Verlet-i/r-respa Is Limited by Nonlinear Instability

Abstract. This paper uncovers additional stability limitations of multiple time stepping (MTS) integrators for molecular dynamics (MD) that attempt to bridge time scales. In particular, it is shown that when constant-energy (NVE) simulations of Newton’s equations of motion are attempted using the MTS integrator Verlet-I/r-RESPA, there are nonlinear instabilities when the longest step size is a third or possibly a fourth of the period of the fastest motion in the system. This is demonstrated both through a thorough set of computer experiments and through the analysis of a nonlinear model problem. The observed and predicted instabilities match exactly. Previous work has identified and explained a linear instability for Verlet-I/r-RESPA 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. These nonlinear instabilities are a severe limitation, and they render Verlet-I/r-RESPA not much better than Verlet/leapfrog when long MD simulations are attempted. The main effect of this nonlinear instability is a mild but systematic drift in the energy that may invalidate long simulations. More aggressive multiple step sizes are possible with mild Langevin coupling, and its combination with the mollified impulse method permits step sizes 3.5 times larger than Verlet-I/r-RESPA while still retaining same accuracy.