Another method which one may encounter is the Euler-Richardson method with

$$\begin{equation} y^{(2)}_{n+1}=y^{(2)}_{n}+h a_{n+1/2}+O(h^2), \tag{32} \end{equation}$$ and $$\begin{equation} \tag{33} y^{(1)}_{n+1}=y^{(1)}_{n}+hy^{(2)}_{n+1/2} +O(h^2). \end{equation}$$