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关于辛算法稳定性的若干注记

尚在久1,2, 宋丽娜3   

  1. 1 HLM, 中国科学院数学与系统科学研究院, 数学研究所, 北京 100190;
    2 中国科学院大学数学科学学院, 北京 100049;
    3 吉林大学 数学学院, 长春 130012
  • 收稿日期:2020-08-11 出版日期:2020-11-15 发布日期:2020-11-15
  • 基金资助:

    国家自然科学基金~(11671392)资助.

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.

Shang Zaijiu, Song Lina. SOME NOTES ON THE STABILITY OF SYMPLECTIC METHODS[J]. Mathematica Numerica Sinica, 2020, 42(4): 405-418.

SOME NOTES ON THE STABILITY OF SYMPLECTIC METHODS

Shang Zaijiu1,2, Song Lina3   

  1. 1 HLM, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
    2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
    3 School of Mathematics, Jilin University, Changchun 130012, China
  • Received:2020-08-11 Online:2020-11-15 Published:2020-11-15
我们讨论辛算法的线性稳定性和非线性稳定性,从动力系统和计算的角度论述了研究辛算法的这两类稳定性问题的重要性,分析总结了相关重要结果.我们给出了解析方法的明确定义,证明了稳定函数是亚纯函数的解析辛方法是绝对线性稳定的.绝对线性稳定的辛方法既有解析方法(如Runge-Kutta辛方法),也有非解析方法(如基于常数变易公式对线性部分进行指数积分而对非线性部分使用其它数值积分的方法).我们特别回顾并讨论了R.I.McLachlan,S.K.Gray和S.Blanes,F.Casas,A.Murua等关于分裂算法的线性稳定性结果,如通过选取适当的稳定多项式函数构造具有最优线性稳定性的任意高阶分裂辛算法和高效共轭校正辛算法,这类经优化后的方法应用于诸如高振荡系统和波动方程等线性方程或者线性主导的弱非线性方程具有良好的数值稳定性.我们通过分析辛算法在保持椭圆平衡点的稳定性,能量面的指数长时间慢扩散和KAM不变环面的保持等三个方面阐述了辛算法的非线性稳定性,总结了相关已有结果.最后在向后误差分析基础上,基于一个自由度的非线性振子和同宿轨分析法讨论了辛算法的非线性稳定性,提出了一个新的非线性稳定性概念,目的是为辛算法提供一个实际可用的非线性稳定性判别法.
In this paper we discuss the linear stability and nonlinear stability of symplectic methods. We illustrate the importance of studying these two types of stability in view of dynamics and its numerical computation and give a brief summary of some relevant results. We give a definition to the notion “analytic method” and show that an analytic symplectic method (e.g., Runge-Kutta symplectic methods) is absolutely linear stable if the stability function of the method is meromorphic on the complex plane. We notice that there are not only analytic methods (e.g., Runge-Kutta methods) but also non-analytic methods (e.g., various exponential integration methods based on constant variational formula) with absolutely linear stability. We review and discuss the main results, initiated by R. I. MacLachlan and S. K. Gray then further developed by S. Blanes, F. Casas and A. Murua, on the linear stability of splitting methods as well as on the construction of arbitrarily high order splitting symplectic methods and more efficient conjugate processed integrators with optimal linear stability by suitably choosing stability polynomial functions. Such optimized integrators show good numerical stability for linear dominated problems with weak nonlinear perturbations such as highly oscillatory systems and wave equations. We discuss the known results on nonlinear stability of symplectic methods by analyzing the stability of elliptic equilibrium, the exponentially slow diffusion of energy surface, and the preservation of the KAM invariant tori. At last we propose a new nonlinear stability notion by analyzing the homoclinic trajectories of the nonlinear oscillator of one degree of freedom on the basis of backward error analysis, to give a practically useful nonlinear stability criterion of symplectic methods.

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[1] Abias L, Sanz-Serna J M. Partitioned Runge-Kutta methods for separable Hamiltonian problems[J]. Math. Comput., 1993, 60:617-634.

[2] Arnold V I. Mathematical Methods of Classical Mechanics[M]. GTM 60, Springer-Verlag, New York, 1978.

[3] Arnold V I. Instability of dynamical systems with several degrees of freedom[J]. Soviet Math. Dokl., 1964, 5:581-585.

[4] Arnold V I, Kozlov V V, Neishtadt A I. Mathematical Aspects of Classical and Celestial Mechanics (Third Edition)[M]. Springer-Verlag Berlin Heidelberg, 2006.

[5] Blanes S, Casas F, Farrés A, Laskar J, Makazaga J, Murua A. New families of symplectic splitting methods for numerical integration in dynamical astronomy[J]. Appl. Numer. Math., 2013, 68:58-72.

