Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

SIAM Journal on Numerical Analysis

Volumn 33, Issue 1, 1996, Pages 368-387

  Jay, Laurent ^a  

^a University of Minnesota   (United States)

Author keywords

Differential algebraic equations; Hamiltonian systems; Index 3; Lobatto methods; Mechanical systems; Partitioned Runge Kutta methods; Symplecticity

Indexed keywords

EID: 0000545837     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/0733019     Document Type: Article

References (41)

1. L. ABIA AND J. M. SANZ-SERNA, Partitioned Runge-Kutta methods for separable Hamiltonian problems, Math. Comput., 60 (1993), pp. 617-634.
2. H. C. ANDERSEN, Rattle: A velocity version of the Shake algorithm for molecular dynamics calculations, J. Comput. Phys., 52 (1983), pp. 24-34.
3. V. I. ARNOLD, Mathematical Methods of Classical Mechanics, Second ed., Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York, 1989.
4. U. M. ASCHER, Two families of symmetric difference schemes for singular perturbation problems, in Numerical Boundary Values ODEs, U. M. Ascher and R. D. Russel, eds., Proceedings of an International Workshop, Vancouver, Canada, July 10-13, 1984, Progress in Scientific Computing 5, Birkhäuser, Boston, 1985, pp. 173-191.
5. U. M. ASCHER, R. M. M. MATTHEIJ, AND R. D. RUSSEL, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1988.
6. U. M. ASCHER AND L. R. PETZOLD, Projected implicit Runge-Kutta methods for differential-algebraic equations, SIAM J. Numer. Anal., 28 (1991), pp. 1097-1120.
7. _, Projected collocation methods for higher-order higher-index differential-algebraic equations, J. Comput. Appl. Math., 43 (1992), pp. 243-259.
8. U. M. ASCHER AND L. R. PETZOLD, Stability of computational methods for constrained dynamics systems, SIAM J. Sci. Comput., 14 (1993), pp. 95-120.
9. K. E. BRENAN, S. L. CAMPBELL, AND L. R. PETZOLD, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland, New York, 1989.
10. J. C. BUTCHER, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, John Wiley and Sons, Chichester, 1987.
11. R. P. K. CHAN, On symmetric Runge-Kutta methods of high order, Computing, 45 (1990), pp. 301-309.
12. P. CHOQUARD, Mécanique Analytique, Vol. 1, Cahiers mathématiques de l'Ecole Polytechnique Fédérale de Lausanne, Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland, 1992.
13. M. P. CALVO AND J. M. SANZ-SERNA, Variable steps for symplectic integrators, in Numerical Analysis 1991, D. F. Griffiths and G. A. Watson, eds., Pitman Research Notes in Math. Series 260, Longman, Essex, England, 1992, pp. 34-48.
14. _, Canonical B-series, Numer. Math., 67 (1994), pp. 161-175.
15. C. FÜHRER AND B. J. LEIMKUHLER, Formulation and numerical solution of the equations of constrained mechanical motion, Research Report, DFVLR-FB 89-08, Institut für Dynamik der Flugsysteme, Oberpaffenhofen, Germany, 1989.
16. E. HAIRER, Backward analysis of numerical integrators and symplectic methods, in Scientific Computation and Differential Equations, K. Burrage et al., eds., Proceedings of the SCADE'93 conference, Auckland, New Zealand, January 1993, Annals of Numerical Mathematics 1, J. C. Baltzer, Amsterdam, 1994, pp. 107-132.
17. E. HAIRER AND L. JAY, Implicit Runge-Kutta methods for higher index differential-algebraic systems, in Contributions in Numerical Mathematics, WSSIAA 2, World Scientific, Singapore, 1993, pp. 213-224.
18. E. HAIRER, CH. LUBICH, AND M. ROCHE, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Math. 1409, Springer-Verlag, Berlin, New York, 1989.
