Skip to main content

Symplectic integrators for Hamiltonian problems: an overview

  • J. M. Sanz-Serna (a1)

In the sciences, situations where dissipation is not significant may invariably be modelled by Hamiltonian systems of ordinary, or partial, differential equations. Symplectic integrators are numerical methods specifically aimed at advancing in time the solution of Hamiltonian systems. Roughly speaking, ‘symplecticness’ is a characteristic property possessed by the solutions of Hamiltonian problems. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions which inherit the property of symplecticness.

Hide All
Abia L. and Sanz-Serna J.M. (1990), ‘Partitioned Runge–Kutta methods for separable Hamiltonian problems’, Applied Mathematics and Computation Reports, Report 1990/8, Universidad de Valladolid.
Aizu K. (1985), ‘Canonical transformation invariance and linear multistep formula for integration of Hamiltonian systems’, J. Comput. Phys. 58, 270274.
Anosov D.V. and Arnold V.I. (eds) (1988), Dynamical Systems I, Springer (Berlin).
Arnold V.I. (ed.) (1988), Dynamical Systems III, Springer (Berlin).
Arnold V.I. (1989), Mathematical Methods of Classical Mechanics, 2nd edition, Springer (New York).
Arnold V.I. and Novikov S.P. (1990), Dynamical Systems IV, Springer, Berlin.
Beyn W.-J. (1991), ‘Numerical methods for dynamical systems’, in Advances in Numerical Analysis, Vol. I (Light W., ed.) Clarendon Press (Oxford) 175236.
Burrage K. and Butcher J.C. (1979), ‘Stability criteria for implicit Runge-Kutta methods’, SIAM J. Numer. Anal. 16, 4657.
Butcher J.C. (1976), ‘On the implementation of implicit Runge–Kutta methods’, BIT 16, 237240.
Butcher J.C. (1987), The Numerical Analysis of Ordinary Differential Equations, John Wiley (Chichester).
Calvo M.P. (1991), Ph.D. Thesis, University of Valladolid (to appear).
Calvo M.P. and Sanz-Serna J.M. (1991a), ‘Order conditions for canonical Runge–Kutta–Nyström methods’, BIT to appear.
Calvo M.P. and Sanz-Serna J.M. (1991b), ‘Variable steps for symplectic integrators’, Applied Mathematics and Computation Reports, Report 1991/3, Universidad de Valladolid.
Calvo M.P. and Sanz-Serna J.M. (1991c), ‘Reasons for a failure. The integration of the two-body problem with a symplectic Runge–Kutta–Nyström code with step-changing facilities’, Applied Mathematics and Computation Reports, Report 1991/7, Universidad de Valladolid.
Candy J. and Rozmus W. (1991), ‘A symplectic integration algorithm for separable Hamiltonian functions’, J. Comput. Phys. 92, 230256.
Channell P. J., ‘Symplectic integration algorithms’, Los Alamos National Laboratory Report, Report AT-6ATN 83–9.
Channell P.J. and Scovel C. (1990), ‘Symplectic integration of Hamiltonian systems’, Nonlinearity 3, 231259.
Cooper G.J. (1987), ‘Stability of Runge–Kutta methods for trajectory problems’, IMA J. Numer. Anal. 7, 113.
Cooper G.J. and Vignesvaran R. (1990), ‘A scheme for the implementation of implicit Runge-Kutta methods’, Computing 45, 321332.
Crouzeix M. (1979), ‘Sur la B-stabilité des méthodes de Runge–Kutta’, Numer. Math. 32, 7582.
Dekker K. and Verwer J.G. (1984), Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North Holland (Amsterdam).
Dormand J.R., El-Mikkawy M.E.A. and Prince P.J. (1987), ‘Families of Runge–Kutta–Nyström formulae’, IMA J. Numer. Anal. 7, 235250.
Eirola T. and Sanz-Serna J.M. (1990), ‘Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods’, Applied Mathematics and Computation Reports, Report 1990/9, Universidad de Valladolid.
Kang Feng (1985), ‘On difference schemes and symplectic geometry’, in Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations (Kang Feng, ed.) Science Press (Beijing) 4258.
