I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then bounces on a boundary.
The problem I have is that the numerics breaks down at the boundary, when the wave reaches this point. I have tried by "trial and error" to just change the way I compute derivatives at this point, but with no luck (I am using finite differences; pseudospectral methods do not work), and I have no idea on how to systematically try to improve the stability and I have no intuition of what can work and what will not work.
Does anyone have any tips on what to try when one encounters such problems? How can I systematically move forward to try to make my numerical scheme stable at the boundary?
edit: Basically, if someone just has a list of ideas to blindly try, that would also be great.