Added reflecting BC for HyperbolicSGN

[cwittens]

No description provided.

[cwittens]

Here are come convergence test results:

test_results = convergence_test(joinpath(examples_dir(), "hyperbolic_serre_green_naghdi_1d", "hyperbolic_serre_green_naghdi_manufactured_reflecting.jl"), [32, 64, 128, 256, 512]);

# using bathymetry_mild_slope:

# for accuracy_order = 2
####################################################################################################
l2
η                        v                        D                        w                        H
N    error     EOC       N    error     EOC       N    error     EOC       N    error     EOC       N    error     EOC
32   1.96e-02  -         32   6.18e-03  -         32   0.00e+00  -         32   1.11e+00  -         32   1.97e-02  -
64   6.60e-03  1.57      64   1.47e-03  2.07      64   0.00e+00  NaN       64   1.12e+00  -0.01     64   6.66e-03  1.57
128  2.27e-03  1.54      128  3.59e-04  2.03      128  0.00e+00  NaN       128  1.13e+00  -0.00     128  2.30e-03  1.53
256  7.93e-04  1.52      256  8.89e-05  2.02      256  0.00e+00  NaN       256  1.13e+00  -0.00     256  8.04e-04  1.52
512  2.79e-04  1.51      512  2.21e-05  2.01      512  0.00e+00  NaN       512  1.13e+00  -0.00     512  2.83e-04  1.51

mean           1.53      mean           2.03      mean           NaN       mean           -0.00     mean           1.53
----------------------------------------------------------------------------------------------------
linf
η                        v                        D                        w                        H
N    error     EOC       N    error     EOC       N    error     EOC       N    error     EOC       N    error     EOC
32   1.28e-01  -         32   9.91e-03  -         32   0.00e+00  -         32   1.72e+00  -         32   1.30e-01  -
64   6.46e-02  0.98      64   2.35e-03  2.08      64   0.00e+00  NaN       64   1.73e+00  -0.01     64   6.57e-02  0.99
128  3.25e-02  0.99      128  5.72e-04  2.04      128  0.00e+00  NaN       128  1.74e+00  -0.00     128  3.30e-02  0.99
256  1.63e-02  1.00      256  1.41e-04  2.02      256  0.00e+00  NaN       256  1.74e+00  -0.00     256  1.65e-02  1.00
512  8.15e-03  1.00      512  4.08e-05  1.79      512  0.00e+00  NaN       512  1.74e+00  -0.00     512  8.28e-03  1.00

mean           0.99      mean           1.98      mean           NaN       mean           -0.00     mean           0.99
----------------------------------------------------------------------------------------------------

# for accuracy_order = 4
####################################################################################################
l2
η                        v                        D                        w                        H
N    error     EOC       N    error     EOC       N    error     EOC       N    error     EOC       N    error     EOC
32   1.22e-02  -         32   5.39e-03  -         32   0.00e+00  -         32   1.14e+00  -         32   1.23e-02  -
64   2.53e-03  2.26      64   7.51e-04  2.84      64   0.00e+00  NaN       64   1.13e+00  0.01      64   2.57e-03  2.26
128  4.80e-04  2.40      128  9.78e-05  2.94      128  0.00e+00  NaN       128  1.13e+00  0.00      128  4.87e-04  2.40
256  8.77e-05  2.45      256  1.25e-05  2.97      256  0.00e+00  NaN       256  1.13e+00  0.00      256  8.89e-05  2.45
512  1.58e-05  2.47      512  2.09e-06  2.58      512  0.00e+00  NaN       512  1.13e+00  0.00      512  1.60e-05  2.47

mean           2.40      mean           2.83      mean           NaN       mean           0.00      mean           2.40
----------------------------------------------------------------------------------------------------
linf
η                        v                        D                        w                        H
N    error     EOC       N    error     EOC       N    error     EOC       N    error     EOC       N    error     EOC
32   9.73e-02  -         32   8.40e-03  -         32   0.00e+00  -         32   1.74e+00  -         32   9.91e-02  -
64   2.93e-02  1.73      64   1.21e-03  2.80      64   0.00e+00  NaN       64   1.74e+00  0.00      64   2.98e-02  1.73
128  7.94e-03  1.88      128  1.60e-04  2.91      128  0.00e+00  NaN       128  1.74e+00  0.00      128  8.06e-03  1.89
256  2.06e-03  1.95      256  2.77e-05  2.54      256  0.00e+00  NaN       256  1.74e+00  -0.00     256  2.09e-03  1.95
512  5.28e-04  1.96      512  3.78e-05  -0.45     512  0.00e+00  NaN       512  1.74e+00  0.00      512  5.32e-04  1.97

mean           1.88      mean           1.95      mean           NaN       mean           0.00      mean           1.89
----------------------------------------------------------------------------------------------------

[cwittens]

This is how I calculated the manufactured solution:

# 1d manufactured solution for hyperbolic Serre-Green-Naghdi equations

using Symbolics

@variables x t
Dt = Differential(t)
Dx = Differential(x)

@variables g pi λ

h = 1 + (2 * t) * (cos(pi*(x)) + x + 2)
vx = (-t * x) * sin(pi*(x))
b = (2 * x) # or b = 0
eta = h
w = -h * Dx(vx) + 3//2 * vx * Dx(b)

