Skip to main content

Abstract

In this chapter we consider symplectic Runge-Kutta (R-K) method.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 209.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. L. Abia and J.M. Sanz-Serna: Partitioned Runge-Kutta methods for separable Hamiltonian problems. Math. Comp., 60:617–634, (1993).

    Article  MathSciNet  MATH  Google Scholar 

  2. J.C. Butcher: The Numerical Analysis of Ordinary Differential Equations. JohnWiley, Chichester, (1987).

    MATH  Google Scholar 

  3. M.P. Calvo and E. Hairer: Further reduction in the number of independent order conditions for symplectic, explicit partitioned Runge-Kutta and Runge-Kutta-Nyström methods. Appl. Numer. Math., 18:107–114, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. A. Chin: Symplectic integrators from composite operator factorization. Physics Letters A, 226:344–348, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  5. G. J. Cooper: Stability of Runge-Kutta methods for trajectory problems. IMA J. Numer. Anal., 7:1–13, (1987).

    Article  MathSciNet  MATH  Google Scholar 

  6. M.P. Calvo and J.M. Sanz-Serna: High-order symplectic Runge-Kutta-Nyström methods. SIAM J. Sci. Comput., 114:1237–1252, (1993).

    Article  MathSciNet  Google Scholar 

  7. M.P. Calvo and J.M. Sanz-Serna: Canonical B-Series. Numer. Math., 67:161–175, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  8. K. Dekker and J.G. Verwer: Stability of Runge-Kutta Methods for Stiff Initial Value Problems. North-Holland, Amsterdam, (1984).

    Google Scholar 

  9. K. Feng: Difference schemes based on variational principle. J. of Appl. and Comput. Math.in Chinese, 2(4):238–262, (1965).

    Google Scholar 

  10. K. Feng: On difference schemes and symplectic geometry. In K. Feng, editor, Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, pages 42–58. Science Press, Beijing, (1985).

    Google Scholar 

  11. K. Feng: Canonical Difference Schemes for Hamiltonian Canonical Differential Equations. In International Workshop on Applied Differential Equations (Beijing, 1985), pages 59–73. World Sci. Publishing, Singapore, (1986).

    Google Scholar 

  12. K. Feng: Difference schemes for Hamiltonian formalism and symplectic geometry. J. Comput. Math., 4:279–289, (1986).

    MathSciNet  MATH  Google Scholar 

  13. K. Feng: Symplectic geometry and numerical methods in fluid dynamics. In F.G. Zhuang and Y.L. Zhu, editors, Tenth International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, pages 1–7. Springer, Berlin, (1986).

    Chapter  Google Scholar 

  14. K. Feng: The Hamiltonian Way for Computing Hamiltonian Dynamics. In R. Spigler, editor, Applied and Industrial Mathematics, pages 17–35. Kluwer, The Netherlands, (1991).

    Google Scholar 

  15. K. Feng: Formal power series and numerical methods for differential equations. In T. Chan and Z.C. Shi, editors, International conf. on scientific computation, pages 28–35. World Scientific, Singapore, (1992).

    Google Scholar 

  16. K. Feng: How to compute property Newton’s equation of motion. In L. A. Ying, B.Y. Guo, and I. Gladwell, editors, Proc of 2nd Conf. on Numerical Method for PDE’s, pages 15–22. World Scientific, Singapore, (1992). Also see Collected Works of Feng Kang. Volume I, II. National Defence Industry Press, Beijing, (1995).

    Google Scholar 

  17. K. Feng: Formal dynamical systems and numerical algorithms. In K. Feng and Z.C Shi, editors, International conf. on computation of differential equationsand dynamical systems, pages 1–10. World Scientific, Singapore, (1993).

    Google Scholar 

  18. K. Feng: Symplectic, contact and volume preserving algorithms. In Z.C. Shi and T. Ushijima, editors, Proc.1st China-Japan conf. on computation of differential equationsand dynamical systems, pages 1–28. World Scientific, Singapore, (1993).

    Google Scholar 

  19. K. Feng: Collected Works of Feng Kang. volume I,II. National Defence Industry Press, Beijing, (1995).

    Google Scholar 

  20. K. Feng: The calculus of generating functions and the formal energy for Hamiltonian systems. J. Comput. Math., 16:481–498, (1998).

    MathSciNet  MATH  Google Scholar 

  21. K. Feng: The step-transition operator for multi-step methods of ODEs. J. Comput. Math., 16(3), (1998).

    Google Scholar 

  22. K. Feng and M.Z. Qin: The symplectic methods for the computation of Hamiltonian equations. In Y. L. Zhu and B. Y. Guo, editors, Numerical Methods for Partial Differential Equations, Lecture Notes in Mathematics 1297, pages 1–37. Springer, Berlin, (1987).

