Verlet Integration - Error Terms

Error Terms

The local error in position of the Verlet integrator is as described above, and the local error in velocity is .

The global error in position, in contrast, is and the global error in velocity is . These can be derived by noting the following:

and

Therefore:

Similarly:

Which can be generalized to (it can be shown by induction, but it is given here without proof):

If we consider the global error in position between and, where, it is clear that:

And therefore, the global (cumulative) error over a constant interval of time is given by:

Because the velocity is determined in a non-cumulative way from the positions in the Verlet integrator, the global error in velocity is also .

In molecular dynamics simulations, the global error is typically far more important than the local error, and the Verlet integrator is therefore known as a second-order integrator.