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Heun's method

In mathematics and computational science, Heun's method, 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

$y\text{'}\left(t\right) = f\left(t,y\left(t\right)\right), qquad qquad y\left(t_0\right)=y_0,$
by way of Heun's method, is to first calculate the intermediate value $tilde\left\{y\right\}_\left\{i+1\right\}$ and then the final approximation $y_\left\{i+1\right\}$ at the next integration point.
$tilde\left\{y\right\}_\left\{i+1\right\} = y_i + h f\left(t_i,y_i\right)$
$y_\left\{i+1\right\} = y_i + frac\left\{h\right\}\left\{2\right\}\left(f\left(t_i, y_i\right) + f\left(t_\left\{i+1\right\},tilde\left\{y\right\}_\left\{i+1\right\}\right)\right).$

Derivation

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

So, the 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 tableau

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