Add 2D compressible Euler with internal energy as prognostic variable

[MarcoArtiano]

The entropy stable and entropy conserving, total energy conserving and KEP schemes are derived in https://arxiv.org/abs/2603.18978.

[codecov]

Codecov Report

❌ Patch coverage is 98.44961% with 2 lines in your changes missing coverage. Please review.
✅ Project coverage is 97.12%. Comparing base (fa8e2d5) to head (fcfec49).
⚠️ Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
...pressible_euler_internal_energy_with_gravity_2d.jl	98.44%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #172      +/-   ##
==========================================
+ Coverage   97.09%   97.12%   +0.03%     
==========================================
  Files          37       38       +1     
  Lines        5021     5150     +129     
==========================================
+ Hits         4875     5002     +127     
- Misses        146      148       +2     

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[ranocha]

Thanks a lot! My main comment is on the naming scheme: Would it make sense to adopt the same naming scheme that you used for the potential temperature equations, e.g., flux_etec (or flux_etec_kep or something like this) and flux_es? The nonconservative parts could get analogous names like flux_nonconservative_es etc.

[MarcoArtiano]

Thanks a lot! My main comment is on the naming scheme: Would it make sense to adopt the same naming scheme that you used for the potential temperature equations, e.g., flux_etec (or flux_etec_kep or something like this) and flux_es? The nonconservative parts could get analogous names like flux_nonconservative_es etc.

Thanks! I agree, it would actually make more sense, so that their properties are immediately clear.

[ranocha]

Thanks! Just a few last comments. I think you also need to update the export statements.

I would also like to see convergence results for the test you added.

Would it make sense to add the baroclinic instability example as well?

[MarcoArtiano]

Thanks! Just a few last comments. I think you also need to update the export statements.

I would also like to see convergence results for the test you added.

Would it make sense to add the baroclinic instability example as well?

The convergence rates match the one in https://arxiv.org/pdf/2509.10311. Here I reported in order pressure, density and vertical velocity perturbations.

We do not support a 2D baroclinic test case yet, therefore we have to wait for the next PR with the 3D equations.