Monitoring energy drift with shadow Hamiltonians. (English) Zbl 1070.65129
Summary: The application of a symplectic integrator to a Hamiltonian system formally conserves the value of a modified, or shadow, Hamiltonian defined by some asymptotic expansion in powers of the step size. An earlier article of R. D. Skeel and D. J. Hardy [SIAM J. Sci. Comput. 23, No. 4, 1172–1188 (2001; Zbl 1002.65135)] describes how it is possible to construct highly accurate shadow Hamiltonian approximations using information readily available from the numerical integration.
This article improves on this construction by giving formulas of order up to 24 (not just up to 8) and by greatly reducing both storage requirements and roundoff error. More significantly, these high order formulas yield remarkable results not evident for 8th order formulas, even for systems as complex as the molecular dynamics of water.
These numerical experiments not only illuminate theoretical properties of shadow Hamiltonians but also give practical information about the accuracy of a simulation. By removing systematic energy fluctuations, they reveal the rate of energy drift for a given step size and uncover the ill effects of using switching functions that do not have enough smoothness.
This article improves on this construction by giving formulas of order up to 24 (not just up to 8) and by greatly reducing both storage requirements and roundoff error. More significantly, these high order formulas yield remarkable results not evident for 8th order formulas, even for systems as complex as the molecular dynamics of water.
These numerical experiments not only illuminate theoretical properties of shadow Hamiltonians but also give practical information about the accuracy of a simulation. By removing systematic energy fluctuations, they reveal the rate of energy drift for a given step size and uncover the ill effects of using switching functions that do not have enough smoothness.
MSC:
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |
| 70-08 | Computational methods for problems pertaining to mechanics of particles and systems |
| 70H05 | Hamilton’s equations |
| 82C80 | Numerical methods of time-dependent statistical mechanics (MSC2010) |
Keywords:
Hamiltonian system; Modified equation; Symplectic integrator; Backward error; Numerical; roundoff error; Numerical experimentsCitations:
Zbl 1002.65135Software:
NAMDReferences:
| [1] | Gans, J.; Shalloway, D., Shadow mass and the relationship between velocity and momentum in symplectic numerical integration, Phys. Rev. E, 61, 4, 4587-4592 (2000) |
| [2] | Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration, (Springer Series in Computational Mathematics, vol. 31 (2002), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0994.65135 |
| [3] | Skeel, R. D.; Hardy, D. J., Practical construction of modified Hamiltonians, SIAM J. Sci. Comput., 23, 4, 1172-1188 (2001) · Zbl 1002.65135 |
| [4] | Borwein, J. M.; Corless, R. M., Emerging tools for experimental mathematics, Am. Math. Monthly, 106, 10, 889-909 (1999) · Zbl 0980.68146 |
| [5] | Sanz-Serna, J.; Calvo, M., Numerical Hamiltonian Problems (1994), Chapman & Hall: Chapman & Hall London · Zbl 0816.65042 |
| [6] | Benettin, G.; Giorgilli, A., On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms, J. Stat. Phys., 74, 1117-1143 (1994) · Zbl 0842.58020 |
| [7] | Moore, B.; Reich, S., Backward error analysis for multi-symplectic integration methods, Numer. Math., 95, 625-652 (2003) · Zbl 1033.65113 |
| [8] | R.D. Engle, Interpolated modified Hamiltonians, Master’s Thesis, University of Illinois at Urbana-Champaign, October 2003. Available from: http://bionum.cs.purdue.edu/Engl03.ps; R.D. Engle, Interpolated modified Hamiltonians, Master’s Thesis, University of Illinois at Urbana-Champaign, October 2003. Available from: http://bionum.cs.purdue.edu/Engl03.ps |
| [9] | Moser, J., Stable and Random Motions in Dynamical Systems (1973), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0271.70009 |
| [10] | Kalé, L.; Skeel, R.; Brunner, R.; Bhandarkar, M.; Gursoy, A.; Krawetz, N.; Phillips, J.; Shinozaki, A.; Varadarajan, K.; Schulten, K., NAMD 2: Greater scalability for parallel molecular dynamics, J. Comput. Phys, 151, 1, 283-312 (1999) · Zbl 0948.92004 |
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