    Chapter  Google Scholar 

  23. K. Feng and M.Z. Qin: Hamiltonian Algorithms for Hamiltonian Dynamical Systems. Progr. Natur. Sci., 1(2):105–116, (1991).

    MathSciNet  Google Scholar 

  24. K. Feng and M.Z. Qin: Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study. Comput. Phys. Comm., 65:173–187, (1991).

    Article  MathSciNet  MATH  Google Scholar 

  25. K. Feng and M. Q. Qin: Symplectic Algorithms for Hamiltonian Systems. Zhejiang Science and Technology Publishing House, Hangzhou, in Chinese, First edition, (2003).

    Google Scholar 

  26. K. Feng and D.L. Wang: A note on conservation laws of symplectic difference schemes for Hamiltonian systems. J. Comput. Math., 9(3):229–237, (1991).

    MathSciNet  MATH  Google Scholar 

  27. K. Feng and D.L. Wang: Symplectic difference schemes for Hamiltonian systems in general symplectic structure. J. Comput. Math., 9(1):86–96, (1991).

    MathSciNet  MATH  Google Scholar 

  28. K. Feng and D.L. Wang: Dynamical systems and geometric construction of algorithms. In Z. C. Shi and C. C. Yang, editors, Computational Mathematics in China, Contemporary Mathematics of AMS Vol 163, pages 1–32. AMS, (1994).

    Google Scholar 

  29. K. Feng and D.L. Wang: On variation of schemes by Euler. J. Comput. Math., 16:97–106, (1998).

    MathSciNet  MATH  Google Scholar 

  30. K. Feng, H.M. Wu, and M.Z. Qin: Symplectic difference schemes for linear Hamiltonian canonical systems. J. Comput. Math., 8(4):371–380, (1990).

    MathSciNet  MATH  Google Scholar 

  31. K. Feng, H.M. Wu, M.Z. Qin and D.L. Wang: Construction of canonical difference schemes for Hamiltonian formalism via generating functions. J. Comput. Math., 7:71–96, (1989).

    MathSciNet  MATH  Google Scholar 

  32. Z. Ge: Symplectic geometry and its application in numerical analysis. PhD thesis, Computer Center, CAS, (1988).

    Google Scholar 

  33. Z. Ge: Generating functions, Hamilton-Jacobi equations and symplectic groupoids on Poisson manifolds. Indiana Univ. Math. J., 39:859, (1990).

    Article  MathSciNet  MATH  Google Scholar 

  34. Z. Ge: Equivariant symplectic difference schemes and generating functions. Physica D, 49:376–386, (1991).

    Article  MathSciNet  MATH  Google Scholar 

  35. Z. Ge: Symplectic integrators for Hamiltonian systems. In W. Cai et al., editor, Numerical Methods in Applied Sciences, pages 97–108, Science Press, New York, (1995).

    Google Scholar 

  36. O. Gonzalez: Time integration and discrete Hamiltonian systems. J. Nonlinear. Sci., 6:449–467, (1996).

    Article  MathSciNet  MATH  Google Scholar 

  37. E. Hairer: Backward analysis of numerical integrators and symplectic methods. Annals of Numer. Math., 1:107–132, (1994).

    MathSciNet  MATH  Google Scholar 

  38. E. Hairer: Variable time step integration with symplectic methods. Appl. Numer. Math., 25:219–227, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  39. E. Hairer: Backward error analysis for multistep methods. Numer. Math., 84:199–232, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  40. E. Hairer: Symmetric projection methods for differential equations on manifolds. BIT, 40:726–734, (2000).

    Article  MathSciNet  MATH  Google Scholar 

  41. E. Hairer: Geometric integration of ordinary differential equations on manifolds. BIT, 41:996–1007, (2001).

    Article  MathSciNet  Google Scholar 

  42. E. Hairer: Global modified Hamiltonian for constrained symplectic integrators. Numer. Math., 95:325–336, (2003).

    Article  MathSciNet  MATH  Google Scholar 

  43. P. Henrici: Discrete Variable Methods in Ordinary Differential Equations. JohnWiley & Sons, Inc., New York, Second edition, (1962).

    MATH  Google Scholar 

  44. E. Hairer and P. Leone: Order barriers for symplectic multi-value methods. In D.F. Grifysis, D.F. Higham, and G.A. Watson, editors, Numerical Analysis 1997 Proc.of the 17-th Dundee Biennial Conference, June 24–27, 1997, Pitman Reserch Notes in math. series 380, pages 133–149, (1997).

