BIT Numerical Mathematics

, Volume 32, Issue 3, pp 539–543 | Cite as

Family of symplectic implicit Runge-Kutta formulae

  • Satoshi Saito
  • Hiroshi Sugiura
  • Taketomo Mitsui
Scientific Notes


A family of formulae for the sympletic IRK method is investigated. Specifically, focus is given to general solutions for formula parameters of IRK under the symplectic and the order conditions. Examples of such formulae are constructed for up to three stages.

AMS Subject Classification

65L06 65L20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abia, L. & Sanz-Serna, J. M.,Partitioned Runge-Kutta methods for separable Hamiltonian problems, Universidad de Valladolid, Applied Mathematics and Computation Reports, Report 1990/8 (1990).Google Scholar
  2. 2.
    Butcher, J. C.,The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester, 1987.Google Scholar
  3. 3.
    Cooper, G. J.,Stability of Runge-Kutta methods for trajectory problems, IMA J. Numer. Anal., 7 (1987), 1–13.Google Scholar
  4. 4.
    Dekker, K. & Verwer, J. G.,Stability of Runge-Kutta Method for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, 1984.Google Scholar
  5. 5.
    Lasagni, F. M.,Canonical Runge-Kutta methods, Z. Angew. Math. Phys., 39 (1988), 952–953.Google Scholar
  6. 6.
    Maeda, S.,Certain kind of Runge-Kutta type formulas and reproduction of orbits of linear system, Trans. IEICE (Japan), J73-A (1990), 1648–1653 (in Japanese).Google Scholar
  7. 7.
    Saito, S., Sugiura, H & Mitsui, T.,Butcher's simplifying assumption for symplectic integrators, BIT 32, (1992), 345–349.Google Scholar
  8. 8.
    Sanz-Serna, J. M.,Runge-Kutta schemes for Hamiltonian systems, BIT, 28 (1988), 877–883.Google Scholar
  9. 9.
    Sanz-Serna, J. M. & Abia, L.,Order conditions for canonical Runge-Kutta schemes, SIAM J. Numer. Anal., 28 (1991), 1081–1096.Google Scholar

Copyright information

© BIT Foundations 1992

Authors and Affiliations

  • Satoshi Saito
    • 1
  • Hiroshi Sugiura
    • 1
  • Taketomo Mitsui
    • 1
  1. 1.Department of Information Engineering, School of EngineeringNagoya UniversityNagoyaJapan

Personalised recommendations