Thus, the midpoint method yields a global error with second-order accuracy for the position and first-order accuracy for the velocity. However, although these methods yield exact results for constant accelerations, the error increases in general with each time step.

One method that avoids this is the so-called half-step method. Here we define $$ \begin{equation} y^{(2)}_{n+1/2}=y^{(2)}_{n-1/2}+h a_{n}+O(h^2), \tag{29} \end{equation} $$ and $$ \begin{equation} y^{(1)}_{n+1}=y^{(1)}_{n}+hy^{(2)}_{n+1/2} +O(h^2). \tag{30} \end{equation} $$ Note that this method needs the calculation of \( y^{(2)}_{1/2} \). This is done using e.g., Euler's method $$ \begin{equation} y^{(2)}_{1/2}=y^{(2)}_{0}+h a_{0}+O(h^2). \tag{31} \end{equation} $$ As this method is numerically stable, it is often used instead of Euler's method.