    Google Scholar 

  45. E. Hairer and Ch. Lubich: The life-span of backward error analysis for numerical integrators. Numer. Math., 76:441–462, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Hochbruck and Ch. Lubich: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal., 34(5), (1997).

    Google Scholar 

  47. W. Huang and B. Leimkuhler: The adaptive Verlet method. SIAM J. Sci. Comput., 18(1):239, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  48. E. Hairer and Ch. Lubich: Invariant tori of dissipatively perturbed Hamiltonian systems under symplectic discretization. Appl. Numer. Math., 29:57–71, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Hochbruck and Ch. Lubich: Exponential integrators for quantum-classical molecular dynamics. BIT, 39:620–645, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  50. E. Hairer and P. Leone: Some properties of symplectic Runge-Kutta methods. New Zealand J. of Math., 29:169–175, (2000).

    MathSciNet  MATH  Google Scholar 

  51. E. Hairer and Ch. Lubich: Energy conservation by Störmer-type numerical integrators. In G.F. Griffiths and G.A. Watson, editors, In Numerical Analysis 1999, pages 169–190. CRC Press LLC, (2000).

    Google Scholar 

  52. E. Hairer and Ch. Lubich: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal., 38:414–441, (2000).

    Article  MathSciNet  MATH  Google Scholar 

  53. J. L. Hong and Y. Liu: Symplectic integration of linear discontinues Hamiltonian systems. Neural Parallel Sci Comput., 8:317–325, (2000).

    MathSciNet  MATH  Google Scholar 

  54. M. Hochbruck and C. Lubich: On magnus integrators for time-dependent Schrödinger equations. SIAM J. Numer. Anal., 41:945–963, (2003).

    Article  MathSciNet  MATH  Google Scholar 

  55. E. Hairer and C. Lubich: symmetric multistep methods over long times. Numer. Math., 97:699–723, (2004).

    Article  MathSciNet  MATH  Google Scholar 

  56. T. Holder, B. Leimkuhler, and S. Reich: Explicit variable step-size and time-reversible integration. Appl. Numer. Math., 39:367–377, (2001).

    Article  MathSciNet  MATH  Google Scholar 

  57. M. Hochbruck, C. Lubich, and H. Selhofer: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19(5):1552–1574, (1998).

    Article  MathSciNet  MATH  Google Scholar 

  58. E. Hairer, Ch. Lubich, and G. Wanner: Geometric Numerical Integration. Number 31 in Springer Series in Computational Mathematics. Springer-Verlag, (2002).

    Google Scholar 

  59. E. Hairer, C. Lubich and G. Wanner: Geometric integration illustrated by the Störmer-Verlet method. Acta Numerica, pages 399–450, (2003).

    Google Scholar 

  60. P. Hydon and E.L. Mansfield: A variational complex for difference equations. Foundations of Computational Mathematics, 4:187–217, (2004).

    Article  MathSciNet  MATH  Google Scholar 

  61. P. Hut, J. Makino and S. McMillan: Building a better leapfrog. Astrophys. J., 443:L93–L96, (1995).

    Article  Google Scholar 

  62. E. Hairer, A. Murua and J.M. Sanz-Serna: The non-existence of symplectic multiderivative Runge-Kutta methods. Preprint, (1993).

    Google Scholar 

  63. E. Hairer, S. P. Nørsett, and G. Wanner: Solving Ordinary Differential Equations I, Nonstiff Problems. Springer-Verlag, Second revised edition, (1993).

    Google Scholar 

  64. D.J. Hardy, D.I. Okunbor, and R.D. Skeel: Symplectic variable step size integration for n-body problems. Appl. Numer. Math., 29:19–30, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  65. W. H. Hundsdorfer and M. N. Spijker: A note on B-stability of Runge-Kutta methods. Numer. Math., 36:319–331, (1981).

    Article  MathSciNet  MATH  Google Scholar 

  66. A. R. Humphries and A. M. Stuart: Runge-Kutta methods for dissipative and gradient dynamical systems. SIAM J. Numer. Anal., 31(5):1452–1485, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  67. E. Hairer and D. Stoffer: Reversible long-term integration with variable stepsizes. SIAM J. Sci. Comput., 18:257–269, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  68. E. Hairer and G. Söderlind: Explicit time reversible adaptive step size control. SIAM J. Sci. Comput., 26:1838–1851, (2005).

