The Five Femtosecond Time Step Barrier

Simulation of the dynamics of biomolecules requires the use of a time step in the range 0.5–1 fs to obtain acceptable accuracy. Nevertheless, the bulk of the CPU time is spent computing interactions, such as those due to long-range electrostatics, which vary hardly at all from one time step to the next. This unnecessary computation is dramatically reduced with the use of multiple time stepping methods, such as the Verlet-I/r-RESPA method, which is based on approximating “slow” forces as widely separated impulses. Indeed, numerical experiments show that time steps of 4 fs are possible for these slow forces but unfortunately also show that a long time step of 5 fs results in a dramatic energy drift. Moreover, this is less pronounced if one uses a yet larger long time step! The cause of the problem can be explained by exact analysis of a simple two degree-of-freedom linear problem, which predicts numerical instability if the time step is just less than half the period of the fastest normal mode. To overcome this, a modification of the impulsive Verlet-I/r-RESPA method is proposed, called the mollified impulse method. The idea is that one modifies the slow part of the potential energy so that it is evaluated at “time averaged” values of the positions, and one uses the gradient of this modified potential for the slow part of the force. Various versions of the algorithm are implemented for water and numerical results are presented.