We note that the truncation error goes like $O(h^4)$ since all the odd terms cancel when we add the two Taylor expansions. We see also that the velocity is not directly included in the equation since the function $x^{(2)}=a(x,t)$ is supposed to be known. If we need the velocity however, we can compute it using the well-known formula $$x^{(1)}_i=\frac{x_{i+1}-x_{i-1}}{2h}+O(h^2).$$ We note that the velocity has a truncation error which goes like $O(h^2)$. In for example so-called Molecular dynamics calculations, since the acceleration is normally known via Newton's second law, there is seldomly a need for computing the velocity.

We note also that the algorithm for the position is not self-starting since, for $i=0$ it depends on the value of $x$ at the fictitious value $x_{-1}$.

We can amend this by introducing the velocity Verlet method.