3
$\begingroup$

I am trying to simulate the following equation using FDTD

$ \left(- \partial^2_t + \partial^2_x + V(x) \right) \psi(x,t) =0 $

subjected to the initial conditions $\psi(x,0) = f(x),~ \partial_t \psi(x,0) = g(x)$, the boundary conditions being that it is completely outgoing on the right boundary, and completely ingoing at the left boundary.

Obviously, I need to implement either absorbing/artificial boundary conditions (ABCs) or something fancy like Perfectly Matching Layers. Focusing on the former, I have tried using Mur's first order ABCs and the standard non-reflecting ABCs (following Everstein) just as it is done for the good old wave equation. However, I am getting a pretty huge reflected wave. Can someone suggest how to approach the problem and minimize reflections at the artificial boundary? Or, how I may construct a better ABC?

Improve this question
$\endgroup$
3
  • $\begingroup$ If it's just about reflections, why not use a complex absorbing potential? I.e. replace your $V(x) -> V(x) + iC(x)$ where $C(x)$ is a negative function which smoothly sets in at the boundary. $\endgroup$
    – davidhigh
    Nov 23, 2021 at 9:56
  • $\begingroup$ @davidhigh I am not at all familiar with this approach. Can you suggest a reference where I can read about it? $\endgroup$
    – noir1993
    Nov 23, 2021 at 14:06
  • $\begingroup$ The method itself also runs under the name absorbing boundary conditions. See here for a review, but you won't need it to apply the method. Just pick $C(x) = -\epsilon \cdot (x-x_B)^2$, e.g., and see how your wave gets absorbed in the region outside the boundary $x_B$. $\endgroup$
    – davidhigh
    Nov 23, 2021 at 15:31

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.