Nonlinear stability analysis of area-preserving integrators

SIAM Journal on Numerical Analysis

Volumn 38, Issue 1, 2001, Pages 129-148

  Skeel, Robert D ^a   Srinivas, K ^b  

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

^b MICROSOFT   (United States)

Author keywords

Hamiltonian; Integrator; Nonlinear; Stability; Symplectic; Twist map

Indexed keywords

EID: 0042694406     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142998349527     Document Type: Article

References (21)

1. V. I. ARNOL'D, Mathematical Methods of Classical Mechanics, 2nd ed., Springer-Verlag, New York, 1989.
2. D. K. ARROWSMITH AND C. M. PLACE, An Introduction to Dynamical Systems, Cambridge University Press, Cambridge, UK, 1990.
3. U. M. ASCHER AND S. REICH, The midpoint scheme and variants for Hamiltonian systems: Advantages and pitfalls, SIAM J. Sci. Comput., 21 (1999), pp. 1045-1065.
4. G. D. BIRKHOFF, Dynamical Systems, AMS, Providence, RI, 1927, rev. ed. 1966.
5. 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), pp. 1785-1791.
6. H. E. CABRAL AND K. R. MEYER, Stability of equilibria and fixed points of conservative systems, Nonlinearity, 12 (1999), pp. 1351-1362.
7. 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.
8. C. LANCZOS, The Variational Principles of Mechanics, 4th ed., Dover, New York, 1986.
9. 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.
10. R. I. McLACHLAN AND P. ATELA, The accuracy of symplectic integrators, Nonlinearity, 5 (1992), pp. 541-562.
11. J. MOSER, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, NJ, 1973.
12. N. M. NEWMARK, A method of computation for structural dynamics, ASCE J. Engrg. Mech. Division, 85 (1959), pp. 67-94.
13. J. M. SANZ-SERNA, Two topics in nonlinear stability, in Advances in Numerical Analysis, W. Light, ed., Clarendon Press, Oxford, 1991, pp. 147-174.
14. J. M. SANZ-SERNA AND F. VADILLO, Nonlinear instability, the dynamic approach, in Numerical Analysis, D. F. Griffiths and G. A. Watson, eds., Pitman Res. Notes Math. Ser. 140, Longman Scientific and Technical, Harlow, UK, 1986, pp. 187-199.
15. T. SCHLICK, M. MANDZIUK, R. D. SKEEL, AND K. SRINIVAS, Nonlinear resonance artifacts in molecular dynamics simulations, J. Comput. Phys., 139 (1998), pp. 1-29.
16. C. L. SIEGEL AND J. K. MOSER, Lectures on Celestial Mechanics, Springer-Verlag, New York, Heidelberg, 1971.
17. J. SIMO, N. TARNOW, AND K. WONG, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Comput. Methods Appl. Mech. Engrg., 100 (1992), pp. 63-116.
18. J. C. SIMO AND N. TARNOW, A Review of Conserving Algorithms for Nonlinear Dynamics, Tech. Rep. SUDAM Report 92-7, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1992.
19. R. D. SKEEL, Symplectic integration with floating-point arithmetic and other approximations, Appl. Numer. Math., 29 (1999), pp. 3-18.
20. 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.
21. G. ZHANG AND T. SCHLICK, Implicit discretization schemes for Langevin dynamics, Molecular Phys., 84 (1995), pp. 1077-1098.