1+1 Radial wave equation- numerical. BC near origin

[irycio]

Homework Statement

Well, I'm not sure if this is a correct subforum to post my problem, but to me it does seem to me as an academic problem. One I can not solve, apparently.

Well, anyway. I'm solving the 1+1 radial wave equation using finite difference. I shifted my grid, so that the origin (r=0) is not one of it's points. I also applied advection boundary condition to prevent reflections.

Now, the problem I have left are the boundary conditions near r=0. Since I'm using five point stencil to approximate spatial derivative, I need to apply them to 2 first points.
I assumed, that since my initial conditions (f(r,t=0)=10.0*exp(-100* r^2)) are symmetric (actually I'm trying to receive an analytic solution f(r,t)=(r-t)/r * 10.0*exp(-100* (r-t)^2), so the initial time derivative is appropriate), so should be the solution (otherwise it would depend on the angles, and I don't want it to).

Hence whenever my five point stencil calls for a non-existent point, I replace it with it's mirror, that lies within my grid.

And so unfortunately that doesn't work as expected. All in all, the "real" solution can be <0 for a given r and t, whereas my solution remains >0 (well, actually at some points the value is equal to, say -0,008, but then around it there are points with value of, say, 0,005, so that it's I believe just some random noise).


Since I checked my code couple of times, checked the initial conditions, I believe the problem lies within the BCs. And so, any help will be much appreciated!

 

[irycio]

Wrong order of convergence while using method of lines

Hi!
Still fighting with radial wave equation :/. I've split it into 2 first order equations in time and am using method of lines to integrate it with RK4 as my time integrator, which is O(h^4). My spatial derivatives are are approximated using 3 point stencil (2nd derivative, O(h^2)) and 2 point centered scheme (1st derivative, present in radial wave eq., O(h^2) as well)). Now, I was comparing my results with analytical solution to check the convergence and eventually it seems that my code is only of a first order accuracy :( - half the step size, double the accuracy. Now, the only place I'm using the method of firs order is the absorbing boundary condition at one of the ends. I used the advection equation there and to avoid complications I used the simplest scheme to approximate a spatial derivative there. However, I start with a gaussian wave packet far from that boundary, so at least at the beginning it shouldn't make any difference.

The question hence is - what can I be doing wrong??