Kang Feng (1986a), ‘Difference schemes for Hamiltonian formalism and symplectic geometry’, J. Comput. Math. 4, 279289.
Kang Feng (1986b), ‘Symplectic geometry and numerical methods in fluid dynamics’, in Tenth International Conference on Numerical Methods in Fluid Dynamics (Lecture Notes in Physics 264) (Zhuang F. G. and Zhu Y. L., eds) Springer (Berlin) 17.
Kang Feng and Meng-zhao Qin (1987), ‘The symplectic methods for the computation of Hamiltonian equations’, in Proceedings of the 1st Chinese Conference for Numerical Methods for PDE's, March 1986, Shanghai (Lecture Notes in Mathematics 1297) (You-lan Zhu and Ben-yu Gou, eds) Springer (Berlin) 137.
Kang Feng, Hua-mo Wu and Meng-zhao Qin (1990), ‘Symplectic difference schemes for linear Hamiltonian canonical systems’, J. Comput. Math. 8, 371380.
Kang Feng, Hua-mo Wu, Meng-zhao Qin and Dao-liu Wang (1989), ‘Construction of canonical difference schemes for Hamiltonian formalism via generating functions’, J. Comput. Math. 7, 7196.
Forest E. and Ruth R. (1990), ‘Fourth-order symplectic integration’, Physica D 43, 105117.
de Frutos J. and Sanz-Serna J.M. (1991), ‘An easily implementable fourth-order method for the time integration of wave problems’, Applied Mathematics and Computation Reports, Report 1991/2, Universidad de Valladolid.
de Frutos J., Ortega T. and Sanz-Serna J.M. (1990), ‘A Hamiltonian, explicit algorithm with spectral accuracy for the “good” Boussinesq system’, Comput. Methods Appl. Mech. Engrg. 80, 417423.
Zhong Ge and Marsden J.E. (1988), ‘Lie–Poisson Hamilton–Jacobi theory and Lie-Poisson integrators’, Phys. Lett. A 133, 134139.
Griffiths D.F. and Sanz-Serna J.M. (1986), ‘On the scope of the method of modified equations‘, SIAM J. Sci. Stat. Comput. 7, 9941008.
Guckenheimer J. and Holmes P. (1983), Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer (New York).
Hairer E., Iserles A. and Sanz-Serna J.M. (1990), ‘Equilibria of Runge–Kutta methods’, Numer. Math. 58, 243254.
Hairer E., Nørsett S.P. and Wanner G. (1987), Solving Ordinary Differential Equations I, Nonstiff Problems, Springer (Berlin).
Iserles A. (1990a), ‘Stability and dynamics of numerical methods for ordinary differential equations’, IMA J. Numer. Anal. 10, 130.
Iserles A. (1990b), ‘Efficient Runge–Kutta methods for Hamiltonian equations’, Numerical Analysis Report DAMTP, Report 1990/NA10, University of Cambridge.
Iserles A. and Nørsett S.P. (1991), Order Stars, Chapman and Hall (London).
Klein F. (1926), Vorlesungen Über die Entwicklung der Mathematik im 19.Jahrhundert, Teil I, Springer (Berlin).
Lasagni F. (1988), ‘Canonical Runge–Kutta methods’, ZAMP 39, 952953.
Lasagni F. (1990), ‘Integration methods for Hamiltonian differential equations’, unpublished manuscript.
Chun-wang Li and Meng-zhao Qin (1988), ‘A symplectic difference scheme for the infinite dimensional Hamiltonian system’, J. Comput. Math. 6, 164174.
MacKay R.S. (1991), ‘Some aspects of the dynamics and numerics of Hamiltonian systems’, in Proceedings of the ‘Dynamics of Numerics and Numerics of Dynamics’ Conference (Broomhead D.S. and Iserles A., eds) Oxford University Press (Oxford) to appear.
MacKay R.S. and Meiss J.D. (eds) (1987), Hamiltonian Dynamical Systems, Adam Hilger (Bristol).
McLachlan R. and Atela P. (1991), ‘The accuracy of symplectic integrators’, Program in Applied Mathematics, Report PAM 76, University of Colorado.
Menyuk C.R. (1984), ‘Some properties of the discrete Hamiltonian method’, Physica D 11, 109129.
Miesbach S. and Pesch H.J. (1990), ‘Symplectic phase flow approximation for the numerical integration of canonical systems’, Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung und Steuerung, Report 233, Technische Universität München.
