RK4 (the classical fourth-order Runge-Kutta method)

The most widely used Runge-Kutta method: four slope samples per step, weighted 1-2-2-1, buying two more orders of accuracy than Euler's method for four times the work per step.

RK4 is the four-stage Runge-Kutta method $k_1=f(t_n,y_n)$ , $k_2=f\!\left(t_n+\tfrac h2,y_n+\tfrac h2k_1\right)$ , $k_3=f\!\left(t_n+\tfrac h2,y_n+\tfrac h2k_2\right)$ , $k_4=f(t_n+h,y_n+hk_3)$ , $y_{n+1}=y_n+\tfrac h6(k_1+2k_2+2k_3+k_4)$ -- local truncation error $O(h^5)$ , global error $O(h^4)$ .

Examples

Halving $h$ for RK4 shrinks the global error by a factor of $2^4=16$ -- far more per halving than Euler's method's own factor of 2.