The Euler method is asymmetric in time, since it uses information about the derivative at the beginning of the time interval. This means that we evaluate the position at $y^{(1)}_1$ using the velocity at $y^{(2)}_0=v_0$. A simple variation is to determine $y^{(1)}_{n+1}$ using the velocity at $y^{(2)}_{n+1}$, that is (in a slightly more generalized form) $$\begin{equation} y^{(1)}_{n+1}=y^{(1)}_{n}+h y^{(2)}_{n+1}+O(h^2) \tag{20} \end{equation}$$ and $$\begin{equation} y^{(2)}_{n+1}=y^{(2)}_{n}+h a_{n}+O(h^2). \tag{21} \end{equation}$$ The acceleration $a_n$ is a function of $a_n(y^{(1)}_{n}, y^{(2)}_{n},t)$ and needs to be evaluated as well. This is the Euler-Cromer method.