao.Symplectic geometric algorithms for Hamiltonian systems[M].Hangzhou:Zhejiang Science and Technology Press,2003(in Chinese)
[3]	Etienne F,Ruth R D.Fourth-order symplectic integration[J].Physica D,1990,43(1):105-117
[4]	Sun Geng.A simple way constructing symplectic Runger-Kutta methods[J].Jounal of Computational Mathmatics,2000,18(1):61-68
[5]	钟万勰.应用力学的辛数学方法[M].北京:高等教育出版社,2006:79-88 Zhong Wanxie.Symplectic solution methodology in apllied mechnics[M].Beijing:Higher Education Press,2006:79-88(in Chinese)
[6]	Chin S A,Scuro S R.Exact evolution of time-reversible symplectic integrators and their phase errors for the harmonic oscillator[J].Physics Letters A,2005,342:397-403
[7]	Grtz Peter.Backward error analysis of symplectic integrators for linear separable Hamiltonian systems[J].Joural of Computational Mathematics,2002,20(5):449-460
[8]	邢誉峰,杨蓉.单步辛算法的相位误差分析及修正[J].力学学报,2007,39(5):668-671 Xing Yufeng,Yang Rong.Phase errors and their correction in symplectic implicit single-step algorithm[J].Chinese Journal of Theoretical and Applied Mechanics,2007,39(5):668-671(in Chinese)
[9]	邢誉峰,冯伟.李级数算法和显式辛算法的相位分析[J].计算力学学报,2009,26(2):167-171 Xing Yufeng,Feng Wei.Phase analysis of Lie series algorithm and explicit symplectic algorithm[J].Chinese Journal of Computational Mechanics,2009,26(2):167-171(in Chinese)