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Comparison of second-order mixed symplectic integrator between semi-implicit Euler method and implicit midpoint rule

Zhong Shuang-Ying Wu Xin

Comparison of second-order mixed symplectic integrator between semi-implicit Euler method and implicit midpoint rule

Zhong Shuang-Ying, Wu Xin
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  • When a Hamiltonian can be split into integrable and nonintegrable parts, the former part is solved analytically, and the latter one is integrated numerically by means of implicit symplectic integrators such as the first-order semi-implicit Euler method or the second-order implicit midpoint rule. These analytical and numerical solutions are used to construct a second-order mixed symplectic integrator with the semi-implicit Euler method and one with the implicit midpoint rule. A theoretical analysis shows that the Euler mixed integrator is inferior to the midpoint one in the sense of numerical stability. Numerical simulations of the circularly-restricted three-body problem also support this fact. It is further shown through numerical integrations of the post-Newtonian Hamiltonian of spinning compact binaries that the qualities of the Euler mixed integrator and the midpoint mixed method do depend on the type of orbits. Especially for chaotic orbits, the Euler mixed integrator often becomes unstable. In addition, the Euler mixed integrator has an advantage over the midpoint mixed method in computational efficiency, and is almost equivalent to the latter in the numerical accuracy if the two mixed integrators are stable. In spite of this, the midpoint mixed integrator is worth recommending for the study of the dynamics of post-Newtonian Hamiltonians of spinning compact binaries.
    • Funds:
    [1]

    Feng K 1986 J. Comput. Math. 4 279

    [2]

    Ruth R D 1983 IEEE Tran. Nucl. Sci. 30 2669

    [3]

    Li R, Wu X 2010 Science China Physics, Mechanics & Astronomy 53 1600

    [4]

    Li R, Wu X 2010 Acta Phys. Sin. 59 7135 (in Chinese) [李 荣、伍 歆 2010 物理学报 59 7135]

    [5]

    Sun W, Wu X, Huang G Q 2011 Res. Astron. Astrophys. 11 (in press)

    [6]

    Chi Y H, Liu X S, Ding P Z 2006 Acta Phys.Sin. 55 6320 (in Chinese) [匙玉华、刘学深、丁培柱 2006 物理学报 55 6320]

    [7]

    Luo X Y, Liu X S, Ding P Z 2007 Acta Phys.Sin. 56 0604 (in Chinese) [罗香怡、刘学深、丁培柱 2007 物理学报 56 0604]

    [8]

    Liu X S, Wei J Y, Ding P Z 2005 Chin. Phys. 14 231

    [9]

    Bian X B, Qiao H X, Shi T Y 2007 Chin. Phys. 16 1822

    [10]

    Cao Y, Yang K Q 2003 Acta Phys. Sin. 52 1984(in Chinese)[曹 禹、杨孔庆 2003 物理学报 52 1985]

    [11]

    Hu W P, Deng Z C 2008 Chin. Phys. B 17 3923

    [12]

    Wisdom J, Holman M 1991 Astron. J. 102 1528

    [13]

    Xu J, Wu X 2010 Res. Astron. Astrophys. 10 173

    [14]

    Zhu J F, Wu X, Ma D Z 2007 Chin. J. Astron. Astrophys. 7 601

    [15]

    Preto M, Saha P 2009 Astrophy. J. 703 1743

    [16]

    Liao X H 1997 Celest. Mech. Dyn. Astron. 66 243

    [17]

    Zhong S Y, Wu X, Liu S Q, Deng X F 2010 Phys. Rev. D 82 124040

    [18]

    Wang Y J, Tang Z M 2001 Acta Phys. Sin. 50 2284(in Chinese)[王永久、唐智明 2001 物理学报 50 2284]

    [19]

    Kidder L E 1995 Phys. Rev. D 52 821

    [20]

    de Andrade V C, Blanchet L, Faye G 2001 Class. Quantum Grav. 18 753

    [21]

    Faye G, Blanchet L, Buonanno A 2006 Phys. Rev. D 74 104033

    [22]

    Damour T, Jaranowski P, Schafer G 2001 Phys. Rev. D 63 044021

    [23]

    Damour T 2001 Phys. Rev. D 64 124013

    [24]

    Buonanno A, Chen Y, Damour T 2006 Phys. Rev. D 74 104005

    [25]

    Damour T, Jaranowski P, Schafer G 2008 Phys. Rev. D 77 064032

    [26]

    Hergt S, Schafer G 2008 Phys. Rev. D 77 104001

    [27]

    Hergt S, Schafer G 2008 Phys. Rev. D 78 124004

    [28]

    Li C B, Chen S, Zhu Y Q 2009 Acta Phys. Sin. 58 2255(in Chinese)[李春彪、陈 谡、朱炴强 2009 物理学报58 2255]

    [29]

