Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 108
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Bogfjellmo, Geir Curry, Charles and Manchon, Dominique 2016. Hamiltonian B-series and a Lie algebra of non-rooted trees. Numerische Mathematik,

    Chen, Zhaoxia Zhang, Ruqiang Shi, Wei and You, Xiong 2016. New optimized symmetric and symplectic trigonometrically fitted RKN methods for second-order oscillatory differential equations. International Journal of Computer Mathematics, p. 1.

    Mei, Lijie and Wu, Xinyuan 2016. The construction of arbitrary order ERKN methods based on group theory for solving oscillatory Hamiltonian systems with applications. Journal of Computational Physics,

    Wang, Bin Iserles, Arieh and Wu, Xinyuan 2016. Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems. Foundations of Computational Mathematics, Vol. 16, Issue. 1, p. 151.

    2016. A Concise Introduction to Geometric Numerical Integration.

    Butcher, J. C. 2015. The cohesiveness of G-symplectic methods. Numerical Algorithms, Vol. 70, Issue. 3, p. 607.

    Butcher, John C. and Imran, Gulshad 2015. Order conditions for G-symplectic methods. BIT Numerical Mathematics, Vol. 55, Issue. 4, p. 927.

    Wang, Lijin Hong, Jialin and Sun, Liying 2015. Modified equations for weakly convergent stochastic symplectic schemes via their generating functions. BIT Numerical Mathematics,

    Butcher, John C. Habib, Yousaf Hill, Adrian T. and Norton, Terence J. T. 2014. The Control of Parasitism in $G$-symplectic Methods. SIAM Journal on Numerical Analysis, Vol. 52, Issue. 5, p. 2440.

    Franco, J.M. and Gómez, I. 2014. Symplectic explicit methods of Runge–Kutta–Nyström type for solving perturbed oscillators. Journal of Computational and Applied Mathematics, Vol. 260, p. 482.

    Peinado, Jesús Alonso, Pedro Ibáñez, Javier Hernández, Vicente and Boratto, Murilo 2014. Solving time-invariant differential matrix Riccati equations using GPGPU computing. The Journal of Supercomputing, Vol. 70, Issue. 2, p. 623.

    Toxvaerd, Søren 2014. Discrete dynamics versus analytic dynamics. The Journal of Chemical Physics, Vol. 140, Issue. 4, p. 044102.

    Wang, Bin and Wu, Xinyuan 2014. A highly accurate explicit symplectic ERKN method for multi-frequency and multidimensional oscillatory Hamiltonian systems. Numerical Algorithms, Vol. 65, Issue. 3, p. 705.

    Zhao, Shan and Wei, G. W. 2014. A unified discontinuous Galerkin framework for time integration. Mathematical Methods in the Applied Sciences, Vol. 37, Issue. 7, p. 1042.

    Franco, J.M. and Gómez, I. 2013. Some procedures for the construction of high-order exponentially fitted Runge–Kutta–Nyström methods of explicit type. Computer Physics Communications, Vol. 184, Issue. 4, p. 1310.

    Franco, J.M. and Gómez, I. 2013. Construction of explicit symmetric and symplectic methods of Runge–Kutta–Nyström type for solving perturbed oscillators. Applied Mathematics and Computation, Vol. 219, Issue. 9, p. 4637.

    Liu, Kai Shi, Wei and Wu, Xinyuan 2013. An extended discrete gradient formula for oscillatory Hamiltonian systems. Journal of Physics A: Mathematical and Theoretical, Vol. 46, Issue. 16, p. 165203.

    Sivak, David A. Chodera, John D. and Crooks, Gavin E. 2013. Using Nonequilibrium Fluctuation Theorems to Understand and Correct Errors in Equilibrium and Nonequilibrium Simulations of Discrete Langevin Dynamics. Physical Review X, Vol. 3, Issue. 1,

    Toxvaerd, So̸ren 2013. Ensemble simulations with discrete classical dynamics. The Journal of Chemical Physics, Vol. 139, Issue. 22, p. 224106.

    Wu, Xinyuan Wang, Bin and Shi, Wei 2013. Efficient energy-preserving integrators for oscillatory Hamiltonian systems. Journal of Computational Physics, Vol. 235, p. 587.

  • Acta Numerica, Volume 1
  • 1992, pp. 243-286

Symplectic integrators for Hamiltonian problems: an overview

  • J. M. Sanz-Serna (a1)
  • DOI:
  • Published online: 01 November 2008

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

K. Aizu (1985), ‘Canonical transformation invariance and linear multistep formula for integration of Hamiltonian systems’, J. Comput. Phys. 58, 270274.

V.I. Arnold and S.P. Novikov (1990), Dynamical Systems IV, Springer, Berlin.

