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.
Read more about this topic: Verlet Integration
Famous quotes containing the words error and/or terms:
“Custom calls me to’t.
What custom wills, in all things should we do’t,
The dust on antique time would lie unswept,
And mountainous error be too highly heaped
For truth to o’erpeer.”
—William Shakespeare (1564–1616)
“One of your biggest jobs as a parent of multiples is no bigger than simply talking to your children individually and requiring that they respond to you individually as well. The benefits of this kind of communication can be enormous, in terms of the relationship you develop with each child, in terms of their language development, and eventually in terms of their sense of individuality, too.”
—Pamela Patrick Novotny (20th century)