Monitoring energy drift with shadow Hamiltonians

Engle, Robert D.; Skeel, Robert D.; Drees, Matthew

doi:10.1016/j.jcp.2004.12.009

Cited by 74 publications

(71 citation statements)

References 8 publications

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“…This is strictly true for certain simple forms of H 0 but not for biomolecular force fields, where the asymptotic expansion eq 12 does not converge. For simulations with small energy drift, however, we find much encouragement in earlier work 25 that Hamiltonians defined by truncating eq 12 can describe the dynamics generated by the integrator extremely accurately. Nevertheless, an important part of this paper is to test numerically whether generalized equipartition holds.…”

Section: Equipartition For the Shadow Hamiltonianmentioning

confidence: 92%

Equipartition and the Calculation of Temperature in Biomolecular Simulations

Eastwood

Stafford

Lippert

et al. 2010

J. Chem. Theory Comput.

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Abstract:Since the behavior of biomolecules can be sensitive to temperature, the ability to accurately calculate and control the temperature in molecular dynamics (MD) simulations is important. Standard analysis of equilibrium MD simulationsseven constant-energy simulations with negligible long-term energy driftsoften yields different calculated temperatures for different motions, however, in apparent violation of the statistical mechanical principle of equipartition of energy. Although such analysis provides a valuable warning that other simulation artifacts may exist, it leaves the actual value of the temperature uncertain. We observe that Tolman's generalized equipartition theorem should hold for long stable simulations performed using velocity-Verlet or other symplectic integrators, because the simulated trajectory is thought to sample almost exactly from a continuous trajectory generated by a shadow Hamiltonian. From this we conclude that all motions should share a single simulation temperature, and we provide a new temperature estimator that we test numerically in simulations of a diatomic fluid and of a solvated protein. Apparent temperature variations between different motions observed using standard estimators do indeed disappear when using the new estimator. We use our estimator to better understand how thermostats and barostats can exacerbate integration errors. In particular, we find that with large (albeit widely used) time steps, the common practice of using two thermostats to remedy so-called hot solvent-cold solute problems can have the counterintuitive effect of causing temperature imbalances. Our results, moreover, highlight the utility of multiple-time step integrators for accurate and efficient simulation.

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Section: Equipartition For the Shadow Hamiltonianmentioning

confidence: 92%

Equipartition and the Calculation of Temperature in Biomolecular Simulations

Eastwood

Stafford

Lippert

et al. 2010

J. Chem. Theory Comput.

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show abstract

“…This unexpected result explains why the order of φ h is actually 2 numerically. Illuminating theoretical and numerical work onH can be found in Engle et al (2005) and Hairer et al (2006). Even though the infinite series inH generally diverges, they show the error in conservation of the first N terms inH, with N determined by h, is exponentially suppressed over exponentially long times.…”

Section: Derivation Of Methods and Basic Propertiesmentioning

confidence: 99%

Symplectic integration for the collisional gravitationalN-body problem

Hernández¹,

Bertschinger²

2015

Mon. Not. R. Astron. Soc.

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We present a new symplectic integrator designed for collisional gravitational N -body problems which makes use of Kepler solvers. The integrator is also reversible and conserves 9 integrals of motion of the N -body problem to machine precision. The integrator is second order, but the order can easily be increased by the method of Yoshida. We use fixed time step in all tests studied in this paper to ensure preservation of symplecticity. We study small N collisional problems and perform comparisons with typically used integrators. In particular, we find comparable or better performance when compared to the 4th order Hermite method and much better performance than adaptive time step symplectic integrators introduced previously. We find better performance compared to SAKURA, a non-symplectic, non-time-reversible integrator based on a different two-body decomposition of the N -body problem. The integrator is a promising tool in collisional gravitational dynamics.

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“…The sum of potential and kinetic energy is given by 2) and this is shown to be independent of t in Appendix B. However, it is known that the Runge-Kutta integration scheme used in this paper is not energy conserving (Engle et al, 2005), hence in figure 6 we plot the percentage energy error and the total vessel energy…”

Section: Energy Budget For Simulationsmentioning

confidence: 96%

The pendulum-slosh problem: Simulation using a time-dependent conformal mapping

Turner

Bridges

Ardakani

2015

Journal of Fluids and Structures

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-AbstractSuspending a rectangular vessel which is partially filled with fluid from a single rigid pivoting pole produces an interesting theoretical model with which to investigate the dynamic coupling between fluid motion and vessel rotation. The exact equations for this coupled system are derived with the fluid motion governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion governed by a modified forced pendulum equation. The nonlinear equations of motion for the fluid are solved numerically via a timedependent conformal mapping, which maps the physical domain to a rectangle in the computational domain with a time dependent conformal modulus. The numerical scheme expresses the implicit free-surface boundary conditions as two explicit partial differential equations which are then solved via a pseudospectral method in space. The coupled system is integrated in time with a fourth-order Runge-Kutta method. The starting point for the simulations is the linear neutral stability contour discovered by Turner, Alemi Ardakani & Bridges (2014, J. Fluid Struct. 52, 166-180). Near the contour the nonlinear results confirm the instability boundary, and far from the neutral curve (parameterised by longer pole lengths) nonlinearity is found to significantly alter the vessel response. Results are also presented for an initial condition given by a superposition of two sloshing modes with approximately the same frequency from the linear characteristic equation. In this case the fluid initial conditions generate large nonlinear vessel motions, which may have implications for systems designed to oscillate in a confined space or on the slosh-induced-rolling of a ship.

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Monitoring energy drift with shadow Hamiltonians

Cited by 74 publications

References 8 publications

Equipartition and the Calculation of Temperature in Biomolecular Simulations

Equipartition and the Calculation of Temperature in Biomolecular Simulations

Symplectic integration for the collisional gravitationalN-body problem

The pendulum-slosh problem: Simulation using a time-dependent conformal mapping

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