A Error for perturbed solution vs Numeric

[member 428835]

Hi PF!

I'm solving an PDE where the analytic solution is called $F(x)$ (unknown). To approximate the analytic solution I made a naive expansion in some small parameter $\epsilon$ such that $F(x) = f_0(x)+\epsilon f_1(x)+O(\epsilon^2)$, where I know $f_0(x)$ and $f_1(x)$. I then solved the PDE numerically, let's call that solution $F_n$. Then the error $(F_n - (f_0(x)+\epsilon f_1(x)))/F_n$ should be $O(\epsilon^2)$. However, when I let $\epsilon=.9$ and then $\epsilon=.8$ my error is still about $0.15$. How can this be?

I should say I know the numeric and asymptotic solutions are correct.

 

[Mark44]

Hi PF!

I'm solving an PDE where the analytic solution is called $F(x)$ (unknown). To approximate the analytic solution I made a naive expansion in some small parameter $\epsilon$ such that $F(x) = f_0(x)+\epsilon f_1(x)+O(\epsilon^2)$, where I know $f_0(x)$ and $f_1(x)$. I then solved the PDE numerically, let's call that solution $F_n$. Then the error $(F_n - (f_0(x)+\epsilon f_1(x)))/F_n$ should be $O(\epsilon^2)$. However, when I let $\epsilon=.9$ and then $\epsilon=.8$ my error is still about $0.15$. How can this be?

I should say I know the numeric and asymptotic solutions are correct.

$\epsilon=.9$ isn't really all that small. Do you get better results for, say, $\epsilon=.1$?

 

[member 428835]

$\epsilon=.9$ isn't really all that small. Do you get better results for, say, $\epsilon=.1$?

So for $\epsilon=0.01$ I'm getting an error of 0.165 but for $\epsilon=0.2$ the error is 0.14. How could this ever be possible?

 

[Mark44]

Maybe your approximate solution isn't that close to the actual solution...