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:
“You can hardly convince a man of an error in a life-time, but must content yourself with the reflection that the progress of science is slow. If he is not convinced, his grandchildren may be. The geologists tell us that it took one hundred years to prove that fossils are organic, and one hundred and fifty more to prove that they are not to be referred to the Noachian deluge.”
—Henry David Thoreau (1817–1862)
“As for the terms good and bad, they indicate no positive quality in things regarded in themselves, but are merely modes of thinking, or notions which we form from the comparison of things with one another. Thus one and the same thing can be at the same time good, bad, and indifferent. For instance music is good for him that is melancholy, bad for him who mourns; for him who is deaf, it is neither good nor bad.”
—Baruch (Benedict)