Time integration algorithm

  The engine of a molecular dynamics program is its time integration algorithm, required to integrate the equation of motion of the interacting particles and follow their trajectory.

Time integration algorithms are based on finite difference methods, where time is discretized on a finite grid, the time step $\Delta t$ being the distance between consecutive points on the grid. Knowing the positions and some of their time derivatives at time t (the exact details depend on the type of algorithm), the integration scheme gives the same quantities at a later time $t+\Delta t$. By iterating the procedure, the time evolution of the system can be followed for long times.

Of course, these schemes are approximate and there are errors associated with them. In particular, one can distinguish between

• Truncation errors, related to the accuracy of the finite difference method with respect to the true solution. Finite difference methods are usually based on a Taylor expansion truncated at some term, hence the name. These errors do not depend on the implementation: they are intrinsic to the algorithm.
• Round-off errors, related to errors associated to a particular implementation of the algorithm. For instance, to the finite number of digits used in computer arithmetics.

Both errors can be reduced by decreasing $\Delta t$.For large $\Delta t$, truncation errors dominate, but they decrease quickly as $\Delta t$ is decreased. For instance, the Verlet algorithm discussed in §2.3.1 has a truncation error proportional to $\Delta t^4$ for each integration time step. Round-off errors decrease more slowly with decreasing $\Delta t$,and dominate in the small $\Delta t$ limit. Using 64-bit precision (corresponding to ``double precision'' when using Fortran on the majority of today's workstations) helps to keep round-off errors at a minimum.

Two popular integration methods for MD calculations are the Verlet algorithm and predictor-corrector algorithms. They are quickly presented in the sections below. For more detailed informations on time integration algorithms, the reader is referred to refs. [3,6] for a general survey, and to ref. [19] for a deeper analysis.