Covariant formulation of the shallow water equations

[tristanmontoya]

New version of #37 with entropy-stable split forms and auxiliary variables. Requires Trixi.jl v0.9.9 (see #59). The following new features are implemented:

• New equation type CovariantShallowWaterEquations2D which implements the split formulation of the covariant derivative using non-conservative terms, but does not use non-conservative terms for the standard formulation. The split formulation conserves total energy when no interface dissipation is added.
• Non-conservative flux-differencing and interface flux kernels for the covariant form using auxiliary variables.
• Auxiliary variables expanded to include more geometric information, including Christoffel symbols (which, for the moment, are computed by applying the LGL collocation derivative operator to the analytical metric tensor components). Note that the split formulation is provably stable regardless of how these geometric terms are computed, although in the future, I plan to add the exact computation of the Christoffel symbols.
• Analysis routines for covariant-form equations take integrals of functions of the form func(u, aux_vars, equations) rather than just func(u, equations), which is needed since entropy is a function of auxiliary variables in this case.
• Standardized initial conditions with automatic conversions to the chosen global coordinate system (i.e. Cartesian or spherical) and set of conservative variables. Currently, we have the Gaussian solid-body rotation, steady geostrophic balance, and Rossby-Haurwitz wave, based on Cases 1, 2, and 6 of the Williamson et al. test suite, respectively.
• Notation is changed to consistently use J rather than sqrtG for the area element.

[codecov]

Codecov Report

Attention: Patch coverage is 95.50562% with 16 lines in your changes missing coverage. Please review.

Project coverage is 90.39%. Comparing base (029c14e) to head (ca50226).
Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
src/equations/covariant_advection.jl	72.22%	5 Missing ⚠️
src/equations/covariant_shallow_water.jl	93.58%	5 Missing ⚠️
src/equations/shallow_water_3d.jl	0.00%	4 Missing ⚠️
src/equations/equations.jl	97.22%	1 Missing ⚠️
...vers/dgsem_p4est/dg_2d_manifold_in_3d_covariant.jl	98.43%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #58      +/-   ##
==========================================
+ Coverage   89.45%   90.39%   +0.94%     
==========================================
  Files          20       22       +2     
  Lines        1801     2104     +303     
==========================================
+ Hits         1611     1902     +291     
- Misses        190      202      +12     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[amrueda]

Many thanks, @tristanmontoya! great work!
I have not checked the math thoroughly, but I leave a couple of comments below.
BTW, thanks for improving the documentation of my elixirs as well!

[amrueda]

LGTM! Thanks, @tristanmontoya!
Some minor comments below:

[tristanmontoya]

LGTM! Thanks, @tristanmontoya! Some minor comments below:

Great, thanks for your review! Going to make those changes and merge now.