Heun's method

In mathematics and computational science, Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule^[1]), or a similar two-stage Runge–Kutta method. It is named after Karl L. W. M. Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods.

The procedure for calculating the numerical solution to the initial value problem via the improved Euler's method is:

$y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0,$

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.

$\tilde{y}_{i+1} = y_i + h f(t_i,y_i)$

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

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

The improved Euler'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

The other method referred to as Heun's method has the Butcher table^[2]:

	0
	2/3	1
		1/4	3/4

References

1. ^ Ascher, Uri M.; Petzold, Linda R. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-412-8 .
2. ^ Leader, Jeffery J. (2004), Numerical Analysis and Scientific Computation, Boston: Addison-Wesley, ISBN 0-201-73499-0 .