Add support for source terms

[JoshuaLampert]

TODO:
BBM-BBM equations with variable bathymetry

• fix manufactured solution
• add test
• proper convergence analysis

Svärd-Kalisch equations with constant baythemtry

• add manufactured solution
• add test
• proper convergence analysis

[JoshuaLampert]

Convergence test for manufactured solution for the BBM-BBM equations with variable bathymetry:

p = 2:

julia> convergence_test("examples/bbm_bbm_variable_bathymetry_1d/bbm_bbm_variable_bathymetry_1d_manufactured.jl", 3, N = 32, accuracy_order = 2)
[...]
l2
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
32   1.32e-02  -         32   9.64e-03  -         32   0.00e+00  -         
64   3.27e-03  2.01      64   2.40e-03  2.01      64   0.00e+00  NaN       
128  8.16e-04  2.00      128  5.99e-04  2.00      128  0.00e+00  NaN       

mean      2.01      mean      2.00      mean      NaN       
----------------------------------------------------------------------------------------------------
linf
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
32   2.42e-02  -         32   1.61e-02  -         32   0.00e+00  -         
64   5.98e-03  2.02      64   4.01e-03  2.01      64   0.00e+00  NaN       
128  1.49e-03  2.00      128  1.00e-03  2.00      128  0.00e+00  NaN       

mean      2.01      mean      2.00      mean      NaN       
----------------------------------------------------------------------------------------------------

p = 4:

julia> convergence_test("examples/bbm_bbm_variable_bathymetry_1d/bbm_bbm_variable_bathymetry_1d_manufactured.jl", 3, N = 32, accuracy_order = 4)
[...]
l2
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
32   1.92e-04  -         32   2.20e-04  -         32   0.00e+00  -         
64   1.21e-05  3.98      64   1.39e-05  3.99      64   0.00e+00  NaN       
128  7.60e-07  3.99      128  8.70e-07  4.00      128  0.00e+00  NaN       

mean      3.99      mean      3.99      mean      NaN       
----------------------------------------------------------------------------------------------------
linf
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
32   2.91e-04  -         32   3.67e-04  -         32   0.00e+00  -         
64   1.87e-05  3.96      64   2.32e-05  3.98      64   0.00e+00  NaN       
128  1.17e-06  4.00      128  1.45e-06  4.00      128  0.00e+00  NaN       

mean      3.98      mean      3.99      mean      NaN       
----------------------------------------------------------------------------------------------------

p = 6:

julia> convergence_test("examples/bbm_bbm_variable_bathymetry_1d/bbm_bbm_variable_bathymetry_1d_manufactured.jl", 3, N = 16, accuracy_order = 6)
[...]
l2
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
16   3.96e-04  -         16   3.86e-04  -         16   0.00e+00  -         
32   6.75e-06  5.87      32   6.77e-06  5.83      32   0.00e+00  NaN       
64   1.11e-07  5.93      64   1.09e-07  5.95      64   0.00e+00  NaN       

mean      5.90      mean      5.89      mean      NaN       
----------------------------------------------------------------------------------------------------
linf
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
16   6.32e-04  -         16   7.15e-04  -         16   0.00e+00  -         
32   1.10e-05  5.84      32   1.23e-05  5.86      32   0.00e+00  NaN       
64   1.78e-07  5.95      64   2.01e-07  5.94      64   0.00e+00  NaN       

mean      5.90      mean      5.90      mean      NaN       
----------------------------------------------------------------------------------------------------

[codecov-commenter]

Codecov Report

Attention: 6 lines in your changes are missing coverage. Please review.

Comparison is base (68e64f4) 96.28% compared to head (617a98f) 95.21%.

