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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