Skip to main content
SpringerLink
Log in

Order conditions for canonical Runge-Kutta-Nyström methods

  • Part II Numerical Mathematics
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

We are concerned with Runge-Kutta-Nyström methods for the integration of second order systems of the special formd 2 y/dt 2=f(y). If the functionf is the gradient of a scalar field, then the system is Hamiltonian and it may be advantageous to integrate it by a so-called canonical Runge-Kutta-Nyström formula. We show that the equations that must be imposed on the coefficients of the method to ensure canonicity are simplifying assumptions that lower the number of independent order conditions. We count the number of order conditions, both for general and for canonical Runge-Kutta-Nyström formulae.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Abia and J. M. Sanz-Serna,Partitioned Runge-Kutta methods for separable Hamiltonian problems, Applied Mathematics and Computation Reports, Report 1990/8, December 1990, Universidad de Valladolid.

  2. V. I. Arnold,Mathematical Methods of Classical Mechanics, 2nd ed., Springer (1989).

  3. J. C. Butcher, private communication (1990).

  4. M. P. Calvo,Canonical Runge-Kutta-Nyström Methods, Ph. D. Thesis, Universidad de Valladolid (in preparation).

  5. M. P. Calvo and J. M. Sanz-Serna,Variable steps for symplectic integrators, Applied Mathematics and Computation Reports, Report 1991/3, April 1991, Universidad de Valladolid.

  6. P. J. Channell and C. Scovel,Symplectic integration of Hamiltonian systems, Nonlinearity 3 (1990), 231–259.

    Article  Google Scholar 

  7. T. Eirola and J. M. Sanz-Serna,Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods, Numer. Math. (to appear).

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

    Google Scholar 

  9. J. de Frutos, T. Ortega & J. M. Sanz-Serna,A Hamiltonian explicit algorithm with spectral accuracy for the “good” Boussinesq system, Comput. Methods Appl. Mech. Engrg. 80 (1990), 417–423.

    Article  Google Scholar 

  10. E. Hairer, private communication (1990).

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

  12. A. Iserles,Efficient Runge-Kutta methods for Hamiltonian problems, Numerical Analysis Reports, DAMTP 1990/NA 10, November 1990, University of Cambridge.

  13. D. E. Knuth,Fundamental Algorithms, 2nd. ed., Addison-Wesley (1973).

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

    Article  MathSciNet  Google Scholar 

  15. R. S. Mackay,Some aspects of the dynamics and numerics of Hamiltonian systems, preprint.

  16. D. Okunbor & R. D. Skeel,An explicit Runge-Kutta-Nyström method is canonical if and only if its adjoint is explicit, preprint.

  17. R. Ruth,A canonical integration technique, IEEE Trans. Nucl. Sci. 30 (1984), 2669–271.

    Google Scholar 

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

    Article  Google Scholar 

  19. J. M. Sanz-Serna,The numerical integration of Hamiltonian systems, Proceedings of the Conference on Computational Differential Equations, Imperial College London, 3rd–7th July 1989, (to appear).

  20. J. M. Sanz-Serna,Two topics in nonlinear stability, in Advances in Numerical Analysis, vol. 1, Will Light ed., Clarendon Press, Oxford, 1991, 147–174.

    Google Scholar 

  21. J. M. Sanz-Serna & L. Abia,Order conditions for canonical Runge-Kutta schemes, SIAM J. Numer. Anal., 28 (1991), 1086–1091.

    Article  Google Scholar 

  22. Y. B. Suris,Canonical transformations generated by methods of Runge-Kutta type for the numerical integration of the system x″, Zh. Vychisl. Mat. i. Mat. Fiz. 29, (1987), 202–211 (in Russian).

    Google Scholar 

  23. Y. B. Suris,Hamiltonian Runge-Kutta type methods and their variational formulation, Mathematical Simulation 2 (1990), 78–87 (in Russian).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada y Computación, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain

    M. P. Calvo & J. M. Sanz-Serna

Authors
  1. M. P. Calvo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. M. Sanz-Serna
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research has been supported by “Junta de Castilla y León” under project 1031-89 and by “Dirección General de Investigación Científica y Técnica” under project PB89-0351.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Calvo, M.P., Sanz-Serna, J.M. Order conditions for canonical Runge-Kutta-Nyström methods. BIT 32, 131–142 (1992). https://doi.org/10.1007/BF01995113

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01995113

AMS

Key words

Search

Navigation