Stability of asynchronous variational integrators

Journal of Computational Physics

Volumn 227, Issue 18, 2008, Pages 8367-8394

  Fong, William ^a   Darve, Eric ^a   Lew, Adrian ^a  

^a Stanford University   (United States)

Author keywords

Molecular dynamics; Multiple time step; Symplectic; Variational integrator

Indexed keywords

STABILITY; VARIATIONAL TECHNIQUES;

ASYNCHRONOUS METHODS; COMPUTATIONAL SAVINGS; LINEAR STABILITY; MECHANICAL SYSTEMS; MULTIPLE TIME SCALE; MULTIPLE TIME STEP; ONE-DIMENSIONAL; SYMPLECTIC; TIME STEP; VARIATIONAL INTEGRATORS;

MOLECULAR DYNAMICS;

EID: 48149095383     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2008.05.017     Document Type: Article

References (49)

1. Barth E., and Schlick T. Extrapolation versus impulse in multiple-timestepping schemes. II. Linear analysis and applications to Newtonian and Langevin dynamics. J. Chem. Phys. 109 5 (1998) 1633-1642
2. Barth E., and Schlick T. Overcoming stability limitations in biomolecular dynamics. I. combining force splitting via extrapolation with Langevin dynamics in LN. J. Chem. Phys. 109 5 (1998) 1617-1632
3. Belytschko T., and Lu Y. Convergence and stability analyses of multi-time step algorithm for parabolic systems. Comput. Methods Appl. Mech. Eng. 102 (1993) 179-198
4. Belytschko T., and Mullen R. Mesh partitions of explicit-implicit time integrators. In: Bathe K.-J., Oden J., and Wunderlich W. (Eds). Formulations and Computational Algorithms in Finite Element Analysis (1976), MIT Press 673-690
5. Benettin G., and Giorgilli A. On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys. 75 5-6 (1994) 1117-1143
6. Biesiadecki J., and Skeel R. Dangers of multiple time steps methods. J. Comput. Phys. 109 (1993) 318-328
7. Daniel W.J.T. Analysis and implementation of a new constant acceleration subcycling algorithm. Int. J. Numer. Methods Eng. 40 (1997) 2841-2855
8. Daniel W.J.T. The subcycled Newmark algorithm. Comput. Mech. 20 (1997) 272-281
9. Daniel W.J.T. A study of the stability of subcycling algorithms in structural dynamics. Comput. Methods Appl. Mech. Eng. 156 1 (1998) 1-13
10. Daniel W.J.T. A partial velocity approach to subcycling structural dynamics. Comput. Methods Appl. Mech. Eng. 192 3/4 (2003) 375-394
11. Y. Feng, K. Han, D. Owen, Asynchronous/multiple time integrators for impact and discrete systems, in: Proceedings of the Eighth International Conference on Computational Plasticity (COMPLAS VIII), Barcelona, Spain, 2005, pp. 5-8.
12. Gravouil A., and Combescure A. Multi-time-step explicit-implicit method for non-linear structural dynamics. Int. J. Numer. Methods Eng. 50 1 (2001) 199-225
13. Gravouil A., and Combescure A. Multi-time-step and two-scale domain decomposition method for non-linear structural dynamics. Int. J. Numer. Methods Eng. 58 10 (2003) 1545-1569
14. Grubmüller H., Heller H., Windemuth A., and Schulten K. Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions. Mol. Sim. 6 (1991) 121-142
15. Hairer E., and Lubich C. The life-span of backward error analysis for numerical integrators. Numer. Math. 76 (1997) 441-462
16. Hairer E., and Lubich C. Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Num. Anal. 38 2 (2001) 414-441
17. Hairer E., Lubich C., and Wanner G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2002), Springer, Verlag
18. Han G., Deng Y., Glimm J., and Martyna G. Error and timing analysis of multiple time-step integration methods for molecular dynamics. Comp. Phys. Commun. 176 (2007) 271-291
19. Hochbruck M., and Lubich C. A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83 3 (1999) 403-426
20. Hughes T., and Liu W. Implicit-explicit finite elements in transient analysis: Stability theory. J. Appl. Mech. 78 (1978) 371-374
21. Hughes T., Pister K., and Taylor R. Implicit-explicit finite elements in nonlinear transient analysis. Comput. Methods Appl. Mech. Eng. 17/18 (1979) 159-182