19. E. HAIRER, S. P. NØRSETT, AND G. WANNER, Solving Ordinary Differential Equations I. Nonstiff Problems, Second revised ed., Computational Mathematics, Vol. 8, Springer-Verlag, Berlin, 1993.
20. E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Computational Mathematics, Vol. 14, Springer-Verlag, Berlin, 1991.
21. L. JAY, Convergence of a class of Runge-Kutta methods for differential-algebraic systems of index 2, BIT, 33 (1993), pp. 137-150.
22. _, Collocation methods for differential-algebraic equations of index 3, Numer. Math., 65 (1993), pp. 407-421.
23. _, Runge-Kutta type methods for index three differential-algebraic equations with applications to Hamiltonian svstems, Ph.D. thesis, Dept. of Math., University of Geneva, Switzerland, 1994.
24. _, Convergence of Runge-Kutta methods for differential-algebraic systems of index 3, Appl. Numer. Math., 17 (1995), pp. 97-118.
25. F. M. LASAGNI, Canonical Runge-Kutta methods, ZAMP, 39 (1988), pp. 952-953.
26. B. J. LEIMKUHLER AND S. REICH, Symplectic integration of constrained Hamiltonian systems, Math. Comput., 63 (1994), pp. 589-606.
27. B. J. LEIMKUHLER AND R. D. SKEEL, Symplectic numerical integrators in constrained Hamiltonian systems, J. Comput. Phys., 112 (1994), pp. 117-125.
28. CH. LUBICH, Integration of stiff mechanical systems by Runge-Kutta methods, ZAMP, 44 (1993), pp. 1022-1053.
29. R. S. MACKAY, Some aspects of the dynamics and numerics of Hamiltonian systems, in The Dynamics of Numerics and the Numerics of Dynamics, D. S. Broomhead and A. Iserles, eds., Clarendon Press, Oxford, 1992, pp. 137-193.
30. A. OSTERMANN, A half-explicit extrapolation method for differential-algebraic systems of index 3, IMA J. Numer. Anal., 10 (1990), pp. 171-180.
31. _, A class of half-explicit Runge-Kutta methods for differential-algebraic systems of index 3, Appl. Numer. Math., 13 (1993), pp. 165-179.
32. L. R. PETZOLD AND F. A. POTRA, ODAE methods for the numerical solution of Euler-Lagrange equations, Appl. Numer. Math., 10 (1992), pp. 397-413.
33. S. REICH, Symplectic integration of constrained Hamiltonian systems by Runge-Kutta methods, Report, Institut für Angewandte Analysis und Stochastik, Berlin, 1993.
34. R. D. SKEEL, J. J. BIESIADECKI, AND D. OKUNBOR, Symplectic integration for macromolecular dynamics, in Proceedings of the International Conference on Computation of Differential Equations and Dynamical Systems, Beijing, China, Sept. 1992, K. Feng and Z.-C. Shi, eds., Series on ApplMath. 4, World Scientific, Singapore, 1993.
35. J. M. SANZ-SERNA, Runge-Kutta schemes for Hamiltonian systems, BIT, 28 (1988), pp. 877-883.
36. J. M. SANZ-SERNA, Symplectic integrators for Hamiltonian problems: An overview, Acta Numerica, 1 (1992), pp. 243-286.
37. SUN GENG, Symplectic partitioned Runge-Kutta methods, J. Comput. Math., to appear.
38. _, Construction of high order Symplectic Runge-Kutta methods, J. Comput. Math., 11 (1993), pp. 250-260.
39. Y. B. SURIS, On canonicality of mappings resulting from Runge-Kutta-type methods during the integration of the system ẍ = -∂U/∂x, Zh. Vychisl. Mat. i Mat. Fiz., 29 (1989), pp. 202-211. (In Russian.) Also in U.S.S.R. Comput. Maths. Phys., 29 (1989), pp. 138-144.
40. Y. B. SURIS, On canonicality of mappings resulting from Runge-Kutta-type methods during the integration of the system ẍ = -∂U/∂x, Zh. Vychisl. Mat. i Mat. Fiz., 29 (1989), pp. 202-211. (In Russian.) Also in U.S.S.R. Comput. Maths. Phys., 29 (1989), pp. 138-144.
41. H. YOSHIDA, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), pp. 262-268.