[6] Blanes S, Casas F, Murua A. On the linear stability of splitting methods. Found[J]. Comput. Math., 2008, 8:357-393.

[7] Blanes S, Casas F, Murua A. Error analysis of splitting methods for the time dependent Schrödinger equation[J]. SIAM J. Sci. Comput., 2011, 33:1525-1548.

[8] Ding X H, Liu H Y, Shang Z J, Sun G. Preservation of stability properties near fixed points of linear Hamiltonian systems by symplectic integrators[J]. Applied Mathematics and Computation, 2011, 217:6105-6114.[arXiv:0802.2121v1[math.NA] 14 Feb 2008].

[9] Ding Z D, Shang Z J. Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems[J]. Science China:Mathematics, 2018, 61(9):1567-1588.

[10] Dujardin D, Faou E. Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential[J]. Numer. Math., 2007, 108(2):223-262.

[11] 冯康. 冯康文集(II)[M]. 北京:国防工业出版社, 1995.

[12] 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.

[13] Feng K, Qin M Z. Symplectic Geometric Algorithms for Hamiltonian Systems. Zhejiang Science and Technology Publishing House[M], Hangzhou and Springer-Verlag Berlin Heidelberg 2010.

[14] Feng K, Shang Z J. Volume-preserving algorithms for source-free dynamical systems[J]. Numer. Math., 1995, 71(4):451-463.

[15] Gauckler L, Hairer E, Lubich C. Dynamics, numerical analysis, and some geometry[J]. Proc. of the International Congress of Mathematians-Rio de Janeiro 2018, Vol. I, Plenary lectures, 523-550.

[16] Hairer E. Backward analysis of numerical integrators and symplectic methods[J]. Annals of Numerical Mathematics, 1994, 1:107-132.

[17] Hairer E, Lubich C. The life-span of backward error analysis for numerical integrators[J]. Numer. Math., 1997, 76:441-462.

[18] Hairer E, Lubich C, Wanner G. Geometric Numerical Integration:Structure-Preserving algorithms for Ordinary Differential Equations (Second Edition)[M]. Springer Series in Computational Mathematics 31, Springer-Verlag Berlin New York, 2006.

[19] Hairer E, Nørsett S P, Wanner G. Solving Ordinary Differential Equations I (Second Revised Edition)[M]. Springer Series in Computational Mathematics 8, Springer-Verlag Berlin, 1993.

[20] Hairer E, Wanner G. Solving Ordinary Differential Equations II (Second Revised Edition)[M]. Springer Series in Computational Mathematics 14, Springer-Verlag Berlin, 1996.

[21] Hochbruck M, Lubich C. A Gautschi-type method for oscillatory second-order differential equations[J]. Numer. Math., 1999, 83:403-426.

[22] Jay L O. Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems[J]. SIAM J. Numer. Anal., 1996, 33(1):368-387.

[23] Jay L O, Petzold L R. Highly Oscillatory Systems and Periodic Stability. Preprint 95-015, Army High Performance Computing Research Center, Stanford, CA, 1995.

[24] Jordan M I. Dynamical, symplectic and stochastic perspectives on gradient-based optimization[J].Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018. Vol.I. Plenary lectures, 523-550.

[25] Laskar J. "Is the solar system stable?"[J] In:Chaos. Vol. 66. Prog. Math. Phys. Birkhäuser/Springer, Basel, 2013, 239-270.

[26] Laskar J, Gastineau M. Existence of collisional trajectories of Mercury, Mars and Venus with the Earth[J]. Nature Letters, Vol. June 2009, 459.

[27] Leimkuhler B, Matthews C, Stoltz G. The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics[J]. IMA J. Numer. Anal, 2016, 36(1):13-79.

[28] Leimkuhler B, Reich S. Simulating Hamiltonian Dynamics[M]. Cambridge Monographs on Applied and Computational Mathematics 14. Cambridge:Cambridge University Press, 2004.

[29] López M A, Sanz-Serna J M, Skeel R D. An explicit symplectic integrator with maximal stability interval[J]. in:Numerical Analysis, A. R. Mitchel 75th Birthday Volume (D. F. Grifiths and G. A. Watson, eds.), World Scientific, Singapore, 1996, 163-176.

[30] McLachlan R I, Gray S K. Optimal stability polynomials for splitting methods, with applications to the time-dependent Schrödinger equation[J]. Appl. Mumer. Math., 1997, 25:275-286.

[31] McLachlan R I, Perlmutter M, Quispel G R W. On the nonlinear stability of symplectic integrators[J]. BIT Numerical Mathematics, 2004, 44:99-117.

