Comparison of Very Smooth Cell-Model Trajectories Using Five Symplectic and Two Runge-Kutta Integrators
Ruby Valley Research Institute Highway Contract 60, Box 601
Ruby Valley, Nevada 89833
E-mail: hooverwilliam@yahoo.com
Received:
Received: 02 April 2015; accepted: 02 April 2015; published online: 03 June 2015
DOI: 10.12921/cmst.2015.21.03.001
Abstract:
Time-reversible symplectic methods, which are precisely compatible with Liouville’s phase-volume-conservation theorem, are often recommended for computational simulations of Hamiltonian mechanics. Lack of energy drift is an apparent advantage of such methods. But all numerical methods are susceptible to Lyapunov instability, which severely limits the maximum time for which chaotic solutions can be “accurate”. The “advantages” of higher-order methods are lost rapidly for typical chaotic Hamiltonians. We illustrate these difficulties for a useful reproducible test case, the two-dimensional one-particle cell model with specially smooth forces. This Hamiltonian problem is chaotic and occurs on a three-dimensional constant-energy shell, the minimum dimension for chaos. We benchmark the problem with quadruple-
precision trajectories using the fourth-order Candy-Rozmus, fifth-order Runge-Kutta, and eighth-order Schlier-Seiter-Teloy integrators. We compare the last, most-accurate particle trajectories to those from six double-precision algorithms, four symplectic and two Runge-Kutta.
Key words:
chaos, classical mechanics, Lyapunov instability, symplectic methods
References:
[1] M. Karplus, ‘Spinach on the Ceiling’: a Theoretical
Chemist’s Return to Biology, Annual Review of Biophysics
and Biomolecular Structure 35,1-47 (2006).
[2] H.A. Posch, W.G. Hoover, and F.J. Vesely, Canonical Dy-
namics ofthe Nosé Oscillator: Stability, Order,andChaos,
Physical ReviewA33, 4253-4265 (1986).
[3] W.G.Hoover and C.G. Hoover, Simulation and Control of
Chaotic Nonequilbrium Systems (World Scientific Publish-
ers, Singapore, 2015).
[4] J.A.Barker and D. Henderson, What is ‘Liquid’ ? Under-
standingthe States of Matter, Reviews of Modern Physics
48,587-671(1976).
[5] B. Leimkuhler and S. Reich, Simulating Hamiltonian Dy-
namics (Cambridge University Press, United Kingdom,
2004).
[6] W.G.Hoover, O. Kum,and N.E. Owens[ now NancyFulda
], “Accurate Symplectic Integrators via Random Sampling”,
theJournalof Chemical Physics 103,1530-1532(1995).
[7] S.K. Gray,D.W. Noid,andB.G.Sumpter, Symplectic Inte-
grators for Large Scale Molecular Dynamics Simulations:
A Comparison of Several Explicit Methods, the Journal of
Chemical Physics 101, 4062-4072 (1994).
[8] E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical
Integration Illustrated by the Störmer-Verlet Method, Acta
Numerica12, 399-450(2003).
[9] D. Levesque and L. Verlet, Molecular Dynamics and Time
Reversibility, Journal of Statistical Physics 72, 519-537
(1993).
[10] G.D. VenneriandW.G. Hoover,Simple Exact Test for Well-
Known Molecular Dynamics Algorithms, Journal of Compu-
tational Physics 73, 468-475 (1987).
[11] H. Yoshida, Construction of Higher Order SymplecticInte-
grators, Physics Letters A 150, 262-268 (1990).
[12] Ch. Schlier and A. Seiter, High-Order Symplectic Integra-
tion: An Assessment, Computer Physics Communications
130,176-189(2000).
[13] Ch. Schlierand, A. Seiter, Symplectic IntegrationofClassical
Trajectories: A Case Study, Journal of Physical Chemistry
102,9399-9404 (1998).
[14] H. Rein and D.S. Spiegel, IAS15: A Fast, Adaptive, High-
Order Integrator for Gravitational Dynamics, Accurate to
Machine Precision Over a Billion Orbits, arχiv 1409.4779,
also available in the Monthly Noticesof the Royal Astonom-
ical Society.
[15] W.G. Hoover, J.C. Sprott, and P.K. Patra, Ergodic Time-
Reversible Chaos for Gibbs’ Canonical Oscillator, (submit-
tedto Physical Review E,2015) = arχiv 1503.06729.
[16] J. Ford,What is Chaos, that We Should beMindful ofit?, in
The NewPhysics, edited by PaulDavies (Cambridge Univer-
sity Press, 1989).
[17] S.Liao, A Comment on the Arguments about the Reliability
and Convergence of Chaotic Simulations,International Jour-
nalof Bifurcation and Chaos,24, 91450119(2014) = arχiv
1401.0256.
