Midpoint Method

The midpoint method is a second-order Runge-Kutta method that evaluates the gradient at the midpoint of the step interval.

$$y_{i+1}=y_i+dt\cdot g\left(t_i+\frac{dt}{2},y_i+\frac{dt}{2}\cdot g(t_i,y_i)\right)$$

The method estimates $y$ at $t_i+dt/2$ using one Euler half-step, then uses that midpoint gradient to advance the full step.

Order

Second-order: Global truncation error $O(d{t}^2)$, Local truncation error $O(d{t}^3)$.

The midpoint method achieves second-order accuracy with only two evaluations of $g$ per step (compared to four for Fourth order Runge-Kutta).

Runge-Kutta methods | Ralston method | Fourth order Runge-Kutta | Order of a method | Explicit Euler method