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), or a similar two-stage Runge–Kutta method. It is named after Karl 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:

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

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.

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

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

where $h$ is the step size and $t_{i+1}=t_{i}+h$.

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Derivation

${\text{Slope}}_{\text{left}}=f(x_{i},y_{i})$

${\text{Slope}}_{\text{right}}=f(x_{i}+h,y_{i}+hf(x_{i},y_{i}))$

${\text{Slope}}_{\text{ideal}}={\frac {1}{2}}({\text{Slope}}_{\text{left}}+{\text{Slope}}_{\text{right}})$

Using the principle that the slope of a line equates to the rise/run, the coordinates at the end of the interval can be found using the following formula:

${\text{Slope}}_{\text{ideal}}={\frac {\Delta y}{h}}$

$\Delta y=h({\text{Slope}}_{\text{ideal}})$

$x_{i+1}=x_{i}+h$, $\textstyle y_{i+1}=y_{i}+\Delta y$

$y_{i+1}=y_{i}+h{\text{Slope}}_{\text{ideal}}$

$y_{i+1}=y_{i}+{\frac {1}{2}}h({\text{Slope}}_{\text{left}}+{\text{Slope}}_{\text{right}})$

$y_{i+1}=y_{i}+{\frac {h}{2}}(f(x_{i},y_{i})+f(x_{i}+h,y_{i}+hf(x_{i},y_{i})))$

The accuracy of the Euler method improves only linearly with the step size is decreased, whereas the Heun Method improves accuracy quadratically . 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 (also known as Ralston's method) has the Butcher tableau:

	0
	2/3	2/3
		1/4	3/4

This method minimizes the truncation error.