Error and timing analysis of multiple time-step integration methods for molecular dynamics

Molecular dynamics simulations of biomolecules performed using multiple time-step integration methods are hampered by resonance instabilities. We analyze the properties of a simple 1D linear system integrated with the symplectic reference system propagator MTS (r-RESPA) technique following earlier work by others. A closed form expression for the time step dependent Hamiltonian which corresponds to r-RESPA integration of the model is derived. This permits us to present an analytic formula for the dependence of the integration accuracy on short-range force cutoff range. A detailed analysis of the force decomposition for the standard Ewald summation method is then given as the Ewald method is a good candidate to achieve high scaling on modern massively parallel machines. We test the new analysis on a realistic system, a protein in water. Under Langevin dynamics with a weak friction coefficient (ζ = 1   ps^-1) to maintain temperature control and using the SHAKE algorithm to freeze out high frequency vibrations, we show that the 5 fs resonance barrier present when all degrees of freedom are unconstrained is postponed to ≈ 12   fs. An iso-error boundary with respect to the short-range cutoff range and multiple time step size agrees well with the analytical results which are valid due to dominance of the high frequency modes in determining integrator accuracy. Using r-RESPA to treat the long range interactions results in a 6× increase in efficiency for the decomposition described in the text.