Beeman's algorithm

Abstract: Attention is drawn to the fact that some of the algorithms used in the simulation of molecular dynamics are less accurate than is commonly believed. In particular, we show that many of the “Verlet-equivalent” integration schemes are not equivalent to the Verlet algorithm, and consequently are not necessarily third order schemes which exhibit exact time-reversal symmetry. Of this class of algorithms, only Beeman's technique is found to generate the optimal positions and velocities for a third order technique. It is also pointed out that the method of constraints introduces errors of O(τ3) into the calculated position, and hence limits the accuracy of simulations that employ this method to second order.

Abstract: Step errors (local errors, called also truncation errors) of the algorithms used in molecular dynamics simulations may result in non-physical correlations between particle velocities, as well as in errors of thermodynamic properties of simulated systems (energy, pressure). The simulations of the Lennard-Jones liquid showed, that the influence is especially high for the Verlet velocity algorithm. Beeman's technique decreases the correlations between the velocities, but at high densities the values of the errors of general averages are close to that of the Verlet method. The influence of step errors can be decreased by about two orders of magnitude by applying the Cowell-Numerov 4th order implicit method (equivalent to the Gear 4th order method treated as an implicit one). The method is very stable (more stable than the Verlet one), and can be highly optimized by restricting iteration to the closest neighbors of a given particle. As a result, the method becomes more efficient than the higher order explicit symplectic methods.