E. Barth and T. Schlick

Effective timestepping algorithms for biomolecular dynamics simulations have been developed which separate temporal and spatial scales of the model and resolve each scale appropriately. Force splitting or multiple timestep (MTS) methods have been discussed for 20 years or more. Essentially, these schemes update fast and slow forces at different frequencies, incorporating the slow forces into the discretization by various approximation techniques. Simple extrapolation formulas for this purpose did not immediately gain prevalence for biomolecular simulations due to an energy drift in the computed trajectories. More popular now are MTS variants that incorporate the slow forces via periodic impulses, and thereby possess the favorable long-time energy preservation of traditional single-timestep methods. In a number of recent works, however, it has been observed that the efficiency gain possible with these impulse approaches is limited by resonance --- a timestep barrier due to a numerical artifact occurring when the timestep is related to the period of the fastest motion present in the dynamics.

In this paper, we study the stability of these two approaches, extrapolation and impulses, to force-splitting in Newtonian and Langevin dynamics. Using a simple linear test system, we consider the energy drift of the former and the resonance-related artifacts of the latter. We show that 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; one such approach is the LN method, detailed in the companion paper.