    Zhang R X, Yang S P 2009 Acta Phys. Sin. 58 2957(in Chinese)[张若洵、杨世平 2009 物理学报58 2957]

    [30]

    Wen S Y, Wang Z, Liu F C 2009 Acta Phys. Sin. 58 3753(in Chinese)[温淑焕、王 哲、刘福才 2009 物理学报58 3753]

    [31]

    Li G L, Chen XY 2010 Chin. Phys. B 19 030507

    [32]

    Liu C X, Liu L 2009 Chin. Phys. B 18 2188

    [33]

    Gu Q L, Gao T G 2009 Chin. Phys. B 18 84

    [34]

    Levin J 2000 Phys. Rev. Lett. 84 3515

    [35]

    Konigsdorffer C, Gopakumar A 2005 Phys. Rev. D 71 024039

    [36]

    Gopakumar A, Konigsdorffer C 2005 Phys. Rev. D 72 121501

    [37]

    Levin J 2006 Phys. Rev. D 74 124027

    [38]

    Schnittman J D, Rasio F A 2001 Phys. Rev. Lett. 87 121101

    [39]

    Cornish N J, Levin J 2002 Phys. Rev. Lett. 89 179001

    [40]

    Wu X, Xie Y 2007 Phys. Rev. D 76, 124004

    [41]

    Sun K H, Liu X, Zhu C X 2010 Chin. Phys. B 19 110510

    [42]

    Wu X, Xie Y 2008 Phys. Rev. D 77 103012

    [43]

    Lubich C, Walther B, Braugmann B 2010 Phys. Rev. D 81 104025

    [44]

    Wu X, Xie Y 2010 Phys. Rev. D 81 084045

    [45]

    Liu F Y, Wu X, Lu B K 2007 Chinese Astorn. Astrophys. 31 172

    [46]

    Murray C D, Dermott S F 1999 Solar System Dynamics. Cambridge Univ. Press, Cambridge, UK.

    [47]

    Zhao H J, Du M L 2007 Acta Phys.Sin. 56 3827 (in Chinese)[赵海军、杜孟利 2007 物理学报 56 3827]

    [48]

    Hartl M D, Buonanno A 2005 Phys. Rev. D 71 024027

    [49]

    Zhong S Y, Wu X 2010 Phys. Rev. D 81 104037

    [50]

    Wang Y, Wu X 2011 Class. Quantum Grav. 28 in press

    [51]

    Wu X, Huang T Y, Zhang H 2006 Phys. Rev. D 74 083001

  • [1]

    Feng K 1986 J. Comput. Math. 4 279

    [2]

    Ruth R D 1983 IEEE Tran. Nucl. Sci. 30 2669

    [3]

    Li R, Wu X 2010 Science China Physics, Mechanics & Astronomy 53 1600

    [4]

    Li R, Wu X 2010 Acta Phys. Sin. 59 7135 (in Chinese) [李 荣、伍 歆 2010 物理学报 59 7135]

    [5]

    Sun W, Wu X, Huang G Q 2011 Res. Astron. Astrophys. 11 (in press)

    [6]

    Chi Y H, Liu X S, Ding P Z 2006 Acta Phys.Sin. 55 6320 (in Chinese) [匙玉华、刘学深、丁培柱 2006 物理学报 55 6320]

    [7]

    Luo X Y, Liu X S, Ding P Z 2007 Acta Phys.Sin. 56 0604 (in Chinese) [罗香怡、刘学深、丁培柱 2007 物理学报 56 0604]

    [8]

    Liu X S, Wei J Y, Ding P Z 2005 Chin. Phys. 14 231

    [9]

    Bian X B, Qiao H X, Shi T Y 2007 Chin. Phys. 16 1822

    [10]

    Cao Y, Yang K Q 2003 Acta Phys. Sin. 52 1984(in Chinese)[曹 禹、杨孔庆 2003 物理学报 52 1985]

    [11]

    Hu W P, Deng Z C 2008 Chin. Phys. B 17 3923

    [12]

    Wisdom J, Holman M 1991 Astron. J. 102 1528

    [13]

    Xu J, Wu X 2010 Res. Astron. Astrophys. 10 173

    [14]

    Zhu J F, Wu X, Ma D Z 2007 Chin. J. Astron. Astrophys. 7 601

    [15]

    Preto M, Saha P 2009 Astrophy. J. 703 1743

    [16]

    Liao X H 1997 Celest. Mech. Dyn. Astron. 66 243

    [17]

    Zhong S Y, Wu X, Liu S Q, Deng X F 2010 Phys. Rev. D 82 124040

    [18]

    Wang Y J, Tang Z M 2001 Acta Phys. Sin. 50 2284(in Chinese)[王永久、唐智明 2001 物理学报 50 2284]

    [19]

    Kidder L E 1995 Phys. Rev. D 52 821

    [20]

    de Andrade V C, Blanchet L, Faye G 2001 Class. Quantum Grav. 18 753

    [21]

