the midpoint method

Here is a different cure for Euler's start-of-step shortsightedness. Instead of averaging the start and end slopes (the way improved Euler does), the midpoint method reasons: the slope that best represents the whole step is probably the slope at its MIDDLE. So it takes a half-step forward to peek at the slope at the midpoint, then goes back and uses that single midpoint slope to make the full step. One well-chosen sample beats one badly-placed one.

Precisely, in two stages: first a half Euler step to the midpoint, y_mid = y_n + (h/2) * f(t_n, y_n), giving an estimate of y at time t_n + h/2. Then evaluate the slope there, k = f(t_n + h/2, y_mid), and use that one midpoint slope for the full step: y_(n+1) = y_n + h * k. The half-step is throwaway scaffolding; the midpoint slope is the real ingredient. Geometrically, you are following a single line whose tilt is read at the centre of the interval, which on a smoothly curving solution is a far better representative slope than the one at the left end.

Like improved Euler, the midpoint method is second order — its global error is of size h^2 — and it also costs two slope evaluations per step. The two are cousins: both are two-stage, second-order Runge-Kutta methods, just built from a different geometric idea (midpoint slope versus averaged endpoint slopes). Neither is dramatically better than the other in general; they are the standard examples used to show that there is a whole family of second-order methods, of which these are two natural members.