    Article  MathSciNet  MATH  Google Scholar 

  69. E. Hairer and G. Wanner: On the Butcher group and general multivalue methods. Computing, 13:1–15, (1974).

    Article  MathSciNet  MATH  Google Scholar 

  70. E. Hairer and G. Wanner: Algebraically stable and implementable Runge-Kutta methods of high order. SIAM J. Numer. Anal., 18:1098–1108, (1981).

    Article  MathSciNet  MATH  Google Scholar 

  71. E. Hairer and G. Wanner: Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. Springer, Berlin, (1991).

    Book  MATH  Google Scholar 

  72. E. Hairer and G. Wanner: Symplectic Runge-Kutta methods with real eigenvalues. BIT, 34:310–312, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  73. E. Hairer and G. Wanner: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd edition, Springer Series in Computational Mathematics 14. Springer-Verlag Berlin, Second edition, (1996).

    Book  Google Scholar 

  74. T. Itoh and K. Abe: Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. of Comp. Phys., 76:85–102, (1988).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  76. L. O. Jay: Lagrangian integration with symplectic methods. Technical Report AHPCRC Preprint 97-009, University of Minnesota, (1997).

    Google Scholar 

  77. L. O. Jay: Structure preservation for constrained dynamics with super partitioned additive Runge-Kutta methods. SIAM J. Sci. Comput., 20(2):416–446, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  78. S. Jiménez: Derivation of the discrete conservation laws for a family of finite difference schemes. Applied Mathematics and Computation, 64:13–45, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  79. Z. Jia and B. Leimkuhler: Geometric integrators for multiple time-scale simulation. J. Phys. A: Math. Gen., 39:5379–5403, (2006).

    Article  MathSciNet  MATH  Google Scholar 

  80. B. Karasözen: Comparison of reversible integrators for a Hamiltonian in normal form. In E. Kreuzer and O. Mahrenholz, editors, Proceedings of the Third International Congress on Industrial and Applied Mathematics, ICIAM 95, Issue 4: Applied Sciences, especially Mechanics (Minisymposia), pages 563–566, (1996).

    Google Scholar 

  81. B. Karasözen: Composite integrators for Bi-Hamiltonian systems. Comp. & Math. with Applic., 32:79–86, (1996).

    Article  MATH  Google Scholar 

  82. B. Karasözen: Numerical Studies on a Bi-Hamiltonian Hénon-Heiles System. Technical Report No 133, Middle East Technical University, Department of Mathematics, Ankara, Turkey, (1996).

    Google Scholar 

  83. B. Karasözen: Reflexive methods for dynamical systems with conserved quantities. Technical Report Nr. 1897, Technische Hochschule Darmstadt, FB Mathematik, (1997).

    Google Scholar 

  84. L. H. Kong, J. L. Hong, and R. X. Liu: Long-term numerical simulation of the interaction between a neutron field and meson field by a symplectic-preserving scheme. J. Phys. A: Math. Theor., 41:255207, (2008).

    Article  MathSciNet  Google Scholar 

  85. U. Kirchgraber: Multi-step methods are essentially one-step methods. Numer. Math., 48:85–90, (1986).

    Article  MathSciNet  MATH  Google Scholar 

  86. J.D. Lambert: Numerical Methods for Ordinary Differential Equations, The Initial Value Problem. Wiley, Chichester, (1991).

    Google Scholar 

  87. F.M. Lasagni: Canonical Runge-Kutta methods. Z. Angew. Math. Phys., 39:952–953, (1988).

    Article  MathSciNet  MATH  Google Scholar 

  88. Y.X. Li, P. Z. Ding, M. X. Jin, and C. X. Wu: Computing classical trajectories of model molecule A 2 B by symplectic algorithm. Chemical Journal of Chinese Universities, 15(8):1181–1186, (2000).

    Google Scholar 

  89. B. J. Leimkuhler: Reversible adaptive regularization: Perturbed Kepler motion and classical atomic trajectories. Phil. Trans. Royal Soc. A, 357:1101, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  90. P. Leone: Symplecticity and Symmetry of General Integration Methods. Thèse, Section de Mathématiques, Université de Genève, Second edition, (2000).

    Google Scholar 

  91. B. J. Leimkuhler and G.W. Patrick: A symplectic integrator for Riemannian manifolds. J. Nonlinear. Sci., 6(4):367–384, (1996).

    Article  MathSciNet  MATH  Google Scholar 

  92. L. Lopez and T. Politi: Applications of the cayley approach in the numerical solution of matrix differential systems on quadratic groups. Appl. Numer. Math., 36:35–55, (2001).

