Hamilton系统的有限元研究

数学物理学报 ›› 2011, Vol. 31 ›› Issue (1): 18-33.

Hamilton系统的有限元研究

陈传淼¹|汤琼²

1.湖南师范大学计算研究所 410081; 2.湖南工业大学信息与计算系湖南株洲 412008

收稿日期:2008-09-15 修回日期:2010-01-25 出版日期:2011-02-25 发布日期:2011-02-25
基金资助:
国家自然科学基金(10771063)和省部共建《高性能计算和随机信息处理》重点实验室资助.

Study of Finite Elements for Hamilton Systems

CHEN Chuan-Miao¹, TANG Qiong²

1.Institute of Computation, Hunan Normal University, Changsha 410081|2.Department of Information and Computation, Hunan University of Thechnology, Hunan Zhuzhou 412008

Received:2008-09-15 Revised:2010-01-25 Online:2011-02-25 Published:2011-02-25
Supported by:
国家自然科学基金(10771063)和省部共建《高性能计算和随机信息处理》重点实验室资助.

摘要/Abstract

摘要：

该文对Hamilton系统的连续有限元法证明了两个优美的性质:在任何情形m次有限元总是能量守恒的, 它对线性系统也是辛的,且对非线性系统每次步进是高精度O(h^2m+1)近似辛的. 在长时间计算中时空平面上轨道和周期的偏离随时间线性增长. 数值实验表明其偏离比其他算法小.

关键词: Hamilton系统, 非线性, 有限元;能量守恒, 辛性质, 长时间误差

Abstract:

Two nice properties of the continuous finite element method for Hamilton systems are proved as follows: in any case the m-degree finite elements always preserve the energy which is sympletic for linear systems and is approximately sympletic with high accuracy O(h^2m+1) in each stepping for nonlinear systems. In long-time computation the deviation of trajectories and their periods in time-space plane will crease linearly with time. Numerical experiments show that their deviations are often smaller than that of other schemes.

Key words: Hamilton systems, Nonlinear, Finite elements, Energy conservation, Sympleticity, Long-time error

中图分类号:

65N30

陈传淼, 汤琼. Hamilton系统的有限元研究[J]. 数学物理学报, 2011, 31(1): 18-33.

CHEN Chuan-Miao, TANG Qiong. Study of Finite Elements for Hamilton Systems[J]. Acta mathematica scientia,Series A, 2011, 31(1): 18-33.

参考文献

[1] 冯康. 冯康全集.北京:国防工业出版社, 1995: 1-2卷

[2] 冯康, 秦孟兆. 哈密尔顿系统的辛几何算法. 杭州: 浙江科学技术出版社, 2003

[3] Sanz-Senna J M, Calvo P M. Numerical Hamilton Problems. London: Chamman Hall, 1994

[4] Marsden J E, Ratiu S T. Introduction to Mechanics and Symmetry. New York: Springer-Verlag, 1994

[5] Stuart A M, Humphries A R. Dynamical Systems and Numerical Analysis. Cambridge: Cambridge University Press, 1996

[6] Hairer E, Lubich C, Wanner G. Geometric Numerical Integration---Structure-Preserving Algorithms for Ordinary Differential Equations. Series in Computational Mathematics 31, Berlin: Springer, 2002

[7] Ge Z, Marsden J F. Lie-Poisson Hamiltonb-Jacobi theory and Lie-Poisson integrators. Phys Lett A, 1988, 133: 134--139

[8] Sanz-Serna J M. Runge-Kutta schemes for Hamiltonian systems. BIT, 1988, 28: 877--883

[9] Kane C, Marsden J E, Ortiz M. Symplectic-energy-momentum preserving variational integrators. Math Phys, 1999, 40: 3353--3371

[10] Gonzalez O, Simo J C. On the stability of symplectic and energy-momentum algorithms for nonlinear Hamiltonian systems with symmetry. Comp Meth Appl Mech and Eng, 1996, 134: 197--222

[11] 郭柏灵. 非线性演化方程. 上海: 上海科技教育出版社, 1995

[12] 钟万勰. 应用力学对偶体系. 北京:科学出版社, 2002

[13] 钟万勰, 姚征. 时间有限元与保辛. 机械强度, 2005, 27: 178--183

[14] 陈传淼. 有限元超收敛构造理论.长沙:湖南科技出版社, 2001

[15] 陈传淼. 科学计算概论. 北京: 科学出版社, 2007

[16] Chen C M, Tang Q. Study of Finite Elements for Nonlinear Hamilton Systems. Changsha: Fifth Intern Conf on PDE and Numerical Analysis, 2006

[17] 李延欣,丁培柱. A_2B模型分子经典轨迹的辛算法计算.高等学校化学学报,1994, 15: 1181

[18] Lin X S, Qi Y Y, He J F, Ding P Z. Recent progress in sympletic algorithms for use in quantum sysrems. Commun Comput Phys, 2007, 2: 1--53

[19] 汤琼, 陈传淼. SchrÖdinger方程的时空有限元与守恒性. 应用数学和力学, 2006, 27: 335--340