Editing Verlet integration (section)

==Error terms==
{{Main article|Truncation error (numerical integration)}}

The global truncation error of the Verlet method is <math>\mathcal O\left(\Delta t^2\right)</math>, both for position and velocity.
This is in contrast with the fact that the local error in position is only <math>\mathcal O\left(\Delta t^4\right)</math> as described above. The difference is due to the accumulation of the local truncation error over all of the iterations.

The global error can be derived by noting the following:

:<math>\operatorname{error}\bigl(x(t_0 + \Delta t)\bigr) = \mathcal O\left(\Delta t^4\right)</math>

and

:<math>x(t_0 + 2\Delta t) = 2x(t_0 + \Delta t) - x(t_0) + \Delta t^2 \ddot{x}(t_0 + \Delta t) + \mathcal O\left(\Delta t^4\right).</math>

Therefore

:<math>\operatorname{error}\bigl(x(t_0 + 2\Delta t)\bigr) = 2\cdot\operatorname{error}\bigl(x(t_0 + \Delta t)\bigr) + \mathcal O\left(\Delta t^4\right) = 3\,\mathcal O\left(\Delta t^4\right).</math>

Similarly:

:<math>\begin{align}
\operatorname{error}\bigl(x(t_0 + 3\Delta t)\bigl) &= 6\,\mathcal O\left(\Delta t^4\right), \\[6px]
\operatorname{error}\bigl(x(t_0 + 4\Delta t)\bigl) &= 10\,\mathcal O\left(\Delta t^4\right), \\[6px]
\operatorname{error}\bigl(x(t_0 + 5\Delta t)\bigl) &= 15\,\mathcal O\left(\Delta t^4\right),
\end{align}</math>

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

:<math>\operatorname{error}\bigl(x(t_0 + n\Delta t)\bigr) = \frac{n(n+1)}{2}\,\mathcal O\left(\Delta t^4\right).</math>

If we consider the global error in position between <math>x(t)</math> and <math>x(t + T)</math>, where <math>T = n\Delta t</math>, it is clear that{{Citation needed|reason=Where is this proof taken from? It is not trivial.|date=July 2018}}

:<math>\operatorname{error}\bigl(x(t_0 + T)\bigr) = \left(\frac{T^2}{2\Delta t^2} + \frac{T}{2\Delta t}\right) \mathcal O\left(\Delta t^4\right),</math>

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

:<math>\operatorname{error}\bigr(x(t_0 + T)\bigl) = \mathcal O\left(\Delta t^2\right).</math>

Because the velocity is determined in a non-cumulative way from the positions in the Verlet integrator, the global error in velocity is also <math>\mathcal O\left(\Delta t^2\right)</math>.

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.