    Article  MathSciNet  MATH  Google Scholar 

  93. H. W. Li and M. Z. Qin: On the formal energy of symplectic R-K method. Math. Num. Sinica, 23:75–92, (2001).

    MathSciNet  Google Scholar 

  94. X.S. Liu, Y.Y. Qi, J. F. He, and P. Z. Ding: Recent progress in symplectic algorithms for use in quantum systems. Communications in Computational Physics, 2(1):1–53, (2007).

    MathSciNet  Google Scholar 

  95. B. Leimkuhler and S. Reich: Symplectic integration of constrained Hamiltonian systems. Math. Comp., 63:589–605, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  96. B. Leimkuhler and S. Reich: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge, First edition, (2005).

    Book  Google Scholar 

  97. B. J. Leimkuhler and E. S. van Vleck: Orthosymplectic integration of linear Hamiltonian systems. Numer. Math., 77:269–282, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  98. J. D. Lambert and I. A. Watson: Symmetric multistep methods for periodic initial value problems. J. Inst. Maths. Applics., 18:189–202, (1976).

    Article  MathSciNet  MATH  Google Scholar 

  99. H. Liu, J.H. Yuan, J.B. Chen, H. Shou, and Y.M. Li: Theory of large-step depth extrapolation. Chinese journal Geophys., 49(6):1779–1793, (2006).

    Google Scholar 

  100. R. I. McLachlan: On the numerical integration of ODE’s by symmetric composition methods. SIAM J. Numer. Anal., 16:151–168, (1995).

    MathSciNet  MATH  Google Scholar 

  101. R. I. McLachlan: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput., 16:151–168, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  102. R. I. McLachlan: More on Symplectic Correctors. In Jerrold E. Marsden, George W. Patrick, and William F. Shadwick, editors, Integration Algorithms and Classical Mechanics, volume 10 of Fields Institute Communications. Fields Institute, American Mathematical Society, July (1996).

    Google Scholar 

  103. R. McLachlan: Splitting methods. Acta Numerica, 11:341–434, (2002).

    Article  MathSciNet  MATH  Google Scholar 

  104. S. Miesbach: Symplektische Phasenflu Kanonischer Differentialgleichungen. Master’s thesis, Technische Universität München, (1989).

    Google Scholar 

  105. S. Miesbach and H.J. Pesch: Symplectic phase flow approximation for the numerical integration of canonical systems. Numer. Math., 61:501–521, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  106. R.I. McLachlan, M. Perlmutter, and G.R.W. Quispel: On the nonlinear stability of symplectic integrators. BIT, 44:99–117, (2004).

    Article  MathSciNet  MATH  Google Scholar 

  107. R. I. McLachlan and G. R. W. Quispel: Generating functions for dynamical systems with symmetries, integrals, and differential invariants. Physica D, 112:298–309, (1998).

    Article  MathSciNet  MATH  Google Scholar 

  108. R.I. McLachlan and G.R.W. Quispel: Numerical integrators that preserve symmetries and reversing symmetries. SIAM J. Numer. Anal., 35:586–599, (1998).

    Article  MathSciNet  MATH  Google Scholar 

  109. R. I. McLachlan and G. R. W. Quispel: Splitting methods. Acta Numerica, 11:341–434, (2002).

    Article  MathSciNet  MATH  Google Scholar 

  110. R.I. McLachlan and G.R.W. Quispel: Geometric integration of conservative polynomial ODEs. Appl. Numer. Math., 45:411–418, (2003).

    Article  MathSciNet  MATH  Google Scholar 

  111. D.I. McLaren and G.R.W. Quispel: Integral-preserving integrators. J. Phys. A: Math. Gen., 37:L489–L495, (2004).

    Article  MathSciNet  MATH  Google Scholar 

  112. R. I. McLachlan, G. R.W. Quispel, and N. Robidoux: A unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions and/or first integrals. Physical Review Letters, 81:2399–2403, (1998).

    Article  MathSciNet  MATH  Google Scholar 

  113. R. I. McLachlan, G. R. W. Quispel, and N. Robidoux: Geometric integration using discrete gradients. Phil. Trans. Royal Soc. A, 357:1021–1046, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  114. R. I. McLachlan, G. R. W. Quispel, and G. S. Turner: Numerical integrators that preserve symmetries and reversing symmetries. SIAM J. Numer. Anal., 35(2):586–599, (1998).

    Article  MathSciNet  MATH  Google Scholar 

  115. R. I. McLachlan and C. Scovel: Equivariant constrained symplectic integration. J. Nonlinear. Sci., 5:233–256, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  116. R. I. McLachlan and C. Scovel: A Survey of Open Problems in Symplectic Integration. In J. E. Mardsen, G. W. Patrick, and W. F. Shadwick, editors, Integration Algorithms and Classical Mechanics, pages 151–180. American Mathematical Society, (1996).

