Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Partitioned Runge-Kutta methods for separable Hamiltonian problems

Authors: L. Abia and J. M. Sanz-Serna
Journal: Math. Comp. 60 (1993), 617-634
MSC: Primary 65L06; Secondary 34A50, 58F05, 70-08, 70H05, 70H15
MathSciNet review: 1181328
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Separable Hamiltonian systems of differential equations have the form $ d{\mathbf{p}}/dt = - \partial H/\partial {\mathbf{q}}$, $ d{\mathbf{q}}/dt = \partial H/\partial {\mathbf{p}}$, with a Hamiltonian function H that satisfies $ H = T({\mathbf{p}}) + V({\mathbf{q}})$ (T and V are respectively the kinetic and potential energies). We study the integration of these systems by means of partitioned Runge-Kutta methods, i.e., by means of methods where different Runge-Kutta tableaux are used for the p and q equations. We derive a sufficient and "almost" necessary condition for a partitioned Runge-Kutta method to be canonical, i.e., to conserve the symplectic structure of phase space, thereby reproducing the qualitative properties of the Hamiltonian dynamics. We show that the requirement of canonicity operates as a simplifying assumption for the study of the order conditions of the method.

References [Enhancements On Off] (What's this?)

  • [1] V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295
  • [2] J. C. Butcher, The numerical analysis of ordinary differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. Runge\mhy Kutta and general linear methods. MR 878564
  • [3] M. P. Calvo and J. M. Sanz-Serna, Variable steps for symplectic integrators, Numerical analysis 1991 (Dundee, 1991) Pitman Res. Notes Math. Ser., vol. 260, Longman Sci. Tech., Harlow, 1992, pp. 34–48. MR 1177227
  • [4] J. de Frutos and J. M. Sanz-Serna, An easily implementable fourth-order method for the time integration of wave problems, J. Comput. Phys. 103 (1992), no. 1, 160–168. MR 1188091,
  • [5] E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1987. Nonstiff problems. MR 868663
  • [6] F. M. Lasagni, Canonical Runge-Kutta methods, Z. Angew. Math. Phys. 39 (1988), no. 6, 952–953. MR 973194,
  • [7] R. S. MacKay, Some aspects of the dynamics and numerics of Hamiltonian systems, The dynamics of numerics and the numerics of dynamics (Bristol, 1990) Inst. Math. Appl. Conf. Ser. New Ser., vol. 34, Oxford Univ. Press, New York, 1992, pp. 137–193. MR 1173232
  • [8] R. Ruth, A canonical integration technique, IEEE Trans. Nuclear Sci. 30 (1984), 2669-2671.
  • [9] J. M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT 28 (1988), no. 4, 877–883. MR 972812,
  • [10] J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview, Acta numerica, 1992, Acta Numer., Cambridge Univ. Press, Cambridge, 1992, pp. 243–286. MR 1165727
  • [11] J. M. Sanz-Serna, The numerical integration of Hamiltonian systems, Computational ordinary differential equations (London, 1989) Inst. Math. Appl. Conf. Ser. New Ser., vol. 39, Oxford Univ. Press, New York, 1992, pp. 437–449. MR 1387155
  • [12] J. M. Sanz-Serna and L. Abia, Order conditions for canonical Runge-Kutta schemes, SIAM J. Numer. Anal. 28 (1991), no. 4, 1081–1096. MR 1111455,
  • [13] Y. B. Suris, Canonical transformations generated by methods of Runge-Kutta type for the numerical integration of the system $ x''= - \partial U/\partial x$, Zh. Vychisl. Mat. i Mat. Fiz. 29 (1987), 202-211. (Russian)
  • [14] Yu. B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpretation, Mat. Model. 2 (1990), no. 4, 78–87 (Russian, with English summary). MR 1064467

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L06, 34A50, 58F05, 70-08, 70H05, 70H15

Retrieve articles in all journals with MSC: 65L06, 34A50, 58F05, 70-08, 70H05, 70H15

Additional Information

Keywords: Hamiltonian systems, symplectic structures, canonical transformations, Runge-Kutta methods, order conditions, bicolor trees, bicolor rooted trees, generating functions
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society