[Users] Boundary issues solving wave equations with MoL

[Roland Haas]

Hello Severin,

sorry for not responding to your email.

About the stable solution: I was wondering if you had a 1d spherically symmetric toy code where you could have tested that your initial data does indeed lead to almost no evolution or if even in a spherical 1d code you would see material accreting or some oscillation on the surface.

Are you evolution equations for the difference to the static background (TOV solution) valid only in the linear regime or also for non-linear deviations? 

My worry would be that even when looking at a difference only, you may be hit by the fact that computing eg a pressure gradient of the TOV solution on the Cactus grid and computing a gravitational "force" will not give perfect balance which would (I expect) show up in your difference equation as driving force potentially shifting where the equilibrium is.

From the look of your plot you seem to be experiencing some sort of instability which causes the amplitude of the oscillations to grow more and more, with smaller timesteps helping a bit but not ultimately curing the issue. You could (his is just a guess) try and play with different MoL integration schemes, eg use the RK2 scheme (very dissipative but stable) to see if this would help (I would stay away from ICN and other non RK schemes as I have no idea how recently those were used for any actual simulations).

In principle you should be able to use MoL for this type of thing. The cases where MoL is difficult to use is trying to use it to evolve a grid scalar or any other quantity that is not defined on the mesh refined grid that Carpet sets up. This affects things like eg integrating this shift at the location of the puncture to follow the puncture (puncture tracking) or trying to integrate particle trajectories (which are basically the same as multiple punctures).

Yours,
Roland

> Thank you Roland for this detailed answer!
> 
> First of all, some things in my thorn have been changed in the last
> weeks. Therefore I attached the updated files to this mail. I
> reformulated the coefficients in my initial data. Now, the code runs
> much better but is still diverging. However the divergence seems to
> come from the center of the star instead. Right now I'm using the
> thornburgnc coordinates and made the spherical part as large as
> possible as this is the currently most stable configuration for my
> code.
> 
> I attached also a plot of the values of my evolved variable for a
> point at r=2 for two different time steps, where you can see the
> behavior of the evolution.
> 
> To be honest I'm not 100% sure what you meant with "stable solutions
> when implemented eg in spherical symmetry?"
> 
> What I'm basically trying to do is to have a wave equation that
> approximates the radial oscillation equations for small amplitudes.
> The coefficients of this wave equation are filled with static values
> using the TOVSolver. Then I only evolve the the differential
> equations, using the MoL thorn. There is no evolution of the GRHydro
> quantities or the ADM quantities themselves. In principal, time
> integration of these equations should be possible.
> 
> I wonder if I'm using the MoL (or other) thorn(s) wrong for this
> purpose?
> 
> Thank you a lot!
> 
> Best regards,
> 
> Severin
> 
> 



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