= Velocity Verlet method
{title2=$x_{n+1}=x_n+h v_n+\tfrac12h^2a_n$}
For an <ordinary differential equation> $\ddot x=f(x,t)$ with <acceleration> independent of <velocity>, the <velocity Verlet method> advances <position> and <velocity> by
$$
a_n=f(x_n,t_n),\qquad x_{n+1}=x_n+h v_n+\frac{h^2}{2}a_n,\qquad
v_{n+1}=v_n+\frac h2\bigl(a_n+f(x_{n+1},t_n+h)\bigr).
$$
Here $h$ is the time step and $t_{n+1}=t_n+h$. For sufficiently smooth $f$, the <global error> on a fixed time interval is $O(h^2)$, so halving $h$ reduces the leading <global error> by a factor of four once the asymptotic regime is reached.
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