Verlet-I/r-RESPA/Impulse is limited by nonlinear instabilities

This paper shows that in molecular dynamics (MD) when constant-energy (NVE) simulations of Newton's equations of motion are attempted using the multiple time stepping (MTS) integrator Verlet-I/r-RESPA/Impulse, there are nonlinear instabilities when the longest step size is a third or possibly a fourth of the period(s) of the fastest motion( s) in the system. This is demonstrated both through a thorough set of computer experiments and through the analysis of a nonlinear model problem. The numerical experiments include not only the unconstrained dynamics simulation of a droplet of flexible water and a flexible protein, but also the constrained dynamics simulation of a solvated protein, representing a range of simulation protocols commonly in use by biomolecular modelers. The observed and predicted instabilities match exactly. Previous work has identified and explained a linear instability for Verlet-I/r-RESPA/Impulse at around half the period of the fastest motion. Mandziuk and Schlick discovered nonlinear resonances in single time stepping MD integrators, but unstable nonlinear resonances for MTS integrators are reported here for the first time. This paper also offers an explanation on the instability of MTS constrained molecular dynamics simulations of explicitly solvated proteins. More aggressive multiple step sizes are possible with mild Langevin coupling or targeted Langevin coupling, and its combination with the mollified Impulse method permits step sizes 3 to 4 times larger than Verlet-I/r-RESPA/Impulse while still retaining some accuracy.