midpoint method

The two-stage Runge-Kutta method that predicts the slope at the step's MIDPOINT (instead of its far end) and uses that single slope for the whole step.

The midpoint method is the two-stage Runge-Kutta method with $k_1=f(t_n,y_n)$ , $k_2=f\!\left(t_n+\tfrac h2,\ y_n+\tfrac h2 k_1\right)$ , $y_{n+1}=y_n+h\,k_2$ -- unlike Heun's method, which AVERAGES a starting slope and an ending slope, the midpoint method uses $k_1$ only to reach the midpoint, then trusts $k_2$ alone for the entire step.

Examples

For $y'=y$ , $y(0)=1$ , $h=0.1$ : $k_1=1$ , the predicted midpoint value is $1+0.05(1)=1.05$ , so $k_2=1.05$ , and $y_1=1+0.1(1.05)=1.105$ .