K. Burrage and J.C. Butcher (1979), ‘Stability criteria for implicit Runge-Kutta methods’, SIAM J. Numer. Anal. 16, 4657.

J.C. Butcher (1976), ‘On the implementation of implicit Runge–Kutta methods’, BIT 16, 237240.

J. Candy and W. Rozmus (1991), ‘A symplectic integration algorithm for separable Hamiltonian functions’, J. Comput. Phys. 92, 230256.

P.J. Channell and C. Scovel (1990), ‘Symplectic integration of Hamiltonian systems’, Nonlinearity 3, 231259.

G.J. Cooper (1987), ‘Stability of Runge–Kutta methods for trajectory problems’, IMA J. Numer. Anal. 7, 113.

G.J. Cooper and R. Vignesvaran (1990), ‘A scheme for the implementation of implicit Runge-Kutta methods’, Computing 45, 321332.

M. Crouzeix (1979), ‘Sur la B-stabilité des méthodes de Runge–Kutta’, Numer. Math. 32, 7582.

J.R. Dormand , M.E.A. El-Mikkawy and P.J. Prince (1987), ‘Families of Runge–Kutta–Nyström formulae’, IMA J. Numer. Anal. 7, 235250.

E. Forest and R. Ruth (1990), ‘Fourth-order symplectic integration’, Physica D 43, 105117.

J. de Frutos , T. Ortega and J.M. Sanz-Serna (1990), ‘A Hamiltonian, explicit algorithm with spectral accuracy for the “good” Boussinesq system’, Comput. Methods Appl. Mech. Engrg. 80, 417423.

Ge Zhong and J.E. Marsden (1988), ‘Lie–Poisson Hamilton–Jacobi theory and Lie-Poisson integrators’, Phys. Lett. A 133, 134139.

D.F. Griffiths and J.M. Sanz-Serna (1986), ‘On the scope of the method of modified equations‘, SIAM J. Sci. Stat. Comput. 7, 9941008.

J. Guckenheimer and P. Holmes (1983), Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer (New York).

E. Hairer , A. Iserles and J.M. Sanz-Serna (1990), ‘Equilibria of Runge–Kutta methods’, Numer. Math. 58, 243254.

E. Hairer , S.P. Nørsett and G. Wanner (1987), Solving Ordinary Differential Equations I, Nonstiff Problems, Springer (Berlin).

A. Iserles (1990a), ‘Stability and dynamics of numerical methods for ordinary differential equations’, IMA J. Numer. Anal. 10, 130.

A. Iserles (1990b), ‘Efficient Runge–Kutta methods for Hamiltonian equations’, Numerical Analysis Report DAMTP, Report 1990/NA10, University of Cambridge.

A. Iserles and S.P. Nørsett (1991), Order Stars, Chapman and Hall (London).

F. Lasagni (1988), ‘Canonical Runge–Kutta methods’, ZAMP 39, 952953.

C.R. Menyuk (1984), ‘Some properties of the discrete Hamiltonian method’, Physica D 11, 109129.

A.I. Neishtadt (1984), ‘The separation of motions in systems with rapidly rotating phase’, J. Appl. Math. Mech 48, 133139.

D.I. Pullin and P.G. Saffman (1991), ‘Long-tune symplectic integration: the example of four-vortex motion’, Proc. Roy. Soc. Lond. A432, 481494.

Qin Meng-zhao and Zhang Mei-qing (1990), ‘Multi-stage symplectic schemes of two kinds of Hamiltonian systems for wave equations’, Comput. Math. Appl. 19, 5162.

R.D. Ruth (1983), ‘A canonical integration techniqe’, IEEE Trans. Nucl. Sci. 30, 26692671.

J.M. Sanz-Serna (1988), ‘Runge–Kutta schemes for Hamiltonian systems’, BIT 28, 877883.

J.M. Sanz-Serna and L. Abia (1991), ‘Order conditions for canonical Runge-Kutta schemes’, SIAM J. Numer. Anal. 28, 10811096.

J.M. Sanz-Serna and D.F. Griffiths (1991), ‘A new class of results for the algebraic equations of implicit Runge-Kutta processes’, IMA J. Numer. Anal, to appear.

J.M. Sanz-Serna and F. Vadillo (1987), ‘Studies in nonlinear instability III: augmented Hamiltonian problems’, SIAM J. Appl. Math. 47, 92108.

R.F. Warming and B.J. Hyett (1974), ‘The modified equation approach to the stability and accuracy analysis of finite difference methods’, J. Comput. Phys. 14, 159179.

H. Yoshida (1990), ‘Construction of higher order symplectic integrators’, Phys. Lett. A150, 262268.

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? *