Linear instability in multiple time stepping

Molecular dynamics for a classical unconstrained simulation requires the solution of Newton's equations of motion:

$$\mathbf{M}\frac{\displaystyle {\mathrm d}^2}{\displaystyle {\mathrm d}t^2}{X}(t) = -\nabla U(X(t))$$ (1)

where $\mathbf{M}$ is a diagonal matrix of atomic masses, $X(t)$ is the collective atomic position vector, and $U$ is the potential energy. $U$ has bonded terms, such as bond springs and angle torsions, and nonbonded terms such as electrostatic and Lennard Jones terms.

Multiple time stepping (MTS) integrators have been used to lengthen the time step for most of the interactions in the equations of motion. These methods evaluate different parts of the force at different frequencies.

One typical MTS integrator is the Verlet-I/r-RESPA multiple time stepping impulse method. In this method the force is split into different components whose dynamics correspond to different time scales, which are then represented as appropriately weighted impulses (with weights determined by consistency). The impulse method is

$$M\frac{\mathrm{d^{2}}}{\mathrm{d}t\mathrm{^{2}}}X=-\sum_{n^{\prime}=-\infty }^{\infty}\delta t \mbox{\boldmath$\delta$} (t-n^{\prime}\Delta t)\nabla U^{\mathrm{fast}}(X)-\sum_{n=-\infty}^{\infty}\Delta t \mbox{\boldmath$\delta$} (t-n\Delta t)\nabla U^{\mathrm{slow}}(X)<tex2html_comment_mark>17$$ (2)

where the partitioning of $U$ into $U^{\mathrm{fast}}$ and $U^{\mathrm{slow}}$ is chosen so that an appropriate time step $\Delta t$ for the slow part of the force is larger than a time step $\delta t$ for the fast part. In the formula, $\delta$ is the Dirac delta function.

For systems with flexible water, with flexibility in the bonds and angles, the fastest motion has a period of 10fs and this method permits an increase from 1 to 3fs in the length of the longest time step $\Delta t,$ with no drift, and to 4fs with little drift. It is completely unstable at 5fs. This can be explained with a linear model problem as a resonance that occurs at about half the period of the fastest motion in the system [83].