Midpoint scheme and variants for Hamiltonian systems: advantages and pitfalls

SIAM Journal of Scientific Computing

Volumn 21, Issue 3, 1999, Pages 1045-1065

  Ascher, Uri M ^a   Reich, Sebastian ^b  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b ZUSE INSTITUTE BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ALGEBRA; DIFFERENTIAL EQUATIONS; INTEGRAL EQUATIONS; MECHANICS; MOLECULAR DYNAMICS;

DECOUPLING; DIFFERENTIAL ALGEBRAIC EQUATIONS; HAMILTONIAN SYSTEMS; HIGHLY OSCILLATORY PROBLEMS; MIDPOINT SCHEME; MULTIBODY SYSTEMS;

EQUATIONS OF MOTION;

EID: 0033293798     PISSN: 10648275     EISSN: None     Source Type: None    
DOI: 10.1137/S1064827597316059     Document Type: Article

References (26)

1. U. ASCHER, Two families of symmetric difference schemes for singular perturbation problems, in Numerical Boundary Value ODEs, U. Ascher and R. Russell, eds., Birkhauser, 1985.
2. U. ASCHER, On symmetric schemes and differential-algebraic equations, SIAM J. Sci. Stat. Comput., 10 (1989), pp. 937-949.
3. U. ASCHER, Stabilization of invariants of discretized differential systems, Numer. Algorithms, 14 (1997), pp. 1-23.
4. U. ASCHER, H. CHIN, L. PETZOLD, AND S. REICH, Stabilization of constrained mechanical systems with DAEs and invariant manifolds, J. Mech. Struct. Machines, 23 (1995), pp. 135-158.
5. U. ASCHER, R. MATTHEIJ, AND R. RUSSELL, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, 1995.
6. U. ASCHER AND S. REICH, On difficulties in integrating highly oscillatory Hamiltonian systems, in Algorithms for Macromolecular Modelling, P. Deuflhard, J. Hermans, B. Leimkuhler, A. Mark, S. Reich, and R. D. Skeel, eds., Springer-Verlag, 1998, pp. 281-296.
7. F. A. BORNEMANN AND CH. SCHUTTE, A Mathematical Approach to Smoothed Molecular Dynamics: Correcting Potentials for Freezing Bond Angles, Konrad-Zuse-Zentrum, Berlin, 1995.
8. G. J. COOPER, Stability of Runge-Kutta methods for trajectory problems, IMA J. Numer. Anal., 7 (1987), pp. 1-13.
9. O. GONZALEZ, Mechanical systems subject to holonomic constraints: Differential-algebraic formulations and conservative integration, Phys. D, submitted.
10. O. GONZALEZ AND J. SIMO, On the stability of symplectic and energy-momentum algorithms for nonlinear Hamiltonian systems with symmetry, Comput. Methods Appl. Mech. Engrg., 134 (1996), pp. 197-222.
11. E. HAIRER AND L. JAY, Implicit Runge-Kutta methods for higher-index differential-algebraic systems, WSSIAA, Contrib. Numer. Math., 2 (1993), pp. 213-224.
12. E. HAIRER AND D. STOFFER, Reversible long-term integration with variable stepsizes, SIAM J. Sci. Comput., 18 (1997), pp. 257-269.
13. E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, New York, Berlin, 1991.
14. L. JAY AND L. PETZOLD, Highly Oscillatory Systems and Periodic-Stability, manuscript, 1995.
15. B. LEIMKUHLER AND R. SKEEL, Symplectic integrators in constrained Hamiltonian systems, J. Comput. Phys., 112 (1994), pp. 117-125.
16. A. J. LICHTENBERG AND M. A. LIEBERMAN, Regular and Stochastic Motions, Springer-Verlag, Berlin, New York, 1983.
17. M. MANDZIUK AND T. SCHLICK, Resonance in the dynamics of chemical systems simulated by the implicit midpoint scheme, Chem. Phys. Lett., 237 (1995), pp. 525-535.
18. A. I. NEISHTADT, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), pp. 133-139.
19. L. R. PETZOLD, L. O. JAY, AND J. YEN, Numerical solution of highly oscillatory ordinary differential equations, Acta Numerica, VI (1997), pp. 437-484.
20. S. REICH, Enhanced energy conserving methods, BIT, 36 (1996), pp. 122-434.
21. H. RUBIN AND P. UNGAR, Motion under a strong constraining force, Comm. Pure Appl. Math., 10 (1957), pp. 65-87.
22. J. M. SANZ-SERNA AND M. P. CALVO, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994.
23. T. SCHLICK, M. MANDZIUK, R. D. SKEEL, AND K. SRINIVAS, Nonlinear resonance artifacts in molecular dynamics simulations, J. Comput. Phys., 139 (1998), pp. 1-19.
24. M. SHIMADA AND H. YOSHIDA, Long term conservation of adiabatic invariants, Publ. Astron. Soc. Japan, 48 (1996), pp. 147-155.
25. R. D. SKEEL, G. ZHANG, AND T. SCHLICK, A family of symplectic integrators: Stability, accuracy, and molecular dynamics applications, SIAM J. Sci. Comput., 18 (1997), pp. 203-222.
26. R. D. SKEEL AND M. ZHANG, Cheap implicit symplectic integrators, Appl. Numer. Math., 25 (1997), pp. 297-302.