# Appropiate Artificial Boundary Conditions for the radial part of the Klein Gordon equation?

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?

• 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. Nov 23, 2021 at 9:56
• @davidhigh I am not at all familiar with this approach. Can you suggest a reference where I can read about it? Nov 23, 2021 at 14:06
• 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 $x_B$. Nov 23, 2021 at 15:31