Nonlinear Resonance Artifacts in Molecular Dynamics Simulations
The intriguing phenomenon of resonance, a pronounced integrator-induced
corruption of a system's dynamics is examined for single molecular systems
subject to the classical equations of motion. This source of timestep limitation
is not well appreciated in general, and certainly analyses of resonance
patterns have been few in connection to biomolecular dynamics. Yet resonances
are present in the commonly used Verlet integrator, in symplectic implicit
schemes, and also limit the scope of current multiple-timestep methods
that are formulated as symplectic and reversible. The only general remedy
to date has been to reduce the timestep. For this purpose, we derive method-dependent
timestep thresholds (e.g., Tables 1 and 2) that serve as useful guidelines
in practice for biomolecular simulations. We also devise closely related
symplectic implicit schemes for which the limitation on the discretization
stepsize is much less severe. Specifically, we design methods to remove
third-order, or both third and fourth-order, resonances. These severe low-order
resonances can lead to instability or very large energies. Our tests on
two simple molecular problems (Morse and Lennard-Jones potentials), as
well as a 22-atom molecule, N-acetylalanyl-N'-methylamide, confirm this
prediction: our methods can delay resonances so that they occur only at
larger timesteps (EW method) or are essentially removed (LIM2 method).
Although stable for large timesteps by this approach, trajectories show
large energy fluctuations, perhaps due to the coupling with other factors
that induce instability in complex nonlinear systems. Thus the methods
developed here may be more useful for conformational sampling of biomolecular
structures. The analysis presented here for the blocked alanine model emphasizes
that one dimensional analysis of resonances can be applied to a more complex
multimode system to analyze resonance behavior, but that resonance due
to frequency coupling is more complex to pinpoint. More generally, instability
apparently due to numerically induced resonances, has been observed in
the application of the implicit midpoint scheme to vibrating structures
and could be expected also in the simulation of nonlinear wave phenomena;
in such applications it is adequate not to resolve the highest frequency
modes, so the proposed methods could be very useful.
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