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

$y\text{'}(t)\; =\; f(t,y(t)),\; qquad\; qquad\; y(t\_0)=y\_0,$

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, 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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