    Google Scholar 

  117. A. Murua: On order conditions for partitioned symplectic methods. SIAM J. Numer. Anal., 34:2204–2211, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  118. A. Murua: Formal series and numerical integrators, part I: Systems of odes and symplectic integrators. Appl. Numer. Math., 29:221–251, (1999).

    Article  MathSciNet  MATH  Google Scholar 

  119. N. Obreschkoff: Neue Quadraturformeln. Abhandlungen pröß Klasse Acad Wiss Mathnatuwiss, 1–20.(1940).

    Google Scholar 

  120. D. Okunbor: Variable step size does not harm second-order integrators for Hamiltonian systems. J. Comput. Appl. Math, 47:273–279, (1993).

    Article  MATH  Google Scholar 

  121. E. I. Okunbor: Energy conserving, Liouville, and symplectic integrators. J. of Comp. Phys., 120(2):375–378, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  122. D. Okunbor and R.D. Skeel: Explicit canonical methods for Hamiltonian systems. Math. Comp., 59:439–455, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  123. G. R. W. Quispel and C. Dyt: Solving ODE’s numerically while preserving symmetries, Hamiltonian structure, phase space volume, or first integrals. In A. Sydow, editor, Proceedings of the 15th IMACS World Congress, pages 601–607. Wissenschaft & Technik, Berlin, (1997).

    Google Scholar 

  124. M. Z. Qin: A symplectic schemes for the Hamiltonian equations. J. Comput. Math., 5:203–209, (1987).

    MathSciNet  MATH  Google Scholar 

  125. M. Z. Qin: Cononical difference scheme for the Hamiltonian equation. Mathematical Methodsand in the Applied Sciences, 11:543–557, (1989).

    Article  MATH  Google Scholar 

  126. M. Z. Qin: Multi-stage symplectic schemes of two kinds of Hamiltonian systems of wave equations. Computers Math. Applic., 19:51–62, (1990).

    MATH  Google Scholar 

  127. M. Z. Qin: Symplectic difference schemes for nonautonomous Hamiltonian systemes. Acta Applicandae Mathematicae, 12(3):309–321, (1996).

    Google Scholar 

  128. M. Z. Qin: A symplectic schemes for the PDE’s. AMS/IP studies in Advanced Mathemateics, 5:349–354, (1997).

    Google Scholar 

  129. G. D. Quinlan and S. Tremaine: Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J., 100:1694–1700, (1990).

    Article  Google Scholar 

  130. G.D. Quinlan and S. Tremaine: Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J., 100:1694–1700, (1990).

    Article  Google Scholar 

  131. G. R. W. Quispel and G. S. Turner: Discrete gradient methods for solving ODE’s numerically while preserving a first integral. Physics Letters A, 29:L341–L349, (1996).

    MathSciNet  MATH  Google Scholar 

  132. M. Z. Qin, D. L. Wang, and M. Q. Zhang: Explicit symplectic difference schemes for separable Hamiltonian systems. J. Comput. Math., 9(3):211–221, (1991).

    MathSciNet  MATH  Google Scholar 

  133. M. Z. Qin and M. Q. Zhang: Explicit Runge-Kutta-like schemes to solve certain quantum operator equations of motion. J. Stat. Phys., 60(5/6):839–843, (1990).

    MathSciNet  MATH  Google Scholar 

  134. M. Z. Qin andW. J. Zhu: Canonical Runge-Kutta-Nyström(RKN) methods for second order ode’s. Computers Math. Applic., 22:85–95, (1991).

    MATH  Google Scholar 

  135. M. Z. Qin and M. Q. Zhang: Symplectic Runge-Kutta Schemes for Hamiltonian System. J. Comput. Math., Supplementary Issue: pages 205–215, (1992).

    Google Scholar 

  136. M. Z. Qin and W. J. Zhu: Construction of higher order symplectic schemes by composition. Computing, 47:309–321, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  137. M. Z. Qin and W. J. Zhu: Multiplicative extrapolatio method for constructing higher order schemes for ODE’s. J. Comput. Math., 12:352–356, (1994).

    MathSciNet  MATH  Google Scholar 

  138. M. Z. Qin, W. J. Zhu, and M. Q. Zhang: Construction of symplectic of a three stage difference scheme for ODE’s. J. Comput. Math., 13:206–210, (1995).

    MathSciNet  MATH  Google Scholar 

  139. S. Reich: Momentum conserving symplectic integrators. Physica D, 76:375–383, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  140. S. Reich: Smoothed dynamics of highly oscillatory Hamiltonian systems. Physica D, 89:28–42, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  141. S. Reich: Enhancing energy conserving methods. BIT, 1:122–134, (1996).

