Symplectic integration with floating-point arithmetic and other approximations

Applied Numerical Mathematics

Volumn 29, Issue 1, 1999, Pages 3-18

  Skeel, Robert D ^a  

^a University of Illinois at Urbana Champaign   (United States)

Author keywords

Floating point arithmetic; Lattice maps; Leapfrog method; Symplectic integration

Indexed keywords

APPROXIMATION THEORY; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

FLOATING-POINT ARITHMETIC; HAMILTONIAN SYSTEMS; LATTICE MAPS; LEAPFROG METHOD; SYMPLECTIC INTEGRATION;

INTEGRATION;

EID: 0032739092     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(98)00033-6     Document Type: Article

References (26)

1. M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987). Reprinted in paperback in 1989 with corrections.
2. G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms, J. Statist. Phys. 74 (1994) 1117-1143.
3. T. Bishop, R.D. Skeel and K. Schulten, Difficulties with multiple timestepping and the fast multipole algorithm in molecular dynamics, J. Comput. Chem. 18 (1997) 1785-1791.
4. P.J. Channell and J.C. Scovel, Symplectic integration of Hamiltonian systems, Nonlinearity 3 (1990) 231-259.
5. D.J. Earn and S. Tremaine, Exact numerical studies of Hamiltonian maps: iterating without roundoff error, Physica D 56 (1992) 1-22.
6. D.J.D. Earn, Symplectic integration without roundoff error, in: V.G. Gurzadyan and D. Pfenniger, eds., Ergodic Concepts in Stellar Dynamics (Springer, Heidelberg, 1994) 122-130; also: Weizmann Institute of Science, August 1993, preprint.
7. Z. Ge and J.E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A 133 (1988) 134-139.
8. H. Grubmüller and P. Tavan, Efficient algorithms for molecular dynamics simulations of proteins: How good are they?, Manuscript, 1994.
9. E. Hairer, Backward error analysis of numerical integrators and symplectic methods, Ann. Numer. Math. 1 (1994) 107-132.
10. E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators, Numer. Math. 74 (1997) 441-462.
11. M. Hénon and C. Heiles, The applicability of the third integral of motion: some numerical experiments, Astron. J. 69 (1964) 73-79.
12. C.F.F. Karney, Long-time correlations in the stochastic regime, Physica D 8 (1983) 360-380.
13. D.H. Lehmer, MTAC 1 (1943) 31. An example from this article was given by W. Kahan in a lecture entitled "The tablemaker's dilemma and other quandaries" at the Second Conference on Mathematical Software at Purdue University, May 29-31, 1974.
14. D.H. Lehmer, MTAC 1 (1943) 31. An example from this article was given by W. Kahan in a lecture entitled "The tablemaker's dilemma and other quandaries" at the Second Conference on Mathematical Software at Purdue University, May 29-31, 1974.
15. B. Mehlig, D.W. Heermann and B.M. Forrest, Hybrid Monte Carlo method for condensed-matter systems, Phys. Rev. B 45 (1992) 679-685.
16. S. Reich, Backward error analysis for numerical integrators, Technical Report, 1997.
17. J. Sanz-Serna and M. Calvo, Numerical Hamiltonian Problems (Chapman and Hall, London, 1994).
18. J.M. Sanz-Serna, Two topics in nonlinear stability, in: W. Light, ed., Advances in Numerical Analysis (Clarendon Press, Oxford, 1991) 147-174.
19. J.M. Sanz-Serna, Symplectic integrators for Hamiltonian problem: an overview, Acta Numer. 1 (1992) 243-286.
20. T. Schlick, M. Mandziuk, R.D. Skeel and K. Srinivas, Nonlinear resonance artifacts in molecular dynamics simulations, J. Comput. Phys. 139 (1998) 1-29.
21. C. Scovel, On symplectic lattice maps, Phys. Lett. A 159 (1991) 396-400.
22. R.D. Skeel and J.B. Keiper, Elementary Numerical Computing with Mathematica (McGraw-Hill, New York, 1993).
23. 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) 203-222.
24. D.M. Stoffer, Some geometric and numerical methods for perturbed integrable systems, Ph.D. Thesis, Swiss Federal Institute of Technology, Zürich (1988).
25. M. Suzuki, Improved Trotter-like formula, Phys. Lett. A 180 (1993).
26. G. Zhang and T. Schlick, Implicit discretization schemes for Langevin dynamics, Mol. Phys. 84 (1995) 1077-1098.