[Users] Poisson equation

[Roland Haas]

Hello Taishi,

the participants discussed your email at yesterday's public Einstein
Toolkit user call, see:
https://docs.einsteintoolkit.org/et-docs/Meeting_agenda

Unfortunately no experts in the CT_MultiLevel code were present.

I am certainly no expert myself but will try to offer some suggestions:

* CT_MultiLevel is indeed compatible (in fact uses and relies) on
  Carpet and the grid setup requires that you cover your solution
  domain with multiple refinement levels. 
* The Poisson equation can be solved and you can see eg an example in
  our gallery: http://einsteintoolkit.org/gallery/poisson/index.html
* Please see the mailing list for previous questions about
  CT_MultiLevel:
  http://lists.einsteintoolkit.org/pipermail/users/2020-August/007562.html
  http://lists.einsteintoolkit.org/pipermail/users/2020-August/007563.html
  http://lists.einsteintoolkit.org/pipermail/users/2020-August/007575.html
* Since CT_MultiLevel only supports Cartesian grids a boundary
  condition like 1/r which typically ends up being implemented as a
  robin type conditions a f(r) + b f'(r) = 0 is (to my understanding)
  not well posed since the r-derivative is not a normal derivative (in
  the normal direction of the boundary) on the x,y,z boundary faces. It
  would require a boundary that is a sphere. The TwoPunctures
  CT_MultiLevel test "cheats" by (also) solving the equation using
  TwoPunctures and then using the known TwoPunctures solution as a
  Dirichlet type boundary condition. You may be able to get away with a
  Dirichlet type b/c using M/r as the value where "M" is the desired
  ADM mass of the system. 

Hopefully someone with more in-depth understanding of CT_MultiLevel
will be able to chip in.

Yours,
Roland

> Dear all
> Hello, I’m a user of EinsteinToolkit.
> Now, I’m trying to solve Poisson's equation using ctthorn as initial data before time evolution on EinsteinToolkit.
> The boundary condition is 1/r.
> 
> I already implemented my thorns for the source term of the equation,
> and solved the equation using "CT_MutiLevel”.
> I checked if the final solution is real solution of the original equation, but
> it is not good solution.
> I changed the resolution, but it is not improved.
> 
> So, I may misunderstand how to use CT_MutiLevel thorns, and I have questions relate to "CT_MutiLevel" :
> 1)
> In order to understand correct behavior, 
> I want to try test simulation using poisson.par, which is one of the example parameter file.
> 
> But, it does not work.
> The output told me the grid structure is inconsistent,
> and I also try another example, but it also has same error.
> Are there other available par file for CT_MutiLevel ?
> 
> 2)
> To solve the elliptic equation with 1/r boundary condition, is CT_MutiLevel the best way ?
> If there are another thorn to solve elliptic equation in EinsteinToolkit, let me know.
> Since I want to use Carpet, the thorn should be compatible with Carpet.
> 
> 3)
> After solving elliptic equation, I want to solve time evolution.
> Since CT_MutiLevel is a solver using MultiGrid method, we must prepare several Carpet grids on whole domain.
> But, is it consist with time evolution ?
> If there is an example par file, let me know.
> 
> Best.
> Taishi.
> 
> ************************************
> Taishi Ikeda
> Physics Department
> Sapienza University of Rome (Italy)
> taishi.ikeda at uniroma1.it
> ************************************
> 
> 


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