    Article  Google Scholar 

  142. S. Reich: Symplectic integration of constrained Hamiltonian systems by composition methods. SIAM J. Numer. Anal., 33:475–491, (1996).

    Article  MathSciNet  MATH  Google Scholar 

  143. S. Reich: Symplectic Methods for Conservative Multibody Systems. In J. E. Mardsen, G. W. Patrick, and W. F. Shadwick, editors, Integration Algorithms and Classical Mechanics, pages 181–192. American Mathematical Society, (1996).

    Google Scholar 

  144. S. Reich: On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems. Numer. Math., 76(2):249–263, (1997).

    Article  MathSciNet  Google Scholar 

  145. S. Reich: Backward error analysis for numerical integrators. SIAM J. Numer. Anal., 36:475–491, (1999).

    Article  MathSciNet  Google Scholar 

  146. R. Ruth: A canonical integration technique. IEEE Trans. Nucl. Sci., 30:26–69, (1983).

    Article  Google Scholar 

  147. J.M. Sanz-Serna and L. Abia: Order conditions for canonical Runge-Kutta schemes. SIAM J. Numer. Anal., 28:1081–1096, (1991).

    Article  MathSciNet  MATH  Google Scholar 

  148. J. M. Sanz-Serna: Runge-Kutta schemes for Hamiltonian systems. BIT, 28:877–883, (1988).

    Article  MathSciNet  MATH  Google Scholar 

  149. J. M. Sanz-Serna and M. P. Calvo: Numerical Hamiltonian Problems. AMMC 7. Chapman & Hall, (1994).

    Google Scholar 

  150. S. Saito, H. Sugiura, and T. Mitsui: Family of symplectic implicit Runge-Kutta formulae. BIT, 32:539–543, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  151. S. Saito, H. Sugiura, and T. Mitsui: Butcher’s simplifying assumption for symplectic integrators. BIT, 32:345–349, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  152. D.M. Stoffer: Variable step size destabilizes the Stöormer/leap-frog/Verlet method. BIT, 33:172–175, (1993).

    Article  MathSciNet  Google Scholar 

  153. D. Stoffer: Variable steps for reversible integration methods. Computing, 55:1–22, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  154. D. Stoffer: On the Qualitative Behaviour of Symplectic Integrators Part I: Perturbed Linear Systems. Numer. Math., 77(4):535–548, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  155. D. Stoffer: On the gualitative behavior of symplectic integrator. II: Integrable systems. J. of Math. Anal. and Applic., 217:501–520, (1998).

    Article  MathSciNet  MATH  Google Scholar 

  156. D. Stoffer: On the qualitative behaviour of symplectic integrators. III: Perturbed integrable systems. J. of Math. Anal. and Appl., 217:521–545, (1998).

    Article  MathSciNet  MATH  Google Scholar 

  157. G. Sun: Construction of high order symplectic Runge-Kutta methods. J. Comput. Math., 11(3):250–260, (1993).

    MathSciNet  MATH  Google Scholar 

  158. G. Sun: Symplectic partitioned Runge-Kutta methods. J. Comput. Math., 11:365–372, (1993).

    MathSciNet  MATH  Google Scholar 

  159. G. Sun: Characterization and construction of linear symplectic R-K methods. J. Comput. Math., 12(2):101–112, (1994).

    MathSciNet  MATH  Google Scholar 

  160. G. Sun: Construction of high order symplectic Partitioned-Runge-Kutta methods. J. Comput. Math., 13(1):40–50, (1995).

    MathSciNet  MATH  Google Scholar 

  161. G. Sun: A simple way constructing symplectic Runge-Kutta methods. J. Comput. Math., 18:61–68, (2000).

    MathSciNet  MATH  Google Scholar 

  162. Y.B. Suris: On the conservation of the symplectic structure in the numerical solution of Hamiltonian systems(in Russian), In: Numerical Solution of Ordinary Differential Equations, ed. S.S. Filippov, Keldysh Institute of Applied Mathematics. USSR Academy of Sciences, Moscow, Second edition, (1988).

    Google Scholar 

  163. Y.B. Suris: The canonicity of mappings generated by Runge-Kutta type methods when integrating the systems \(\ddot x = - \frac{{\partial U}}{{\partial x}}.\). U.S.S.R. Comput. Maths. Math. Phys., 29:138–144, (1989).

    Article  MathSciNet  MATH  Google Scholar 

  164. Y.B. Suris: Hamiltonian methods of Runge-Kutta type and their variational interpretation. Math. Model., 2:78–87, in Russian, (1990).

