Velocity Verlet method ($x_{n+1}=x_n+h{v}_n+\frac{1}{2}{h}^2{a}_n$)

For an ordinary differential equation $\ddot{x}=f(x,t)$ with acceleration independent of velocity, the velocity Verlet method advances position and velocity byHere $h$ 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.