    Faye G, Blanchet L, Buonanno A 2006 Phys. Rev. D 74 104033

    [22]

    Damour T, Jaranowski P, Schafer G 2001 Phys. Rev. D 63 044021

    [23]

    Damour T 2001 Phys. Rev. D 64 124013

    [24]

    Buonanno A, Chen Y, Damour T 2006 Phys. Rev. D 74 104005

    [25]

    Damour T, Jaranowski P, Schafer G 2008 Phys. Rev. D 77 064032

    [26]

    Hergt S, Schafer G 2008 Phys. Rev. D 77 104001

    [27]

    Hergt S, Schafer G 2008 Phys. Rev. D 78 124004

    [28]

    Li C B, Chen S, Zhu Y Q 2009 Acta Phys. Sin. 58 2255(in Chinese)[李春彪、陈 谡、朱炴强 2009 物理学报58 2255]

    [29]

    Zhang R X, Yang S P 2009 Acta Phys. Sin. 58 2957(in Chinese)[张若洵、杨世平 2009 物理学报58 2957]

    [30]

    Wen S Y, Wang Z, Liu F C 2009 Acta Phys. Sin. 58 3753(in Chinese)[温淑焕、王 哲、刘福才 2009 物理学报58 3753]

    [31]

    Li G L, Chen XY 2010 Chin. Phys. B 19 030507

    [32]

    Liu C X, Liu L 2009 Chin. Phys. B 18 2188

    [33]

    Gu Q L, Gao T G 2009 Chin. Phys. B 18 84

    [34]

    Levin J 2000 Phys. Rev. Lett. 84 3515

    [35]

    Konigsdorffer C, Gopakumar A 2005 Phys. Rev. D 71 024039

    [36]

    Gopakumar A, Konigsdorffer C 2005 Phys. Rev. D 72 121501

    [37]

    Levin J 2006 Phys. Rev. D 74 124027

    [38]

    Schnittman J D, Rasio F A 2001 Phys. Rev. Lett. 87 121101

    [39]

    Cornish N J, Levin J 2002 Phys. Rev. Lett. 89 179001

    [40]

    Wu X, Xie Y 2007 Phys. Rev. D 76, 124004

    [41]

    Sun K H, Liu X, Zhu C X 2010 Chin. Phys. B 19 110510

    [42]

    Wu X, Xie Y 2008 Phys. Rev. D 77 103012

    [43]

    Lubich C, Walther B, Braugmann B 2010 Phys. Rev. D 81 104025

    [44]

    Wu X, Xie Y 2010 Phys. Rev. D 81 084045

    [45]

    Liu F Y, Wu X, Lu B K 2007 Chinese Astorn. Astrophys. 31 172

    [46]

    Murray C D, Dermott S F 1999 Solar System Dynamics. Cambridge Univ. Press, Cambridge, UK.

    [47]

    Zhao H J, Du M L 2007 Acta Phys.Sin. 56 3827 (in Chinese)[赵海军、杜孟利 2007 物理学报 56 3827]

    [48]

    Hartl M D, Buonanno A 2005 Phys. Rev. D 71 024027

    [49]

    Zhong S Y, Wu X 2010 Phys. Rev. D 81 104037

    [50]

    Wang Y, Wu X 2011 Class. Quantum Grav. 28 in press

    [51]

    Wu X, Huang T Y, Zhang H 2006 Phys. Rev. D 74 083001

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  • Received Date:  25 November 2010
  • Accepted Date:  21 December 2010
  • Published Online:  15 September 2011

Comparison of second-order mixed symplectic integrator between semi-implicit Euler method and implicit midpoint rule

  • 1. Department of Physics, Nanchang University, Nanchang 330031, China

Abstract: When a Hamiltonian can be split into integrable and nonintegrable parts, the former part is solved analytically, and the latter one is integrated numerically by means of implicit symplectic integrators such as the first-order semi-implicit Euler method or the second-order implicit midpoint rule. These analytical and numerical solutions are used to construct a second-order mixed symplectic integrator with the semi-implicit Euler method and one with the implicit midpoint rule. A theoretical analysis shows that the Euler mixed integrator is inferior to the midpoint one in the sense of numerical stability. Numerical simulations of the circularly-restricted three-body problem also support this fact. It is further shown through numerical integrations of the post-Newtonian Hamiltonian of spinning compact binaries that the qualities of the Euler mixed integrator and the midpoint mixed method do depend on the type of orbits. Especially for chaotic orbits, the Euler mixed integrator often becomes unstable. In addition, the Euler mixed integrator has an advantage over the midpoint mixed method in computational efficiency, and is almost equivalent to the latter in the numerical accuracy if the two mixed integrators are stable. In spite of this, the midpoint mixed integrator is worth recommending for the study of the dynamics of post-Newtonian Hamiltonians of spinning compact binaries.

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