Runge-Kutta method

A family of one-step methods that sample the slope at several cleverly chosen points within a single step, then combine them into one weighted-average step.

A Runge-Kutta method advances $(t_n,y_n)$ to $(t_{n+1},y_{n+1})$ by evaluating $f$ at $s$ stage points $k_1,\ldots,k_s$ -- each later stage using the earlier ones' own slopes to predict where in the step it should be evaluated -- then setting $y_{n+1}=y_n+h\sum_i b_i k_i$ for fixed weights $b_i$ . Euler's method is the $s=1$ member of this family; Heun's method and the midpoint method are its two-stage ( $s=2$ ) members; RK4 is its most common four-stage member.

Examples

Euler's method is $s=1$ : it just uses $k_1=f(t_n,y_n)$ directly, with no earlier stages to build from.