Integrating a wavelike equation with absorbing boundary conditions

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

• I don’t quite follow why the advection equation appears? Are you solving the wave equation or advection-diffusion? This is for finite volume method, but may provide some insight danieljfarrell.github.io/FVM Jan 15 at 13:44
• @ boyfarrel I am solving the wave equation, but using the advection equation to account for the non-reflecting boundary conditions.
– Joel
Jan 15 at 13:51
• core.ac.uk/download/pdf/206022852.pdf hopefully useful Jan 15 at 21:59
• @boyfarrell Thanks I will check it out.
– Joel
Jan 16 at 8:33
• I'm always a bit skeptical about the use of absorbing boundary conditions (at least if it's not pedagogical). Why can't you just increase your grid? Should be easily possible as it is 1D. But if you need it anyways, here and here are two threads to start with. Jan 18 at 8:13