I am trying to numerically solve the following equation:
$\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}+V(x) \phi(x, t)=0$ On some domain, with: $\phi(x, 0) = I(x)$
$\frac{\partial \phi(x, 0)}{\partial t} = 0$
And with non-reflecting (absorbing) boundary conditions.
I am using finite difference method, so I get the following:
$\phi_{i}^{j+1}=-\phi_{i}^{j-1}+2 \phi_{i}^{j}+r^2\left(\phi_{i+1}^{j}-2 \phi_{i}^{j}+\phi_{i-1}^{j}\right) - (\Delta t)^2V_{i}\phi_{i}^{j}\,\,\, (1)$, where $j$ is used for time and $i$ for position, $r = \frac{\Delta t}{\Delta x}$ and $\Delta t$, $\Delta x$ are the time and space steps.
For $j = 0$, I use the $\frac{\partial \phi(x, 0)}{\partial t} = 0$ condition and the central difference approximation, $\left(\frac{\partial \phi}{\partial t}\right)_{j} = \frac{\phi_{j+1}-\phi_{j-1}}{2 \Delta x}$, to get:
$\phi_{i}^{1}= \phi_{i}^{0}+\frac{r^2}{2}\left(\phi_{i+1}^{0}-2 \phi_{i}^{0}+\phi_{i-1}^{0}\right) - \frac{(\Delta t)^2}{2}V_{i}\phi_{i}^{0} \,\,\, (1')$
To accommodate the boundary conditions I use the advection equation $\frac{\partial \phi}{\partial t}=- \frac{\partial \phi}{\partial x}$, again I use the central difference approximation, and I substitute into (1) to get:
$(1+r) \phi_{i, j+1}=2 r^{2} \phi_{i-1, j}+2\left(1-r^{2}\right) \phi_{i, j}-(1-r) \phi_{i, j-1}\,\,\, (2)$.
For $j = 0$ I use the same idea as before and I get:
$\phi_{i}^{1} = (1 - r^2) \phi_{i}^{0}+r^2\phi_{i-1}^{0}-\frac{(\Delta t)^2}{2}V_{i}\phi_{i}^{0}\,\,\, (2')$.
Now I am trying to implement this using python, and I use the following logic:
For the interior points I use equations (1), and (1'). For the boundary points I use equations (2), and (2'). But the results that I get are those for reflecting boundary conditions not absorbing ones.
I am using python, and if necessary I can post the code that I have written.
Any help or advice is very much appreciated. I also would appreciate any resources or references. Thanks