Moser J. (1973), Stable and Random Motions in Dynamical Systems, Princeton University Press (Princeton).
Neishtadt A.I. (1984), ‘The separation of motions in systems with rapidly rotating phase’, J. Appl. Math. Mech 48, 133139.
Neri F. (1987), ‘Lie algebras and canonical integration’, University of Maryland Technical Report.
Okunbor D. and Skeel R.D. (1990), ‘An explicit Runge–Kutta–Nyström method is canonical if and only if its adjoint is explicit’, Working Document 90–3, Department of Computer Science, University of Illinois at Urbana-Champaign.
Okunbor D. and Skeel R.D. (1991), ‘Explicit canonical methods for Hamiltonian systems’, Working Document 91–1, Department of Computer Science, University of Illinois at Urbana-Champaign.
Pullin D.I. and Saffman P.G. (1991), ‘Long-tune symplectic integration: the example of four-vortex motion’, Proc. Roy. Soc. Lond. A 432, 481494.
Meng-zhao Qin (1988), ‘Leap-frog schemes of two kinds of Hamiltonian systems for wave equations’, Math. Num. Sin. 10, 272281.
Meng-zhao Qin and Mei-qing Zhang (1990), ‘Multi-stage symplectic schemes of two kinds of Hamiltonian systems for wave equations’, Comput. Math. Appl. 19, 5162.
Ruth R.D. (1983), ‘A canonical integration techniqe’, IEEE Trans. Nucl. Sci. 30, 26692671.
Sanz-Serna J.M. (1988), ‘Runge–Kutta schemes for Hamiltonian systems’, BIT 28, 877883.
Sanz-Serna J.M. (1989) ‘The numerical integration of Hamiltonian systems’, in Proc. 1989 London Numerical ODE Conference, Oxford University Press (Oxford) to appear.
Sanz-Serna J.M. (1990), ‘Numerical ordinary differential equations vs. dynamical systems’, Applied Mathematics and Computation Reports, Report 1990/3, Universidad de Valladolid.
Sanz-Serna J.M. (1991a), ‘Two topics in nonlinear stability’, in Advances in Numerical Analysis, Vol. I (Light W., ed.) Clarendon Press (Oxford) 147174.
Sanz-Serna J.M. (1991b), ‘Symplectic Runge–Kutta and related methods: recent results’, Applied Mathematics and Computation Reports, Report 1991/5, Universidad de Valladolid.
Sanz-Serna J.M. and Abia L. (1991), ‘Order conditions for canonical Runge-Kutta schemes’, SIAM J. Numer. Anal. 28, 10811096.
Sanz-Serna J.M. and Griffiths D.F. (1991), ‘A new class of results for the algebraic equations of implicit Runge-Kutta processes’, IMA J. Numer. Anal, to appear.
Sanz-Serna J.M. and Vadillo F. (1986), ‘Nonlinear instability, the dynamic approach’, in Numerical Analysis (Griffiths D.F. and Watson G.A., eds) Longman (London) 187199.
Sanz-Serna J.M. and Vadillo F. (1987), ‘Studies in nonlinear instability III: augmented Hamiltonian problems’, SIAM J. Appl. Math. 47, 92108.
Suris Y.B. (1989), ‘Canonical transformations generated by methods of Runge-Kutta type for the numerical integration of the system x″ = −∂UxZh. Vychisl. Mat. i Mat. Fiz. 29, 202211 (in Russian). An English translation by D.V. Bobyshev, edited by R.D. Skeel, is available as Working Document 91–2, Numerical Computing Group, Department of Computer Science, University of Illinois at Urbana-Champaign.
Suris Y.B. (1990), ‘Hamiltonian Runge–Kutta type methods and their variational formulation’, Math. Sim. 2, 7887 (in Russian).
Warming R.F. and Hyett B.J. (1974), ‘The modified equation approach to the stability and accuracy analysis of finite difference methods’, J. Comput. Phys. 14, 159179.
Yoshida H. (1990), ‘Construction of higher order symplectic integrators’, Phys. Lett. A 150, 262268.
Yuhua Wu (1988), ‘The generating function for the solution of ODE's and its discrete methods’, Comput. Math. Appl. 15, 10411050.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 101 *
Loading metrics...

Abstract views

Total abstract views: 523 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd November 2017. This data will be updated every 24 hours.