Partitioned Runge-Kutta methods for separable Hamiltonian problems

Abia, L.; Sanz-Serna, J. M.

doi:10.1090/S0025-5718-1993-1181328-1

Partitioned Runge-Kutta methods for separable Hamiltonian problems
HTML articles powered by AMS MathViewer

by L. Abia and J. M. Sanz-Serna PDF

Math. Comp. 60 (1993), 617-634 Request permission

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

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, DOI 10.1007/978-1-4757-2063-1
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
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
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, DOI 10.1016/0021-9991(92)90331-R
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, DOI 10.1007/978-3-662-12607-3
F. M. Lasagni, Canonical Runge-Kutta methods, Z. Angew. Math. Phys. 39 (1988), no. 6, 952–953. MR 973194, DOI 10.1007/BF00945133
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

A canonical integration technique

30

J. M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT 28 (1988), no. 4, 877–883. MR 972812, DOI 10.1007/BF01954907
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, DOI 10.1017/s0962492900002282
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
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, DOI 10.1137/0728058

Canonical transformations generated by methods of Runge-Kutta type for the numerical integration of the system

29

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