Files	Patch %	Lines
src/equations/bbm_bbm_1d.jl	83.33%	5 Missing ⚠️
src/equations/bbm_bbm_variable_bathymetry_1d.jl	97.14%	1 Missing ⚠️

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Additional details and impacted files

@@            Coverage Diff             @@
##             main      #65      +/-   ##
==========================================
- Coverage   96.28%   95.21%   -1.07%     
==========================================
  Files          13       13              
  Lines         968     1045      +77     
==========================================
+ Hits          932      995      +63     
- Misses         36       50      +14     

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[JoshuaLampert]

Convergence test for Svärd-Kalisch equations with constant bathymetry:

p = 2

julia> convergence_test("examples/svaerd_kalisch_1d/svaerd_kalisch_1d_manufactured.jl", 5, N = 16, accuracy_order = 2, tspan = (0.0, 0.1))
[...]
l2
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
16   5.71e-02  -         16   3.49e-03  -         16   0.00e+00  -         
32   1.46e-02  1.97      32   8.47e-04  2.04      32   0.00e+00  NaN       
64   3.66e-03  1.99      64   2.10e-04  2.01      64   0.00e+00  NaN       
128  9.16e-04  2.00      128  5.24e-05  2.00      128  0.00e+00  NaN       
256  2.87e-04  1.68      256  1.31e-05  2.00      256  0.00e+00  NaN       

mean      1.91      mean      2.01      mean      NaN       
----------------------------------------------------------------------------------------------------
linf
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
16   1.08e-01  -         16   5.91e-03  -         16   0.00e+00  -         
32   2.65e-02  2.03      32   1.44e-03  2.04      32   0.00e+00  NaN       
64   6.56e-03  2.01      64   3.58e-04  2.01      64   0.00e+00  NaN       
128  1.72e-03  1.93      128  8.93e-05  2.00      128  0.00e+00  NaN       
256  7.87e-04  1.13      256  2.23e-05  2.00      256  0.00e+00  NaN       

mean      1.78      mean      2.01      mean      NaN       
----------------------------------------------------------------------------------------------------

p = 4

julia> convergence_test("examples/svaerd_kalisch_1d/svaerd_kalisch_1d_manufactured.jl", 3, N = 16, accuracy_order = 4, tspan = (0.0, 0.1))
l2
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
16   5.52e-03  -         16   1.11e-04  -         16   0.00e+00  -         
32   3.65e-04  3.92      32   7.08e-06  3.98      32   0.00e+00  NaN       
64   2.77e-05  3.72      64   4.44e-07  3.99      64   0.00e+00  NaN       

mean      3.82      mean      3.98      mean      NaN       
----------------------------------------------------------------------------------------------------
linf
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
16   8.34e-03  -         16   1.65e-04  -         16   0.00e+00  -         
32   5.66e-04  3.88      32   1.09e-05  3.91      32   0.00e+00  NaN       
64   5.39e-05  3.39      64   6.93e-07  3.98      64   0.00e+00  NaN       

mean      3.64      mean      3.95      mean      NaN       
----------------------------------------------------------------------------------------------------

p = 6

julia> convergence_test("examples/svaerd_kalisch_1d/svaerd_kalisch_1d_manufactured.jl", 2, N = 16, accuracy_order = 6, tspan = (0.0, 0.1))
l2
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
16   6.94e-04  -         16   9.23e-06  -         16   0.00e+00  -         
32   1.26e-05  5.79      32   1.59e-07  5.86      32   0.00e+00  NaN       

mean      5.79      mean      5.86      mean      NaN       
----------------------------------------------------------------------------------------------------
linf
η                        v                        D                        
N    error     EOC       N    error     EOC       N    error     EOC       
16   1.17e-03  -         16   1.60e-05  -         16   0.00e+00  -         
32   2.28e-05  5.68      32   2.77e-07  5.85      32   0.00e+00  NaN       

mean      5.68      mean      5.85      mean      NaN       
----------------------------------------------------------------------------------------------------

With increasing mesh refinement, the convergence order is not recovered, probably due to higher errors in the time integration. The time step restriction also seems to be really strict (adaptive time stepping chooses quite quite small time steps ~3e-4 for p = 4 and N = 64). I'll investigate time step restrictions later in a bit more detail. In principle, the manufactured solution seems to work for the Svärd-Kalisch equations with constant bathymetry. Unfortunately, I haven't been able to reconstruct a manufactured solution for the Svärd-Kalisch equations with variable bathymetry yet. I try to add a convergence test for the Svärd-Kalisch equations with variable bathymetry in a subsequent PR.

[ranocha]

This looks great 👍 (I didn't check the exact form of the source terms - but your results are convincing 🙂)