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Hamilton系统下基于相位误差的精细辛算法

刘晓梅 周钢 朱帅

刘晓梅, 周钢, 朱帅. Hamilton系统下基于相位误差的精细辛算法[J]. 应用数学和力学, 2019, 40(6): 595-608. doi: 10.21656/1000-0887.390249
引用本文: 刘晓梅, 周钢, 朱帅. Hamilton系统下基于相位误差的精细辛算法[J]. 应用数学和力学, 2019, 40(6): 595-608. doi: 10.21656/1000-0887.390249
LIU Xiaomei, ZHOU Gang, ZHU Shuai. A Highly Precise Symplectic Direct Integration Method Based on Phase Errors for Hamiltonian Systems[J]. Applied Mathematics and Mechanics, 2019, 40(6): 595-608. doi: 10.21656/1000-0887.390249
Citation: LIU Xiaomei, ZHOU Gang, ZHU Shuai. A Highly Precise Symplectic Direct Integration Method Based on Phase Errors for Hamiltonian Systems[J]. Applied Mathematics and Mechanics, 2019, 40(6): 595-608. doi: 10.21656/1000-0887.390249

Hamilton系统下基于相位误差的精细辛算法

doi: 10.21656/1000-0887.390249
基金项目: 国家自然科学基金(50876066)
详细信息
    作者简介:

    刘晓梅(1982—),女,讲师,博士(通讯作者. E-mail: xmliu@sspu.edu.cn).

  • 中图分类号: O241;O302

A Highly Precise Symplectic Direct Integration Method Based on Phase Errors for Hamiltonian Systems

