[Users] Boundary issues solving wave equations with MoL

[Haas, Roland]

Hello Severin,

sorry for the very long delay.

I am no real expert on this others on this list may be better suited.

My worry is that you are assuming that outside of the star (and on the
star surface I guess) all time derivatives (and the spatial one in drPi
it seems) vanish. This would be true of the TOV solution was actually
numerically stationary. For the "usual" Valencia formulations of
hydrodynamics this is not quite true though for (at least two reasons):
the grid is Cartesian typically leading to interpolation errors when
interpolation the correct static TOV data found in the initial data
thorn into the grid, this leads to oscillations which can move the
stellar surface. Your code may be able to deal with those (and the
Tuebingen group are after all experts on this).

The second issue that over time atmosphere actually accretes onto the
star thus increasing its mass. This happens b/c while there is a
analytic stationary solution the handling of eg the pressure gradient
and gravity in the ET (one is a flux, the other is a source term) tends
to make it very difficult for the code to find a numerical equilibrium
solution so there is going to be some dynamics present in the
simulation.

Does you scheme lead to stable solutions when implemented eg in
spherical symmetry?

None of this really helps you actually fix the issue I am afraid. The
usual solution seems to be to not impose and b/c on the stellar surface
but only very far away and accept the fact that the surface will blur
out. A sharp surface can (apparently, I have never tried this) be
obtained using Wolfgang Kastaun's scheme (which I assume your group is
well familiar with) in the Pizza code:

https://publikationen.uni-tuebingen.de/xmlui/bitstream/handle/10900/49026/pdf/dissertation_kastaun.pdf?sequence=1&isAllowed=y

Yours,
Roland

> Dear all,
> 
> I'm trying to solve a wave equation that describes radial oscillations of a TOV star, depending on radius r and t in Llama multipatch coordinates (Thornburg04).
> 
> The equation has a structure of
> 
> Ẍ=AX′+BX″+CX\ddot{X} = A X' + B X'' + C X
> 
> I rewrote it as set of coupled equations to evolve it with the MoL thorn:
> 
> X˙=Π
> 
> H˙=Π′\dot{H} = \Pi'
> 
> Π˙=AH+BH′+CX\dot{\Pi} = AH + B H' +C X
> 
> or as noted in my code as it is attached:
> 
> Pidot = A*H + B*drH + C*Xi
> Hdot = drPi
> Xidot = Pi
> 
> For MoL registered as evolved variables are Xi with rhs Xidot, Pi with rhs Pidot and H with rhs Hdot.
> 
> The spatial derivatives drH and drPi are also calculated during the schedule of MoL_CalcRHS in my thorn.
> 
> Boundary conditions are applied for r>R and an interval near the TOV radius r=R.
> 
> 
> if (grid_r(i,j,k) >= (TOV_surface - rprec) .AND. grid_r(i,j,k) <= (TOV_surface + rprec) )then
>            H(i,j,k) = 0
>            drPi(i,j,k) = 0
>            else if ( grid_r(i,j,k) > (TOV_surface + rprec) ) then
>            Xi(i,j,k) = 0
>            Pi(i,j,k) = 0
>            H(i,j,k) = 0
> 
> end if
> 
> if (grid_r(i,j,k) > (TOV_surface + rprec) )then
>            Xidot(i,j,k) = 0
>            Hdot(i,j,k) = 0
>            Pidot(i,j,k) = 0
>            else if (grid_r(i,j,k) >= (TOV_surface - rprec) .AND. grid_r(i,j,k) <= (TOV_surface + rprec) )then
>            Hdot(i,j,k) = 0
>            Pidot(i,j,k) = B(i,j,k)*drH(i,j,k)  + C(i,j,k)*Xi(i,j,k)
> 
> end if
> 
> 
> Problems occur close to the surface of the star. My evolved variables start do diverge close to the surface after a few iterations. Looking at the data it seems that the divergence is founded by the values of H. I tried following things to encircle the issues:
> 
>   * If I put Pidot = B*drH + C*Xi the values seem to be ok, whereas for
>     Pidot = A*H the divergence appears. The A-factor only amplifies this
>     behavior. If I put Pidot = H it behaves the same, but much slower.
>   * If I use drXi instead of H (as it is commented out), it does not
>     make any difference.
>   * If I enlarge the size of the interval (e.g. TOV_surface + 5*rprec)
>     the same divergence appears, but shifted towards grid points next to
>     the interval. Also the divergence appears a few iterations later.
>   * If I change the drXi or H values close to the surface manually (e.g.
>     using a backsided differentiation, or put specific values by hand)
>     it also shifts the divergence (like above).
> 
> So far I don't know how to remedy this issue, but maybe I'm overlooking something obvious.
> 
> Does anyone have an idea on what I could try?
> 
> Thanks a lot!
> 
> Best regards and merry Christmas,
> 
> Severin Frank
> 



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