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

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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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