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?