However, we do not know the value of \( y_{i+1/2} \). Here comes thus the next approximation, namely, we use Euler's method to approximate \( y_{i+1/2} \). We have then $$ \begin{equation} y_{(i+1/2)}=y_i + \frac{h}{2}\frac{dy}{dt} = y(t_i) + \frac{h}{2}f(t_i,y_i). \tag{39} \end{equation} $$ This means that we can define the following algorithm for the second-order Runge-Kutta method, RK2. $$ \begin{equation} k_1=hf(t_i,y_i), \tag{40} \end{equation} $$ $$ \begin{equation} k_2=hf(t_{i+1/2},y_i+k_1/2), \tag{41} \end{equation} $$ with the final value $$ \begin{equation} y_{i+i}\approx y_i + k_2 +O(h^3). \tag{42} \end{equation} $$

The difference between the previous one-step methods is that we now need an intermediate step in our evaluation, namely \( t_i+h/2 = t_{(i+1/2)} \) where we evaluate the derivative \( f \). This involves more operations, but the gain is a better stability in the solution.