[18] W. G. Hoover and C. G. Hoover, What is Liquid? Lya-
punov Instability Reveals Symmetry-Breaking Irreversibility
Hidden within Hamilton’s Many-Body Equations of Motion,
Condensed Matter Physics 18, 13003-1-13 (2015) = arχiv
1405.2485.
Time-reversible symplectic methods, which are precisely compatible with Liouville’s phase-volume-conservation theorem, are often recommended for computational simulations of Hamiltonian mechanics. Lack of energy drift is an apparent advantage of such methods. But all numerical methods are susceptible to Lyapunov instability, which severely limits the maximum time for which chaotic solutions can be “accurate”. The “advantages” of higher-order methods are lost rapidly for typical chaotic Hamiltonians. We illustrate these difficulties for a useful reproducible test case, the two-dimensional one-particle cell model with specially smooth forces. This Hamiltonian problem is chaotic and occurs on a three-dimensional constant-energy shell, the minimum dimension for chaos. We benchmark the problem with quadruple-
precision trajectories using the fourth-order Candy-Rozmus, fifth-order Runge-Kutta, and eighth-order Schlier-Seiter-Teloy integrators. We compare the last, most-accurate particle trajectories to those from six double-precision algorithms, four symplectic and two Runge-Kutta.
Key words:
chaos, classical mechanics, Lyapunov instability, symplectic methods
References:
[1] M. Karplus, ‘Spinach on the Ceiling’: a Theoretical
Chemist’s Return to Biology, Annual Review of Biophysics
and Biomolecular Structure 35,1-47 (2006).
[2] H.A. Posch, W.G. Hoover, and F.J. Vesely, Canonical Dy-
namics ofthe Nosé Oscillator: Stability, Order,andChaos,
Physical ReviewA33, 4253-4265 (1986).
[3] W.G.Hoover and C.G. Hoover, Simulation and Control of
Chaotic Nonequilbrium Systems (World Scientific Publish-
ers, Singapore, 2015).
[4] J.A.Barker and D. Henderson, What is ‘Liquid’ ? Under-
standingthe States of Matter, Reviews of Modern Physics
48,587-671(1976).
[5] B. Leimkuhler and S. Reich, Simulating Hamiltonian Dy-
namics (Cambridge University Press, United Kingdom,
2004).
[6] W.G.Hoover, O. Kum,and N.E. Owens[ now NancyFulda
], “Accurate Symplectic Integrators via Random Sampling”,
theJournalof Chemical Physics 103,1530-1532(1995).
[7] S.K. Gray,D.W. Noid,andB.G.Sumpter, Symplectic Inte-
grators for Large Scale Molecular Dynamics Simulations:
A Comparison of Several Explicit Methods, the Journal of
Chemical Physics 101, 4062-4072 (1994).
[8] E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical
Integration Illustrated by the Störmer-Verlet Method, Acta
Numerica12, 399-450(2003).
[9] D. Levesque and L. Verlet, Molecular Dynamics and Time
Reversibility, Journal of Statistical Physics 72, 519-537
(1993).
[10] G.D. VenneriandW.G. Hoover,Simple Exact Test for Well-
Known Molecular Dynamics Algorithms, Journal of Compu-
tational Physics 73, 468-475 (1987).
[11] H. Yoshida, Construction of Higher Order SymplecticInte-
grators, Physics Letters A 150, 262-268 (1990).
[12] Ch. Schlier and A. Seiter, High-Order Symplectic Integra-
tion: An Assessment, Computer Physics Communications
130,176-189(2000).
[13] Ch. Schlierand, A. Seiter, Symplectic IntegrationofClassical
Trajectories: A Case Study, Journal of Physical Chemistry
102,9399-9404 (1998).
[14] H. Rein and D.S. Spiegel, IAS15: A Fast, Adaptive, High-
Order Integrator for Gravitational Dynamics, Accurate to
Machine Precision Over a Billion Orbits, arχiv 1409.4779,
also available in the Monthly Noticesof the Royal Astonom-
ical Society.
[15] W.G. Hoover, J.C. Sprott, and P.K. Patra, Ergodic Time-
Reversible Chaos for Gibbs’ Canonical Oscillator, (submit-
tedto Physical Review E,2015) = arχiv 1503.06729.
[16] J. Ford,What is Chaos, that We Should beMindful ofit?, in
The NewPhysics, edited by PaulDavies (Cambridge Univer-
sity Press, 1989).
[17] S.Liao, A Comment on the Arguments about the Reliability
and Convergence of Chaotic Simulations,International Jour-
nalof Bifurcation and Chaos,24, 91450119(2014) = arχiv
1401.0256.
[18] W. G. Hoover and C. G. Hoover, What is Liquid? Lya-
punov Instability Reveals Symmetry-Breaking Irreversibility
Hidden within Hamilton’s Many-Body Equations of Motion,
Condensed Matter Physics 18, 13003-1-13 (2015) = arχiv
1405.2485.
