Time-Stepping Methods

Runge-Kutta Method

Midpoint Runge-Kutta

The midpoint Runge-Kutta method is given by $k_1=hf(t_n,y_n)$ $k_2=hf(t_n+\frac{h}{2},y_n+\frac{k_1}{2})$ $y_{n+1}=y_n+k_2$

with a LTE of $O(h^3)$.

Higher-Order Runge-Kutta Method

For a higher order Runge-Kutta method, the best known one is

$k_1=hf(t_n,y_n)$ $k_2=hf(t_n+\frac{h}{2},y_n+\frac{k_1}{2})$ $k_3=hf(t_n+\frac{h}{2},y_n+\frac{k_2}{2})$ $k_4=hf(t_n+h,y_n+k_3)$ $y_{n+1}=y_n+\frac{k_1}{6}+\frac{k_2}{3}+\frac{k_3}{3}+\frac{k_4}{6}$ with a LTE of $O(h^5)$.