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.