Heun's method

Remarks

Proof idea: by Taylor's theorem, $f\big(t + h, y + h f\big) = f + h(f_t + f_y f) + O(h^2)$ at $(t, y)$, so the increment is $h f + \frac{h^2}{2}(f_t + f_y f) + O(h^3)$, which matches the Taylor expansion of $y(t + h)$ up to $h^2$. In Butcher form it has $c = (0, 1)$, $a_{21} = 1$ and $b = (\tfrac12, \tfrac12)$. Example: for $y' = y$, $y(0) = 1$ and $h = 0.1$, one step gives $\tilde y_1 = 1.1$ and $y_1 = 1 + 0.05 (1 + 1.1) = 1.105$, against $e^{0.1} \approx 1.10517$; the Euler method gives $1.1$. If $f$ does not depend on $y$, the method is the trapezoidal rule for $\int f(t)\,dt$.

Going deeper

For a student

Heun's method is also called the improved Euler method or the explicit trapezoidal method. It is the simplest predictor–corrector scheme: an explicit Euler predictor followed by one correction with the trapezoidal rule. Like the midpoint method, it is a two-stage explicit Runge–Kutta method of order $2$. The difference between its result and the Euler predictor gives a cheap estimate of the local error for step size control.