22. J. Izaguirre, Q. Ma, T. Matthey, J. Willcock, T. Slabach, B. Moore, G. Viamontes, Overcoming instabilities in Verlet-I/r-RESPA with the mollified impulse method, in: T. Schlick, H. Gan (Eds.), Computational Methods for Macromolecules: Challenges and Applications, Proceedings of the Third International Workshop on Algorithms for Macromolecular Modeling, Springer Verlag, 2000.
23. Izaguirre J., Reich S., and Skeel R. Longer time steps for molecular dynamics. J. Chem. Phys. 110 20 (1999) 9853-9864
24. Kane C., Marsden J., Ortiz M., and West M. Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49 10 (2000) 1295-1325
25. Leimkuhler B., and Reich S. Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics (2005), Cambridge University Press
26. A. Lew, Variational time integrators in computational solid mechanics, Ph.D. thesis, Caltech, 2003.
27. Lew A., Marsden J., Ortiz M., and West M. Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167 2 (2003) 85-146
28. Lew A., Marsden J., Ortiz M., and West M. Variational time integrators. Int. J. Numer. Methods Eng. 60 (2004) 153-212
29. A. Lew, M. Ortiz, Asynchronous Variational Integrators in Geometry, Mechanics and Dynamics, Volume dedicated to J. Marsden in his 60th birthday, Springer, 2002.
30. MacKay R. Some aspects of the dynamics of Hamiltonian systems. In: Broomhead D., and Iserles A. (Eds). The Dynamics of Numerics and the Numerics of Dynamics (1992), Clarendon Press, Oxford 137-193
31. Mandziuk M., and Schlick T. Resonance in the dynamics of chemical systems simulated by the implicit-midpoint scheme. Chem. Phys. Lett. 237 (1995) 525
32. Marsden J., Pekarsky S., Shkoller S., and West M. Variational methods, multisymplectic geometry and continuum mechanics. J. Geomet. Phys. 38 (2001) 253-284
33. Marsden J., and West M. Discrete mechanics and variational integrators. Acta Numer. (2001) 357-514
34. Miller III T., Eleftheriou M., Pattnaik P., Ndirango A., and Martyna G. Symplectic quaternion scheme for biophysical molecular dynamics. J. Chem. Phys. 116 (2002) 8649
35. Minary P., Tuckerman M., and Martyna G. Long time molecular dynamics for enhanced conformational sampling in biomolecular systems. Phys. Rev. Lett. 93 (2004) 150201
36. Müller S., and Ortiz M. On the gamma-convergence of discrete dynamics and variational integrators. J. Nonlinear Sci. 14 (2004) 279-296
37. Neal M., and Belytschko T. Explicit-explicit subcycling with non-integer time step ratios for structural dynamic systems. Comput. Struct. 6 (1989) 871-880
38. Reich S. Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36 5 (1999) 1549-1570
39. Sandu A., and Schlick T. Masking resonance artifacts in force-splitting methods for biomolecular simulations by extrapolative Langevin dynamics. J. Comp. Phys. 151 (1999) 74-113
40. Sanz-Serna J.M. Symplectic integrators for Hamiltonian problems - an overview. Acta Numer. (1992) 243-286
41. Schlick T., Mandziuk M., Skeel R., and Srinivas K. Nonlinear resonance artifacts in molecular dynamics simulations. J. Comput. Phys. 139 (1998) 1
42. Skeel R., and Srinivas K. Nonlinear stability analysis of area-preserving integrators. SIAM J. Numer. Anal. 38 1 (2000) 129-148
43. Skeel R., Zhang G., and Schlick T. A family of symplectic integrators stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput. 18 1 (1997) 203-222
44. Skeel R.D., and Srinivas K. Nonlinear stability analysis of area-preserving integrators. SIAM J. Numer. Anal. 38 1 (2000) 129-148
45. Smolinski P., and Wu Y.-S. An implicit multi-time step integration method for structural dynamics problems. Comput. Mech. 22 (1998) 337-343
46. Suris Y. Hamiltonian methods of Runge-Kutta type and their variational interpretation. Math. Model. 2 (1990) 78-87
47. Tuckerman M., Berne B., and Martyna G. Reversible multiple time scale molecular dynamics. J. Chem. Phys. 97 (1992) 1990-2001
48. Wendlandt J.M., and Marsden J.E. Mechanical integrators derived from a discrete variational principle. Physica D 106 (1997) 223-246
49. M. West, Variational integrators, Ph.D. thesis, Caltech, 2004.