• 论文 • 上一篇    下一篇

哈密尔顿系统的有限元法

汤琼1, 陈传淼2, 刘罗华3   

  1. 1. 湖南工业大学 理学院, 湖南株洲 412008;
    2. 湖南师范大学数学与计算机科学学院, 长沙 410081;
    3. 湖南工业大学 理学院, 湖南株洲 412008
  • 收稿日期:2008-09-08 出版日期:2009-12-15 发布日期:2009-12-30
  • 基金资助:

    国家自然科学基金(10771063, 60874025)及湖南省自然科学基金资助项目(09JJ3007)资助项目

汤琼, 陈传淼, 刘罗华. 哈密尔顿系统的有限元法[J]. 计算数学, 2009, 31(4): 393-406.

Tang Qiong, Chen Chuanmiao, Liu Luohua. FINITE ELEMENT METHODS FOR HAMILTONIAN SYSTEMS[J]. Mathematica Numerica Sinica, 2009, 31(4): 393-406.

FINITE ELEMENT METHODS FOR HAMILTONIAN SYSTEMS

Tang Qiong1, Chen Chuanmiao2, Liu Luohua3   

  1. 1. College of Science, Hunan University of Technology, Hunan, Zhuzhou 412008, China;
    2. Department of Mathematics and Computer Science, Hunan Normal University, Hunan, Changsha 410081, China;
    3. College of Science, Hunan University of Technology, Hunan, Zhuzhou 412008, China
  • Received:2008-09-08 Online:2009-12-15 Published:2009-12-30
利用常微分方程的连续有限元法, 结合函数的M-型展开, 对非线性哈密尔顿系统证明了连续一、二次有限元分在3阶量、5阶量意义下近似保辛, 且保持能量守恒.在数值实验中结合庞加莱截面, 哈密尔顿混沌数值试验结果与理论相吻合.

 

By applying the continuous finite element methods for ordinary differential equations and combine M-type function unfold, the linear element are proved an approximately symplectic method which is accurate of third order to their symplectic structure and the quadratic element are proved an approximately symplectic method which is accurate of fifth order to their symplectic structure, as well as energy conservative. Combine Poincarê section, the numerical results of Hamiltonian chaos agree with the theory.

 

MR(2010)主题分类: 

()

[1] 冯康. 冯康文集[M]. 北京: 国防工业出版社, 1995.
[2] 冯康, 秦孟兆. 哈密尔顿系统的辛几何算法[M]. 杭州: 浙江科学 技术出版社, 2003.
[3] 陈传淼, 黄云清. 有限元高精度理论[M]. 长沙: 湖南科学技术出 版社, 1995.
[4] 陈传淼. 有限元超收敛构造理论[M]. 长沙: 湖南科学技术出版社, 2001.
[5] Bridges T J, Reich S. Multisymplectic intergrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity[J], Phys Lett A, 2001, 284: 184-193.
[6] Sanz-Serna J M, Calvo M P. Numerical Hamiltonian Problems[M]. Great Britain by St Edmundsbury Press, 1994.
[7] Kane C, Marsden J E, Ortiz M. Symplectic-Energy-Momentum Preserving Variational Integrators[J]. Math Phys, 1999, 40: 3353-3371.
[8] 王雨顺, 王斌, 秦孟兆. 2+1维 Sine-Gordon 方程多辛格式的复合构造[J]. 中国科学(A辑), 2003, 33: 272-281.
[9] 丁培柱, 李延欣等. 微观化学反应动力学经典轨迹的辛算法计算[J]. 中国学术期刊文摘"科技快报", 1996, 2(2): 111.
[10] 钟万勰, 姚征. 时间有限元与保辛[J]. 机械强度, 2005, 27(2): 178-183.
[11] 钟万勰, 孙雁. 小参数摄动法与保辛[J]. 动力学与控制学报, 2005, 3: 1-6.
[12] 顾雁. 量子混沌[M].上海:上海科技教育出版社, 1996.
[13] Tang Q, Chen C M. Continuous finite element methods for Hamiltonian systems. Applied Mathematics and Mechanics[J], 2007, 28(8): 1071-1080.

[1] 尚在久, 宋丽娜. 关于辛算法稳定性的若干注记[J]. 计算数学, 2020, 42(4): 405-418.
[2] 刘子源, 梁家瑞, 钱旭, 宋松和. 带乘性噪声的空间分数阶随机非线性Schrödinger方程的广义多辛算法[J]. 计算数学, 2019, 41(4): 440-452.
[3] 郭峰. 非线性耦合Schrödinger-KdV方程组的一个局部能量守恒格式[J]. 计算数学, 2018, 40(3): 313-324.
[4] 林超英, 黄浪扬, 赵越, 朱贝贝. 带三次项的非线性四阶Schrödinger方程的一个局部能量守恒格式[J]. 计算数学, 2015, 37(1): 103-112.
阅读次数
全文


摘要