Extrapolation versus impulse in multiple-timestepping schemes. II. Linear analysis and applications to Newtonian and Langevin dynamics.

Force splitting or multiple timestep (MTS) methods are effective techniques that accelerate biomolecular dynamics simulations by updating the fast and slow forces at different frequencies. Since simple extrapolation formulas for incorporating the slow forces into the discretization produced notable energy drifts, symplectic MTS variants based on periodic impulses became more popular. However the efficiency gain possible with these impulse approaches is limited by a timestep barrier due to resonance-a numerical artifact occurring when the timestep is related to the period of the fastest motion present in the dynamics. This limitation is lifted substantially for MTS methods based on extrapolation in combination with stochastic dynamics, as demonstrated for the LN method in the companion paper for protein dynamics. To explain our observations on those complex nonlinear systems, we examine here the stability of extrapolation and impulses to force-splitting in Newtonian and Langevin dynamics. We analyze for a simple linear test system the energy drift of the former and the resonance-related artifacts of the latter technique. We show that two-class impulse methods are generally stable except at integer multiples of half the period of the fastest motion, with the severity of the instability worse at larger timesteps. Extrapolation methods are generally unstable for the Newtonian model problem, but the instability is bounded for increasing timesteps. This boundedness ensures good long-timestep behavior of extrapolation methods for Langevin dynamics with moderate values of the collision parameter. We thus advocate extrapolation methods for efficient integration of the stochastic Langevin equations of motion, as in the LN method described in paper I.

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