Use ForwardDiff.jl to compute Christoffel symbols exactly

[tristanmontoya]

Previously, the collocation derivative was applied to the metric tensor components to evaluate the Christoffel symbols approximately. Since the rest of the geometric information for the covariant solver is computed analytically for the spherical shell, I would like the Christoffel symbols to also be exact. Because the expressions get quite messy when differentiating by hand, an easy way to compute them is to use automatic differentiation. Since reverse-mode automatic differentiation is not needed here and Trixi.jl already has ForwardDiff.jl as a dependency, I've used ForwardDiff.jl to do forward-mode automatic differentiation.

If the Christoffel symbols needed to be recomputed frequently (e.g. in an adaptive simulation, or for a time-varying metric), perhaps there would be an argument for coding them by hand for efficiency purposes, but, since for now they are only calculated once per simulation and stored at each node, using automatic differentiation makes the most sense to me.

Now that we are computing the Christoffel symbols exactly, the sensitivity of the covariant basis vector components described in #49 can be seen from the fact that ForwardDiff returns NaNs when using GlobalSphericalCoordinates and an even number of cells per dimension on each face of the cubed sphere, presumably due to this pole problem. The default option GlobalCartesianCoordinates avoids this problem, and should therefore always be used instead of GlobalSphericalCoordinates.

Note: The test values for the spherical shallow water equations have been updated accordingly. This will therefore affect the test values for the new cases added in #73. Also note that the coveralls tests fail due to there being less code to cover now, thus leading to the percentage of covered code dropping.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 90.29%. Comparing base (a496119) to head (8121b38).

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #78      +/-   ##
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- Coverage   90.39%   90.29%   -0.11%     
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  Files          22       22              
  Lines        2104     2082      -22     
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- Hits         1902     1880      -22     
  Misses        202      202              

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