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:
“We call contrary to nature what happens contrary to custom; nothing is anything but according to nature, whatever it may be, Let this universal and natural reason drive out of us the error and astonishment that novelty brings us.”
—Michel de Montaigne (1533–1592)
“Man is the only animal that can remain on friendly terms with the victims he intends to eat until he eats them.”
—Samuel Butler (1835–1902)