Editing Verlet integration (section)

===Discretisation error===
The time symmetry inherent in the method reduces the level of local errors introduced into the integration by the discretization by removing all odd-degree terms, here the terms in <math>\Delta t</math> of degree three. The local error is quantified by inserting the exact values <math>\mathbf x(t_{n-1}), \mathbf x(t_n), \mathbf x(t_{n+1})</math> into the iteration and computing the [[Taylor expansion]]s at time <math>t = t_n</math> of the position vector <math>\mathbf{x}(t \pm \Delta t)</math> in different time directions:
:<math>\begin{align}
\mathbf{x}(t + \Delta t)
&= \mathbf{x}(t) + \mathbf{v}(t)\Delta t + \frac{\mathbf{a}(t) \Delta t^2}{2}
+ \frac{\mathbf{b}(t) \Delta t^3}{6} + \mathcal{O}\left(\Delta t^4\right)\\
\mathbf{x}(t - \Delta t)
&= \mathbf{x}(t) - \mathbf{v}(t)\Delta t + \frac{\mathbf{a}(t) \Delta t^2}{2}
- \frac{\mathbf{b}(t) \Delta t^3}{6} + \mathcal{O}\left(\Delta t^4\right),
\end{align}</math>
where <math>\mathbf{x}</math> is the position, <math>\mathbf{v} = \dot{\mathbf x}</math> the velocity, <math>\mathbf{a} = \ddot{\mathbf x}</math> the acceleration, and <math>\mathbf{b} = \dot{\mathbf a} = \overset{\dots}{\mathbf x}</math> the [[Jerk (physics)|jerk]] ([[third derivative]] of the position with respect to the time).

Adding these two expansions gives
:<math>\mathbf{x}(t + \Delta t) = 2\mathbf{x}(t) - \mathbf{x}(t - \Delta t) + \mathbf{a}(t) \Delta t^2 + \mathcal{O}\left(\Delta t^4\right).</math>
We can see that the first- and third-order terms from the Taylor expansion cancel out, thus making the Verlet integrator an order more accurate than integration by simple Taylor expansion alone.

Caution should be applied to the fact that the acceleration here is computed from the exact solution, <math>\mathbf a(t) = \mathbf A\bigl(\mathbf x(t)\bigr)</math>, whereas in the iteration it is computed at the central iteration point, <math>\mathbf a_n = \mathbf A(\mathbf x_n)</math>. In computing the global error, that is the distance between exact solution and approximation sequence, those two terms do not cancel exactly, influencing the order of the global error.