[32] McLachlan R I, Quispel G R W. Splitting methods[J]. Acta Numerica, 2002, (11):341-434.

[33] McLachlan R I, Sun Y J, Tse P S P. Linear stability of partitioned Runge-Kutta methods[J]. SIAM J. Numer. Anal. 2011, 49(1):232-263.

[34] Moser J. New aspects in the theory of stability of Hamiltonian systems[J]. Commun. Pure Appl. Math., 1958, XI:81-114.

[35] Poincaré H. Sur le probléme des trois corps et les équations de la dynamique[J]. Acta Math., 1890, 13:1-271.

[36] Qin H, Guan X. Variational symplectic integrator for long time simulations of the guiding-certer motion of charged particles in general mananetic fields[J]. Phys. Rev. Lett., 2008, 100:035006.

[37] Sanz-Serna J M. Two topics in nonlinear stability[J]. Advances in Numerical Analysis, W. Light, ed., Clarendon Press, Oxford, 1991, 147-174.

[38] Sanz-Serna J M, Calvo M P. Numerical Hamiltonian problems[M]. Vol. 7. Applied Mathematics and Mathematical Computation. London:Chapman & Hall, 1994.

[39] Sanz-Serna J M, Vadillo F. Nonlinear instability, the dynamic approach[J]. Numerical Analysis, D. F. Griffiths and G. A. Watson, eds., Pitman Res. Notes Math. Ser. 140, Longman Scientific and Technical, Harlow, UK, 1986, 187-199.

[40] Schlick T, Mandziuk M, Skeel R D, Srinivas K. Nonlinear resonance artifacts in molecular dynamics simulations[J]. J. Comput. Phys., 1998, 139:1-29.

[41] Shang Z J. KAM theorem of symplectic algorithms for Hamiltonian systems[J]. Numer. Math., 1999, 83:477-496.

[42] Shang Z J. Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems[J]. Nonlinearity, 2000, 13:299-308.

[43] Shang Z J. Stability analysis of symplectic integrator[R]. Report at the Oberwolfach Workshop on Geometric Numerical Integration, Mathematisches Forschungsinstitut Oberwolfach, Germany, March 2006, 19-25.

[44] Siegel C L, Moser J K. Lectures on Celestial Mechanics[M]. Springer-Verlag, New York, Heidelberg, 1971.

[45] Skeel R D, Srinivas K. Nonlinear Stability Analysis of Area-Preserving Integrators[J]. SIAM Journal on Numerical Analysis, 2000, 38(1):129-148.

[46] Skeel R D, Zhang G, Schlick T. A family of symplectic integrators:Stability, accuracy, and molecular dynamics applications[J]. SIAM J. Sci. Comput., 1997, 18(1997):203-222.

[47] Skokos Ch, Gerlach E, Bodyfelt J K, Papamikos G, Eggl S. High order three part split symplectic integrators:efficient techniques for the long time simulation of the disordered discrete nonlinear Schrödinger equation[J]. Phys. Lett. A, 2014, 378:1809-1815.

[48] 宋丽娜. 哈密尔顿系统辛几何算法的稳定性及相关问题的研究[D]. 博士学位论文, 中国科学院数学与系统科学研究院/中国科学院大学, 北京, 2009.

[49] Sun G. Symplectic partitioned Runge-Kutta methods[J]. J. Comput. Math. 1993, 11(4):365-372.

[50] Sun G. Construction of high order symplectic PRK methods[J]. J. Comput. Math. 1995, 13(1):40-50.

[51] Suzuki M. General theory higher order decomposition of exponential operators and symplectic integrators[J]. Phys. Lett. A, 1992, 165:387-395.

[52] Tang Y F. The symplecticity of multi-step methods[J]. Computers Math. Applic., 1993, 25:83-90.

[53] Wang L S, Wang Y. Preservation of equilibria for symplectic methods applied to Hamiltonian systems[J]. Acta Mathematicae Applicatae Sinica (English Series), 2010, 26(2):219-228.

[54] Wu X Y, Liu K, Shi W. Structure-Preserving Algorithms for Oscillatory Differential Equations II[M]. Springer-Verlag Berlin Heidelberg and Science Press, Beijing, China, 2015.

[55] Wu X Y, Wang B. Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations[M]. Science Press Beijing and Springer Nature Singapore Pte Ltd. 2018.

[56] Wu X Y, You X, Wang B. Structure-Preserving Algorithms for Oscillatory Differential Equations[M]. Science Press Beijing and Springer-Verlag Berlin Heidelberg, 2013.

[57] Yoshida H. Construction of higher order symplectic integrators[J]. Phys. Lett. A, 1990, 150:262-268.
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