Add Legendre-Gauss basis for DGSEM and implement solver support for 1D/2D TreeMesh

[sloede]

No description provided.

[github-actions]

Review checklist

This checklist is meant to assist creators of PRs (to let them know what reviewers will typically look for) and reviewers (to guide them in a structured review process). Items do not need to be checked explicitly for a PR to be eligible for merging.

Purpose and scope

• The PR has a single goal that is clear from the PR title and/or description.
• All code changes represent a single set of modifications that logically belong together.
• No more than 500 lines of code are changed or there is no obvious way to split the PR into multiple PRs.

Code quality

• The code can be understood easily.
• Newly introduced names for variables etc. are self-descriptive and consistent with existing naming conventions.
• There are no redundancies that can be removed by simple modularization/refactoring.
• There are no leftover debug statements or commented code sections.
• The code adheres to our conventions and style guide, and to the Julia guidelines.

Documentation

• New functions and types are documented with a docstring or top-level comment.
• Relevant publications are referenced in docstrings (see example for formatting).
• Inline comments are used to document longer or unusual code sections.
• Comments describe intent ("why?") and not just functionality ("what?").
• If the PR introduces a significant change or new feature, it is documented in NEWS.md with its PR number.

Testing

• The PR passes all tests.
• New or modified lines of code are covered by tests.
• New or modified tests run in less then 10 seconds.

Performance

• There are no type instabilities or memory allocations in performance-critical parts.
• If the PR intent is to improve performance, before/after time measurements are posted in the PR.

Verification

• The correctness of the code was verified using appropriate tests.
• If new equations/methods are added, a convergence test has been run and the results
  are posted in the PR.

Created with ❤️ by the Trixi.jl community.

[sloede]

Convergence test results:

julia>  convergence_test("examples/tree_2d_dgsem/elixir_euler_convergence_gauss.jl", 5, tspan=(0.0, 0.1)

####################################################################################################
l2
rho                 rho_v1              rho_v2              rho_e
error     EOC       error     EOC       error     EOC       error     EOC
6.84e-07  -         8.45e-07  -         8.45e-07  -         2.69e-06  -
3.49e-08  4.29      4.30e-08  4.30      4.30e-08  4.30      1.56e-07  4.11
2.25e-09  3.95      2.81e-09  3.94      2.81e-09  3.94      9.84e-09  3.98
1.36e-10  4.05      1.70e-10  4.04      1.70e-10  4.04      6.13e-10  4.00
8.57e-12  3.99      1.08e-11  3.98      1.08e-11  3.98      3.83e-11  4.00

mean      4.07      mean      4.06      mean      4.06      mean      4.03
----------------------------------------------------------------------------------------------------
linf
rho                 rho_v1              rho_v2              rho_e
error     EOC       error     EOC       error     EOC       error     EOC
2.36e-06  -         3.08e-06  -         3.08e-06  -         1.30e-05  -
1.74e-07  3.77      2.15e-07  3.84      2.15e-07  3.84      9.16e-07  3.82
1.30e-08  3.74      1.61e-08  3.74      1.61e-08  3.74      6.05e-08  3.92
6.76e-10  4.26      8.31e-10  4.28      8.31e-10  4.28      3.50e-09  4.11
4.40e-11  3.94      5.59e-11  3.89      5.59e-11  3.89      2.27e-10  3.95

mean      3.93      mean      3.94      mean      3.94      mean      3.95
----------------------------------------------------------------------------------------------------
Dict{Symbol, Any} with 3 entries:
  :variables => ("rho", "rho_v1", "rho_v2", "rho_e")
  :l2        => [4.07106, 4.06455, 4.06455, 4.02501]
  :linf      => [3.92877, 3.93774, 3.9377, 3.95104]

[codecov]

Codecov Report

❌ Patch coverage is 97.35450% with 5 lines in your changes missing coverage. Please review.
✅ Project coverage is 97.13%. Comparing base (79cf51e) to head (88ac730).
⚠️ Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
src/solvers/dgsem_tree/dg_1d.jl	93.33%	2 Missing ⚠️
src/solvers/dgsem_tree/dg_2d.jl	95.56%	2 Missing ⚠️
src/solvers/dgsem/basis_gauss_legendre.jl	99.10%	1 Missing ⚠️

Additional details and impacted files

@@           Coverage Diff            @@
##             main    #1965    +/-   ##
========================================
  Coverage   97.13%   97.13%            
========================================
  Files         595      596     +1     
  Lines       45802    45943   +141     
========================================
+ Hits        44486    44623   +137     
- Misses       1316     1320     +4     

Flag	Coverage Δ
unittests	97.13% <97.35%> (+<0.01%)	⬆️

Flags with carried forward coverage won't be shown. Click here to find out more.

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

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.

[DanielDoehring]

That's really nice! As a smoke test, I compared the solution of the 1D advection test case to a 1D Gauss-Legendre solver I once implemented in SimpleDiscontinuousGalerkin.jl using the same setup and the two solutions have differences of the order of 1e-16 👍

Ah that is nice!

[JoshuaLampert]

Only one small final comment.

[jlchan]

This is very cool! One suggestion - this should work for VolumeIntegralWeakForm but not VolumeIntegralFluxDifferencing yet since that would require some modifications of calc_volume_integral! to incorporate surface corrections. Might make sense to note that in NEWS.md?

[DanielDoehring]

This is very cool! One suggestion - this should work for VolumeIntegralWeakForm but not VolumeIntegralFluxDifferencing yet since that would require some modifications of calc_volume_integral! to incorporate surface corrections. Might make sense to note that in NEWS.md?

Yes, makes sense to me - I would probably also delete the derivative_matrix_split as well to really make sure that this is not used right now.

[ranocha]

Can we use a uniform naming pattern - either _gauss or _gauss_legendre, please?

[ranocha]

Suggested change

	basis = LobattoLegendreBasis(RealT, polydeg)
	basis = LobattoLegendreBasis(RealT, polydeg),

[ranocha]

This can be very confusing, since it allows you to pass

DGSEM(; polydeg = 3, basis = GaussLegendreBasis(5))

[DanielDoehring]

I agree. I will add a PR which adds a new constructor for this.