Funds: The National Natural Science Foundation of China(50876066)
  • 摘要: Hamilton系统是一类重要的动力系统,辛算法(如生成函数法、SRK法、SPRK法、多步法等)是针对Hamilton系统所设计的具有保持相空间辛结构不变或保Hamilton函数不变的算法.但是,时域上,同阶的辛算法与Runge-Kutta法具有相同的数值精度,即辛算法在计算过程中也存在相位误差,导致时域上解的数值精度不高.经过长时间计算后,计算结果在时域上也会变得“面目全非”.为了提高辛算法在时域上解的精度,将精细算法引入到辛差分格式中,提出了基于相位误差的精细辛算法(HPD-symplectic method),这种算法满足辛格式的要求,因此在离散过程中具有保Hamilton系统辛结构的优良特性.同时,由于精细化时间步长,极大地减小了辛算法的相位误差,大幅度提高了时域上解的数值精度,几乎可以达到计算机的精度,误差为O(10-13).对于高低混频系统和刚性系统,常规的辛算法很难在较大的步长下同时实现对高低频精确仿真,精细辛算法通过精细计算时间步长,在大步长情况下,没有额外增加计算量,实现了高低混频的精确仿真.数值结果验证了此方法的有效性和可靠性.
  • [1] FENG K. Difference schemes for Hamiltonian formalism and symplectic geometry[J]. Journal of Computational Mathematics,1986,4(3): 279-289.
    [2] GRTZ P. Backward error analysis of symplectic integrators for linear separable Hamiltonian systems[J]. Journal of Computational Mathematics,2002,20(5): 449-460.
    [3] 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))
    [4] MONOVASILIS T, KALOGIRATOU Z, SIMOS T E. Symplectic partitioned Runge-Kutta methods with minimal phase-lag[J]. Computer Physics Communications,2010,181(7): 1251-1254.
    [5] SIMOS T E. A two-step method with vanished phase-lag and its first two derivatives for the numerical solution of the Schrdinger equation[J]. Journal of Mathematical Chemistry,2011,49(10): 2486-2518.
    [6] MONOVASILIS T, KALOGIRATOU Z, SIMOS T E. Two new phase-fitted symplectic partitioned Runge-Kutta methods[J]. International Journal of Modern Physics C,2011,22(12): 1343-1355.
    [7] 陈璐, 王雨顺. 保结构算法的相位误差分析及其修正[J]. 计算数学, 2014,36(3): 271-290.(CHEN Lu, WANG Yushun. Phase error analysis and correction of structure preserving algorithms[J]. Mathematica Numerica Sinica,2014,36(3): 271-290.(in Chinese))
    [8] VYVER H V D. A symplectic Runge-Kutta-Nystrom method with minimal phase-lag[J]. Physics Letters A,2007,367(1/2): 16-24.
    [9] MONOVASILIS Th, KALOGIRATOU Z, SIMOS T E. Exponentially fitted symplectic Runge-Kutta-Nystrm methods[J]. Applied Mathematics & Information Sciences,2013,7(1): 81-85.
    [10] 刘晓梅, 周钢, 王永泓, 等. 辛算法的纠飘研究[J]. 北京航空航天大学学报, 2013,39(1): 22-26.(LIU Xiaomei, ZHOU Gang, WANG Yonghong, et al. Rectifying drifts of symplectic algorithm[J]. Journal of Beijing University of Aeronautics and Astronautics,2013,39(1): 22-26.(in Chinese))
    [11] MONOVASILIS Th. Symplectic partitioned Runge-Kutta methods with the phase-lag property[J]. Applied Mathematics and Computation,2012,218(18): 9075-9084.
    [12] XI X P, SIMOS T E. A new high algebraic order four stages symmetric two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrdinger equation and related problems[J]. Journal of Mathematical Chemistry,2016,54(7): 1417-1439.
    [13] 朱帅, 周钢, 刘晓梅, 等. 精细辛有限元方法及其相位误差研究[J]. 力学学报, 2016,48(2): 399-405.(ZHU Shuai, ZHOU Gang, LIU Xiaomei, et al. Precise symplectic time finite element method and the study of phase error[J]. Chinese Journal of Theoretical and Applied Mechanics,2016,48(2): 399-405.(in Chinese))
    [14] 钟万勰. 结构动力方程的精细时程积分法[J]. 大连理工大学学报, 1994,37(2): 131-136.(ZHONG Wanxie. On precise time-integration method for structural dynamics[J]. Journal of Dalian University of Technology,1994,37(2): 131-136.(in Chinese))
    [15] 钟万勰, 吴峰, 孙雁, 等. 保辛水波动力学[J]. 应用数学和力学, 2018,39(8): 855-874.(ZHONG Wanxie, WU Feng, SUN Yan, et al. Symplectic water wave dynamics[J]. Applied Mathematics and Mechanics,2018,39(8): 855-874.(in Chinese))
    [16] 富明慧, 李勇息. 求解病态线性方程组的预处理精细积分法[J]. 应用数学和力学, 2018,39(4): 462-469.(FU Minghui, LI Yongxi. A preconditioned precise integration method for solving ill-conditioned linear equations[J]. Applied Mathematics and Mechanics,2018,39(4): 462-469.(in Chinese))
    [17] HUANG Y A, DENG Z C, YAO L X. An improved symplectic precise integration method for analysis of the rotating rigid-flexible coupled system[J]. Journal of Sound and Vibration,2007,299(1/2): 229-246.
    [18] 曾进, 周钢. 精细辛算法[J]. 上海交通大学学报, 1997,31(9): 31-33.(ZENG Jin, ZHOU Gang. Precise symplectic algorithm[J]. Journal of Shanghai Jiaotong University,1997,31(9): 31-33.(in Chinese))
    [19] 黄永安, 尹周平, 邓子辰, 等. 多体动力学的几何积分方法研究进展[J]. 力学进展, 2009,39(1): 44-57.(HUANG Yongan, YIN Zhouping, DENG Zicheng, et al. Progress in geometric integration method for multibody dynamics[J]. Advances in Mechanics,2009,39(1): 44-57.(in Chinese))
    [20] 徐明毅, 张勇传. 精细辛算法的高效格式和简化计算[J]. 力学与实践, 2005,27(1): 55-57.(XU Mingyi, ZHANG Yongchuan. Efficient format and simple computation of precise symplectic integration method[J]. Mechanics in Engineering, 2005,27(1): 55-57.(in Chinese))
    [21] BRUSA L, NIGRO L. A one-step method for direct integration of structural dynamic equations[J]. International Journal for Numerical Methods in Engineering,1980,15(5): 685-699.
    [22] DAVID C, ERNST H, LUBICH C. Numerical energy conservation for multi-frequency oscillatory differential equations[J]. BIT Numerical Mathematics,2005,45(2): 287-305.
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出版历程
  • 收稿日期:  2018-09-21
  • 修回日期:  2018-10-11
  • 刊出日期:  2019-06-01

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