Heun's method

In mathematics and computational science, Heun's method (also called the modified Euler's method or the explicit trapezoidal rule), named after Karl L. W. M. Heun, is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It can be seen as an extension of the Euler method into a two-stage second-order Runge–Kutta method.

The procedure for calculating the numerical solution to the initial value problem

$y'(t)=f(t,y(t)),\qquad \qquad y(t_{0})=y_{0},$

by way of Heun's method, is to first calculate the intermediate value ${\tilde {y}}_{i+1}$ and then the final approximation $y_{i+1}$ at the next integration point.

${\tilde {y}}_{i+1}=y_{i}+hf(t_{i},y_{i})$

$y_{i+1}=y_{i}+{\frac {h}{2}}(f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1})).$

Derivation

The scheme can be compared with the implicit trapezoidal method, but with $f(t_{i+1},y_{i+1})$ replaced by $f(t_{i+1},{\tilde {y}}_{i+1})$ in order to make it explicit. ${\tilde {y}}_{i+1}$ is the result of one step of Euler's method on the same initial value problem.

So, Heun's method is a predictor-corrector method with forward Euler's method as predictor and trapezoidal method as corrector.

Runge–Kutta method

Heun's method is a two-stage Runge–Kutta method, and can be written using the Butcher tableau (after John C. Butcher):

	0
	1	1
		1/2	1/2