WIP: Add non-conservative 2D Euler equations with gravity potential

[amrueda]

This set of equations extends the CompressibleEulerEquations2D by incorporating geopotential energy into the total energy rho_e. This ensures that the energy equation remains a conservation law, even in the presence of gravitational effects. Additionally, the source terms in the momentum equations are treated as non-conservative terms rather than point-wise source terms, following:

• Souza, A. N., He, J., Bischoff, T., Waruszewski, M., Novak, L., Barra, V., ... & Schneider, T. (2023). The flux‐differencing discontinuous galerkin method applied to an idealized fully compressible nonhydrostatic dry atmosphere. Journal of Advances in Modeling Earth Systems, 15(4), e2022MS003527. https://doi.org/10.1029/2022MS003527.
• Waruszewski, M., Kozdon, J. E., Wilcox, L. C., Gibson, T. H., & Giraldo, F. X. (2022). Entropy stable discontinuous Galerkin methods for balance laws in non-conservative form: Applications to the Euler equations with gravity. Journal of Computational Physics, 468, 111507. https://doi.org/10.1016/j.jcp.2022.111507.

We implement the non-conservative term flux_nonconservative_waruszewski, which provides well-balancedness (to machine precision accuracy) for isothermal hydrostatic equilibria:

Other standard benchmark tests are provided as well:

RisingBubble_afterBugFix_lmars.mp4

[codecov]

Codecov Report

❌ Patch coverage is 84.77612% with 51 lines in your changes missing coverage. Please review.
✅ Project coverage is 96.31%. Comparing base (aa057e7) to head (2825d7e).
⚠️ Report is 12 commits behind head on main.

Files with missing lines	Patch %	Lines
src/equations/compressible_euler_gravity_2d.jl	84.77%	51 Missing ⚠️

Additional details and impacted files

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- Coverage   97.08%   96.31%   -0.78%     
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  Files          37       38       +1     
  Lines        5010     5345     +335     
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+ Hits         4864     5148     +284     
- Misses        146      197      +51     

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[tristanmontoya]

@amrueda What still needs to be done for this PR to be ready?

[MarcoArtiano]

Thanks, I left a few comments.

[MarcoArtiano]

Do we need to support StructuredMesh?

[amrueda]

For simple cases, it's nice to be able to test with the TreeMesh or StructuredMesh solver :)

[MarcoArtiano]

I don't remember if we already discussed about that. I'm sorry.

Since the equations already mentions that this is with gravity, I would naturally assume that the rho_e is indeed the total energy, without the need to specify.

[MarcoArtiano]

In atmospheric flows I think we could avoid this a bit more expensive boundary condition and use the other more simple already implemented.

I'm note sure if actually there are differences and one has an advantage over the other.

[amrueda]

I have added this BC because it's very robust and has been shown to be entropy stable. See, e.g.,

Hindenlang, F. J., Gassner, G. J., & Kopriva, D. A. (2020, August). Stability of wall boundary condition procedures for discontinuous Galerkin spectral element approximations of the compressible Euler equations. In Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018: Selected Papers from the ICOSAHOM Conference, London, UK, July 9-13, 2018 (pp. 3-19). Cham: Springer International Publishing.

The classical reflective slip-wall BC that uses the numerical flux function is only provably ES for some Riemann solvers, such as LLF, HLL, HLLC (see paper). However, there's no guarantee that it will also be ES for other Riemann solvers. I don't know if LMARS provides entropy stability, for example.

You are right that this is probably not critical in most atmospheric science cases, but the additional robustness might be important in some cases. For instance, when simulating the blast from a volcano eruption ;)

I don't think this BC is intrinsically more expensive than the reflective boundary condition that uses the surface numerical flux function. In particular, the numerical flux function needs to compute the maximum wave speed and pressure, which is essentially what is needed for this BC.