The fourth-order Runge-Kutta, RK4, which we will employ in the solution of various differential equations below, has the following algorithm $$ k_1=hf(t_i,y_i) \hspace{0.5cm} k_2=hf(t_i+h/2,y_i+k_1/2) $$ $$ k_3=hf(t_i+h/2,y_i+k_2/2)\hspace{0.5cm} k_4=hf(t_i+h,y_i+k_3) $$ with the final result $$ y_{i+1}=y_i +\frac{1}{6}\left( k_1 +2k_2+2k_3+k_4\right). $$ Thus, the algorithm consists in first calculating \( k_1 \) with \( t_i \), \( y_1 \) and \( f \) as inputs. Thereafter, we increase the step size by \( h/2 \) and calculate \( k_2 \), then \( k_3 \) and finally \( k_4 \). The global error goes as \( O(h^4) \).