Heun's method

In mathematics and computational science, Heun's method may refer to the improved^[1] or modified Euler's method (that is, the explicit trapezoidal rule^[2]), 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:

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

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

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

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

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