Given the theta method, $y_{n+1}=y_n+\theta hf_n+(1-\theta)hf_{n+1}$
Question: a) How do you calculate local error? b) If the method is 0-stable, what is the order of the method?
I know how to calculate local truncation error (LTE)
To do LTE I would just Taylor expand around $y_n$ and I would result with,(not doing the whole derivation)
$(\theta-\frac{1}{2})h^2y''(t_n)+(\frac{1}{2}\theta-\frac{1}{3})h^3y'''+0(h^4)$
I'm just not sure if this helps me get the local error.
a) The only thing I have for this is essentially something I did for Euler method where you get the numerical solution and subtract the true solution.
$\bar{y_n}-y(t_n)$
Where $y(t_n)$ is the true solution. The only problem is I don't know if this correct, honestly I'm lost with the local error.
b) For the order if it's o-stable, I'm assuming it would a 1st order because the theta method is a 1st order numerical method? Still not too sure.
Any help would be great! Thanks.