    MathSciNet  MATH  Google Scholar 

  165. Y. F. Tang: The symplecticity of multi-step methods. Computers Math. Applic., 25:83–90, (1993).

    Article  MATH  Google Scholar 

  166. Y. F. Tang: The necessary condition for Runge-Kutta schemes to be symplectic for Hamiltonian systems. Computers Math. Applic., 26:13–20, (1993).

    Article  MATH  Google Scholar 

  167. Y. F. Tang: Formal energy of a symplectic scheme for Hamiltonian systems and its applications. Computers Math. Applic., 27:31–39, (1994).

    Google Scholar 

  168. R. de Vogelaere: Methods of integration which preserve the contact transformation property of the Hamiltonian equations. Report No. 4, Dept. Math., Univ. of Notre Dame, Notre Dame, Ind., Second edition, (1956).

    Google Scholar 

  169. D. L. Wang: Semi-discrete Fourier spectral approximations of infinite dimensional Hamiltonian systems and conservations laws. Computers Math. Applic., 21:63–75, (1991).

    Article  MATH  Google Scholar 

  170. D. L. Wang: Symplectic difference schemes for Hamiltonian systems on Poisson manifolds. J. Comput. Math., 9(2):115–124, (1991).

    MathSciNet  MATH  Google Scholar 

  171. D. L. Wang: Poisson difference schemes for Hamiltonian systems on Poisson manifolds. J. Comput. Math., 9:115–124, (1991).

    MathSciNet  MATH  Google Scholar 

  172. D. L.Wang: Decomposition vector fields and composition of algorithms. In Proceedings of International Conference on computation of differential equations and dynamical systems, Beijing, 1993. World Scientific, (1993).

    Google Scholar 

  173. D. L. Wang: Some acpects of Hamiltonian systems and symplectic defference methods. Physica D, 73:1–16, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  174. H. Yoshida: Conserved quantities of symplectic integrators for Hamiltonian systems. Preprint, (1990).

    Google Scholar 

  175. M. Q. Zhang and M. Z. Qin: Explicit symplectic schemes to solve vortex systems. Comp. & Math. with Applic., 26(5):51, (1993).

    Article  MathSciNet  MATH  Google Scholar 

  176. W. Zhu and M. Qin: Applicatin of higer order self-adjoint schemes of PDE’s. Computers Math. Applic., 26(3):15–26, (1993).

    Article  MathSciNet  MATH  Google Scholar 

  177. W. Zhu and M. Qin: Constructing higer order schemes by formal power series. Computers Math. Applic., 25(12):31–38, (1993).

    Article  MathSciNet  MATH  Google Scholar 

  178. W. Zhu and M. Qin: Order conditionof two kinds of canonical difference schemes. Computers Math. Applic., 25(6):61–74, (1993).

    Article  MathSciNet  MATH  Google Scholar 

  179. W. Zhu and M. Qin: Poisson schemes for Hamiltonian systems on Poisson manifolds. Computers Math. Applic., 27:7–16, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  180. W. Zhu and M. Qin: Reply to “comment on Poisson schemes for Hamiltonian systems on Poisson manifolds”. Computers Math. Applic., 29(7):1, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  181. W. Zhu and M. Qin: Simplified order conditions of some canonical difference schemes. J. Comput. Math., 13(1):1–19, (1995).

    MathSciNet  MATH  Google Scholar 

  182. K. Zare and V. Szebehely: Time transformations in the extended phase-space. Celest. Mech., 11:469–482, (1975).

    Article  MathSciNet  MATH  Google Scholar 

  183. M. Q. Zhang and R. D. Skeel: Symplectic integrators and the conservation of angular momentum. J. Comput. Chem., 16:365–369, (1995).

    Article  Google Scholar 

  184. M. Q. Zhang and R. D. Skeel: Cheap implicit symplectic integrators. Appl. Numer. Math., 25(2):297, (1997).

    Article  MathSciNet  MATH  Google Scholar 

  185. H. P. Zhu and J. K. Wu: Generalized canonical transformations and symplectic algorithm of the autonomous Birkhoffian systems. Progr. Natur. Sci., 9:820–828, (1999).

    MathSciNet  Google Scholar 

  186. W. Zhu, X. zhao, and Y Tang: Numerical methods with a high order of accuracy applied in the quantum system. J. Chem. Phys., 104(6):2275–2286, (1996).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Feng, K., Qin, M. (2010). Symplectic Runge-Kutta Methods. In: Symplectic Geometric Algorithms for Hamiltonian Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01777-3_8

Download citation

Publish with us

Policies and ethics