I don't understand how to find the correct step-size $h$ for the Euler method. My script says the following:
One method consists in computing the numerical solution for an arbitrary $h$ and then $2h$. The Richardson extrapolation gives an estimate of $e = \max_t|y(t,2h)-y(t,h)|$ of the error. When the error is smaller than the tolerance, we keep the result and start from $2y(t,h)-y(t,h)$. If the error is larger we restart with $h/2$ until we reach the tolerance.
( $y(t,2h)$ means approximation with $2h$)
I don't understand why the Richardson extrapolation is mentioned. For what do I have to use it? Can I not just calculate $y(t,2h)$ and $y(t,h)$ and see the error?
Best Answer
You can just check the error, and not bother with Richardson extrapolation, that's fine. But once you've paid the computational price of calculating the error, you might as well use Richardson extrapolation to reduce the error further; it costs you essentially nothing.