Implement synchronized shifting for child elements

[bennibolm]

In this PR, I implemented a first version where freshly refined elements are shifted simultaneously (with the same theta) towards the mean value of their parent element (if necessary).
For now, I haven't found an example where the implementation without simultaneous shifting preserves conservation and the one without doesn't. I'll test some more setups.
see trixi-framework#2396

[github-actions]

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[bennibolm]

All tests are working. They are only shown as failing because the codecov upload fails since it's a branch on my fork.

[bennibolm]

My first implementation is ready. Now, I first compute the (nonnegative) mean solution of elements that are going to be refined before the refinement process and then pass that information to the limiter. Then, all the 4 child elements are shifted w.r.t. this mean and with the same theta.
I'm already using this implementation in a branch of mine where I use mortars with subcell limiting. There it works very good.
Unfortunately, I'm struggling with finding a good example without subcell limiting. What were elixirs and setups where you found positivity issues with the solution transfer during the refinement/coarsening process?

[patrickersing]

@bennibolm Thanks a lot for moving this forward!

[patrickersing]

Unfortunately, I'm struggling with finding a good example without subcell limiting. What were elixirs and setups where you found positivity issues with the solution transfer during the refinement/coarsening process?

I remember that there were positivity issues in the elixir p4est_2d_dgsem/elixir_euler_double_mach_amr.jl. The simulation didn't crash, but you should see negative value if you check just before the limiter call in the refinement routine.

[bennibolm]

Thanks for all the comments. I will apply the suggestions and/or answer in the next few days.

I remember that there were positivity issues in the elixir p4est_2d_dgsem/elixir_euler_double_mach_amr.jl. The simulation didn't crash, but you should see negative value if you check just before the limiter call in the refinement routine.

Okay, thanks. I will try this.

[bennibolm]

I tested the p4est_2d_dgsem/elixir_euler_double_mach_amr.jl for the positivity issues. As I found out before and also wrote somewhere, the issue appears during the initialization process of the amr callback - i.e. when the initial refinement level is lower than some amr level, and we need refinement at the start.
In this specific example, we get some negative values for pressure after refinement. Applying the limiter fixes this issue - in both cases: with joint limiting for refined elements and without.

[patrickersing]

LGTM! Nice that you also managed to apply this strategy for the coarsening step.