Predictor–corrector method

Euler trapezoidal example

Example of an Euler - trapezoidal predictor-corrector method.

In this example ${\displaystyle h}$  = ${\displaystyle \Delta {t}}$ , ${\displaystyle t_{i+1}=t_{i}+\Delta {t}=t_{i}+h}$ 

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

First calculate an initial guess value ${\displaystyle {\tilde {y}}_{g}}$  via Euler:

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

Next, improve the initial guess through iteration of the trapezoidal rule. This iteration process normally converges quickly.

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

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

...

${\displaystyle {\tilde {y}}_{g+n}=y_{i}+{\frac {h}{2}}(f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{g+n-1})).}$ 

This iteration process is repeated until some fixed value n or until the guesses converge to within some error tolerance e :

${\displaystyle |{\tilde {y}}_{g+n}-{\tilde {y}}_{g+n-1}|\leq e}$ 

then use the final guess as the next step:

${\displaystyle y_{i+1}={\tilde {y}}_{g+n}.}$ 

Note that the overall error is unrelated to convergence in the algorithm but instead to the step size and the core method, which in this example is a trapezoidal, (linear) approximation of the actual function. The step size h ( ${\displaystyle \Delta {t}}$  ) needs to be relatively small in order to get a good approximation. See also stiff equation.