Explicit canonical methods for Hamiltonian systems
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- by Daniel Okunbor and Robert D. Skeel PDF
- Math. Comp. 59 (1992), 439-455 Request permission
Abstract:
We consider canonical partitioned Runge-Kutta methods for separable Hamiltonians $H = T(p) + V(q)$ and canonical Runge-Kutta-Nyström methods for Hamiltonians of the form $H = \frac {1}{2}{p^{\text {T}}}{M^{ - 1}}p + V(q)$ with M a diagonal matrix. We show that for explicit methods there is great simplification in their structure. Canonical methods of orders one through four are constructed. Numerical experiments indicate the suitability of canonical numerical schemes for long-time integrations.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
- P. J. Channell and C. Scovel, Symplectic integration of Hamiltonian systems, Nonlinearity 3 (1990), no. 2, 231–259. MR 1054575
- Etienne Forest and Ronald D. Ruth, Fourth-order symplectic integration, Phys. D 43 (1990), no. 1, 105–117. MR 1060047, DOI 10.1016/0167-2789(90)90019-L
- George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler, Computer methods for mathematical computations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977. MR 0458783
- Herbert Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, Mass., 1951. MR 0043608
- Michel Hénon and Carl Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J. 69 (1964), 73–79. MR 158746, DOI 10.1086/109234
- 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
- Daniel Okunbor and Robert D. Skeel, An explicit Runge-Kutta-Nyström method is canonical if and only if its adjoint is explicit, SIAM J. Numer. Anal. 29 (1992), no. 2, 521–527. MR 1154280, DOI 10.1137/0729032 M.-Z. Qin and M.-Q. Zhang, Symplectic Runge-Kutta schemes for Hamiltonian systems, preprint, 1989. R. D. Ruth, A canonical integration technique, IEEE Trans. Nuclear Sci. 30 (1983), 2669-2671.
- 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
- Yu. B. Suris, Some properties of methods for the numerical integration of systems of the form $\ddot x=f(x)$, Zh. Vychisl. Mat. i Mat. Fiz. 27 (1987), no. 10, 1504–1515, 1598 (Russian). MR 918549
- Yu. B. Suris, On the canonicity of mappings that can be generated by methods of Runge-Kutta type for integrating systems $\ddot x=-\partial U/\partial x$, Zh. Vychisl. Mat. i Mat. Fiz. 29 (1989), no. 2, 202–211, 317 (Russian); English transl., U.S.S.R. Comput. Math. and Math. Phys. 29 (1989), no. 1, 138–144 (1990). MR 987190, DOI 10.1016/0041-5553(89)90058-X
- Haruo Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A 150 (1990), no. 5-7, 262–268. MR 1078768, DOI 10.1016/0375-9601(90)90092-3
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Math. Comp. 59 (1992), 439-455
- MSC: Primary 70-08; Secondary 65L06, 70H05
- DOI: https://doi.org/10.1090/S0025-5718-1992-1136225-3
- MathSciNet review: 1136225