Allow a-posteriori corrections for du

[bennibolm]

See #2948.
To fix this, I added a SubcellLimiterIDPCorrection function which corrects du accordingly.
Additionally, I added a check whether SubcellLimiterIDPCorrection is the first stage callback in the custom SSP time integration method. It doesn't make sense if it's not.

[github-actions]

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[bennibolm]

I tested the resulting numbers of entropy_timederivative and got the correct results: Without any limiting I get the same numbers as the pure DG simulation; Also, I get the same numbers as the pure LGL FV simulation without enabled correction stage.

[bennibolm]

Problem: The amount of limiting depends on the time step. We don't have a fully semidiscrete scheme.

[codecov]

Codecov Report

❌ Patch coverage is 83.33333% with 3 lines in your changes missing coverage. Please review.
✅ Project coverage is 92.12%. Comparing base (bc7858a) to head (8da8fd7).

Files with missing lines	Patch %	Lines
src/time_integration/methods_SSP.jl	60.00%	2 Missing ⚠️
src/callbacks_step/analysis.jl	83.33%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #2953      +/-   ##
==========================================
- Coverage   97.11%   92.12%   -5.00%     
==========================================
  Files         622      622              
  Lines       48235    48249      +14     
==========================================
- Hits        46843    44446    -2397     
- Misses       1392     3803    +2411     

Flag	Coverage Δ
unittests	92.12% <83.33%> (-5.00%)	⬇️

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[amrueda]

Thanks, @bennibolm... And sorry for the delay!
This is a nice and elegant fix to the problem.
LGTM... But there are some tests failing.

[amrueda]

Suggested change

	# Apply `perform_idp_correction!` to `du` instead of `u`:
	# Pass `du` and `1.0` instead of `u` and `dt`
	perform_idp_correction!(du, 1.0, mesh, equations, solver, cache)
	# Apply `perform_idp_correction!` to `du` instead of `u`:
	# Pass `du` and `1.0` instead of `u` and `dt`. This returns the correct
	# semi-discrete correction: du_i = du_i^(FV) + \sum_j f^(antidiffusive)_ij
	perform_idp_correction!(du, 1.0, mesh, equations, solver, cache)

[amrueda]

On a second pass, I realized the following:
At each analysis callback, du is computed and subsequently "corrected" with the IDP routines using the bounds that were computed for the explicit Euler step of the previous RK stage. This is a bit weird because there is no guarantee that those are meaningful bounds for the solution at the time of the analysis callback. Also, there is no guarantee that the low-order method can fulfill those bounds. I'd suggest computing the bounds before du_correction_subcell_limiting!.

[amrueda]

Is this always the case? if not, we'll have to do something different here or force it to be the first.

[amrueda]

I think that we have to re-compute the bounds before the call to du_correction_subcell_limiting!. The most consistent way to do this is, probably, to force the call of the analysis callback to be after the call to the time-step callback...

[bennibolm]

At each analysis callback, du is computed and subsequently "corrected" with the IDP routines using the bounds that were computed for the explicit Euler step of the previous RK stage

Oh yeah, you are right. In that case, I actually would prefer to close this PR and don't add additional computations. As we discussed back then, the IDP limiting is simply not a pure semi-discrete method and the entropy analysis doesn't make full sense.
What do you say? @amrueda