Second-Order Finite Volume Integral in 1D

[DanielDoehring]

This picks up the #433 draft by Gregor and updates it to the current structure. Could serve as a starting point to examine the effect of using an O2 FV scheme in the subcell capturing VolumeIntegralShockCapturingHG.

2D/3D will probably follow in some time, but this should also be trivial so if anyone else would like to pick this up, feel free to do so.

Also, this is helpful if one would like to bring a classic second order finite volume solver into Trixi at some point.

Xref: #425

Convergence studies:

Monotonized Central

l2
rho                 rho_v1              rho_e
error     EOC       error     EOC       error     EOC
4.82e-04  -         5.31e-04  -         7.90e-04  -
1.17e-04  2.04      1.28e-04  2.05      1.92e-04  2.04
2.96e-05  1.99      3.31e-05  1.95      4.87e-05  1.98
7.38e-06  2.00      8.33e-06  1.99      1.22e-05  1.99

mean      2.01      mean      2.00      mean      2.00
----------------------------------------------------------------------------------------------------
linf
rho                 rho_v1              rho_e
error     EOC       error     EOC       error     EOC
1.45e-03  -         1.42e-03  -         2.57e-03  -
3.64e-04  2.00      3.86e-04  1.88      6.72e-04  1.93
9.48e-05  1.94      1.02e-04  1.92      1.62e-04  2.06
2.55e-05  1.90      2.76e-05  1.89      4.00e-05  2.01

mean      1.94      mean      1.90      mean      2.00
----------------------------------------------------------------------------------------------------

minmod

l2
rho                 rho_v1              rho_e
error     EOC       error     EOC       error     EOC
2.90e-03  -         2.98e-03  -         2.25e-03  -
9.25e-04  1.65      9.51e-04  1.65      6.72e-04  1.74
2.93e-04  1.66      3.01e-04  1.66      2.02e-04  1.74
9.19e-05  1.67      9.43e-05  1.68      5.99e-05  1.75

mean      1.66      mean      1.66      mean      1.74
----------------------------------------------------------------------------------------------------
linf
rho                 rho_v1              rho_e
error     EOC       error     EOC       error     EOC
6.92e-03  -         6.62e-03  -         6.60e-03  -
2.72e-03  1.35      2.70e-03  1.29      2.31e-03  1.52
1.07e-03  1.35      1.07e-03  1.33      8.00e-04  1.53
4.19e-04  1.35      4.22e-04  1.35      2.81e-04  1.51

mean      1.35      mean      1.32      mean      1.52
----------------------------------------------------------------------------------------------------

superbee

l2
rho                 rho_v1              rho_e
error     EOC       error     EOC       error     EOC
2.07e-03  -         2.19e-03  -         1.73e-03  -
1.12e-03  0.88      1.15e-03  0.93      7.28e-04  1.24
5.49e-04  1.03      5.56e-04  1.05      3.12e-04  1.22
2.06e-04  1.41      2.07e-04  1.42      1.10e-04  1.50

mean      1.11      mean      1.13      mean      1.32
----------------------------------------------------------------------------------------------------
linf
rho                 rho_v1              rho_e
error     EOC       error     EOC       error     EOC
4.50e-03  -         4.96e-03  -         3.87e-03  -
3.26e-03  0.47      3.39e-03  0.55      2.17e-03  0.83
2.29e-03  0.51      2.34e-03  0.53      1.33e-03  0.71
1.25e-03  0.88      1.26e-03  0.90      6.58e-04  1.01

mean      0.62      mean      0.66      mean      0.85
----------------------------------------------------------------------------------------------------

van Leer

l2
rho                 rho_v1              rho_e
error     EOC       error     EOC       error     EOC
7.99e-04  -         8.34e-04  -         9.04e-04  -
2.22e-04  1.85      2.30e-04  1.86      2.25e-04  2.01
6.06e-05  1.87      6.26e-05  1.88      5.68e-05  1.98
1.63e-05  1.90      1.67e-05  1.90      1.44e-05  1.98

mean      1.87      mean      1.88      mean      1.99
----------------------------------------------------------------------------------------------------
linf
rho                 rho_v1              rho_e
error     EOC       error     EOC       error     EOC
2.73e-03  -         2.53e-03  -         2.84e-03  -
8.73e-04  1.64      8.63e-04  1.55      8.09e-04  1.81
2.90e-04  1.59      2.89e-04  1.58      2.33e-04  1.80
9.80e-05  1.57      9.80e-05  1.56      6.98e-05  1.74

mean      1.60      mean      1.56      mean      1.78
----------------------------------------------------------------------------------------------------

Not really sure why superbee is so bad.

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[codecov]

Codecov Report

Attention: Patch coverage is 94.05941% with 6 lines in your changes missing coverage. Please review.

Project coverage is 96.33%. Comparing base (480aea9) to head (7b97473).

Files with missing lines	Patch %	Lines
src/equations/linear_scalar_advection_1d.jl	0.00%	1 Missing ⚠️
src/equations/linear_scalar_advection_2d.jl	0.00%	1 Missing ⚠️
src/equations/linear_scalar_advection_3d.jl	0.00%	1 Missing ⚠️
src/solvers/dg.jl	90.00%	1 Missing ⚠️
src/solvers/dgsem_tree/dg_1d.jl	97.06%	1 Missing ⚠️
src/solvers/dgsem_tree/finite_volume_O2.jl	98.08%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #2022      +/-   ##
==========================================
- Coverage   96.34%   96.33%   -0.01%     
==========================================
  Files         470      472       +2     
  Lines       37497    37598     +101     
==========================================
+ Hits        36125    36220      +95     
- Misses       1372     1378       +6     

Flag	Coverage Δ
unittests	96.33% <94.06%> (-0.01%)	⬇️

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[amrueda]

Thanks for implementing this, @DanielDoehring!

No clue why superbee is so bad. I think the implementation is correct.

Most of my comments only suggest minor changes. My only concern is that the central reconstruction for the subcells at the element border can lead to overshoots and crashes. Would it be easy to add the possibility to use a "safer" scheme at the boundary subcells?

[DanielDoehring]

Here is the convergence study for the version with constant reconstruction of the boundary subcells:

####################################################################################################
l2
rho                 rho_v1              rho_e               
error     EOC       error     EOC       error     EOC       
3.01e-04  -         4.39e-04  -         8.69e-04  -         
1.11e-04  1.43      1.28e-04  1.78      3.38e-04  1.36      
4.01e-05  1.47      4.95e-05  1.37      1.44e-04  1.23      
1.87e-05  1.10      2.28e-05  1.12      6.89e-05  1.06      
9.20e-06  1.02      1.12e-05  1.03      3.40e-05  1.02      

mean      1.26      mean      1.32      mean      1.17      
----------------------------------------------------------------------------------------------------
linf
rho                 rho_v1              rho_e               
error     EOC       error     EOC       error     EOC       
1.40e-03  -         1.30e-03  -         4.53e-03  -         
6.30e-04  1.15      6.70e-04  0.95      2.28e-03  0.99      
2.94e-04  1.10      3.40e-04  0.98      1.15e-03  0.99      
1.48e-04  0.98      1.69e-04  1.01      5.85e-04  0.97      
7.37e-05  1.01      8.41e-05  1.01      2.93e-04  1.00      

mean      1.06      mean      0.99      mean      0.99      
----------------------------------------------------------------------------------------------------

[torrilhon]

Please note #1880 ...