PI = λ // 3 * (eta / h) * (1 - eta / h)
# taken from https://arxiv.org/pdf/2408.02665 eq. (89)
eq1 = Dt(h) + Dx(h * vx)
eq2 = Dt(vx) + (g * h * Dx(h) + h * vx * Dx(vx) + Dx(h * PI) + (g * h + 3 // 2 * h / eta * PI) * Dx(b)) / h
eq3 = Dt(w) + vx * Dx(w) - λ * (1 - eta / h) / h
eq4 = Dt(eta) + Dx(eta) * vx + 3//2 * Dx(b) * vx - w

s1 = simplify(expand_derivatives(eq1))
s2 = simplify(expand_derivatives(eq2))
s3 = 0
s4 = simplify(expand_derivatives(eq3))
s5 = simplify(expand_derivatives(eq4))

println("s1 = $s1\n")
println("s2 = $s2\n")
println("s3 = zero(s1)\n")
println("s4 = $s4\n")
println("s5 = $s5\n")

[github-actions]

Benchmark Results (Julia v1.10)

Time benchmarks

	main	78a807e...	main / 78a807e...
bbm_1d/bbm_1d_basic.jl - rhs!:	13.7 ± 0.27 μs	13.8 ± 0.28 μs	0.995 ± 0.028
bbm_1d/bbm_1d_fourier.jl - rhs!:	0.529 ± 0.0093 ms	0.534 ± 0.0093 ms	0.992 ± 0.025
bbm_bbm_1d/bbm_bbm_1d_basic_reflecting.jl - rhs!:	0.0805 ± 0.00043 ms	0.0805 ± 0.0004 ms	1 ± 0.0073
bbm_bbm_1d/bbm_bbm_1d_dg.jl - rhs!:	0.0344 ± 0.0005 ms	0.0366 ± 0.00044 ms	0.94 ± 0.018
bbm_bbm_1d/bbm_bbm_1d_relaxation.jl - rhs!:	26.9 ± 1.8 μs	27.3 ± 0.41 μs	0.986 ± 0.067
bbm_bbm_1d/bbm_bbm_1d_upwind_relaxation.jl - rhs!:	0.0485 ± 0.0009 ms	0.0485 ± 0.00054 ms	1 ± 0.022
hyperbolic_serre_green_naghdi_1d/hyperbolic_serre_green_naghdi_dingemans.jl - rhs!:	4.19 ± 0.03 μs	4.31 ± 0.04 μs	0.972 ± 0.011
kdv_1d/kdv_1d_basic.jl - rhs!:	1.45 ± 0.013 μs	1.42 ± 0.02 μs	1.02 ± 0.017
kdv_1d/kdv_1d_implicit.jl - rhs!:	1.39 ± 0.02 μs	1.43 ± 0.02 μs	0.973 ± 0.019
serre_green_naghdi_1d/serre_green_naghdi_well_balanced.jl - rhs!:	0.192 ± 0.0058 ms	0.2 ± 0.0081 ms	0.96 ± 0.048
svaerd_kalisch_1d/svaerd_kalisch_1d_dingemans_relaxation.jl - rhs!:	0.142 ± 0.0025 ms	0.149 ± 0.0045 ms	0.952 ± 0.033
time_to_load	1.97 ± 0.0053 s	1.99 ± 0.0034 s	0.993 ± 0.0031

Memory benchmarks

	main	78a807e...	main / 78a807e...
bbm_1d/bbm_1d_basic.jl - rhs!:	1 allocs: 4.12 kB	1 allocs: 4.12 kB	1
bbm_1d/bbm_1d_fourier.jl - rhs!:	1 allocs: 4.12 kB	1 allocs: 4.12 kB	1
bbm_bbm_1d/bbm_bbm_1d_basic_reflecting.jl - rhs!:	5 allocs: 1.17 kB	5 allocs: 1.17 kB	1
bbm_bbm_1d/bbm_bbm_1d_dg.jl - rhs!:	10 allocs: 8.62 kB	10 allocs: 8.62 kB	1
bbm_bbm_1d/bbm_bbm_1d_relaxation.jl - rhs!:	2 allocs: 8.25 kB	2 allocs: 8.25 kB	1
bbm_bbm_1d/bbm_bbm_1d_upwind_relaxation.jl - rhs!:	2 allocs: 8.25 kB	2 allocs: 8.25 kB	1
hyperbolic_serre_green_naghdi_1d/hyperbolic_serre_green_naghdi_dingemans.jl - rhs!:	0 allocs: 0 B	0 allocs: 0 B
kdv_1d/kdv_1d_basic.jl - rhs!:	0 allocs: 0 B	0 allocs: 0 B
kdv_1d/kdv_1d_implicit.jl - rhs!:	0 allocs: 0 B	0 allocs: 0 B
serre_green_naghdi_1d/serre_green_naghdi_well_balanced.jl - rhs!:	0.075 k allocs: 0.66 MB	0.075 k allocs: 0.66 MB	1
svaerd_kalisch_1d/svaerd_kalisch_1d_dingemans_relaxation.jl - rhs!:	0.042 k allocs: 0.315 MB	0.042 k allocs: 0.315 MB	1
time_to_load	0.153 k allocs: 14.5 kB	0.153 k allocs: 14.5 kB	1

[coveralls]

Pull Request Test Coverage Report for Build 16544586009

Details

• 29 of 29 (100.0%) changed or added relevant lines in 1 file are covered.
• No unchanged relevant lines lost coverage.
• Overall coverage increased (+0.02%) to 98.314%

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Totals
Change from base Build 16506099434:	0.02%
Covered Lines:	2274
Relevant Lines:	2313

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💛 - Coveralls

[JoshuaLampert]

Thanks! This looks good to me. Just for clarification: Do you have a formal proof of entropy conservation for this semidiscretization? It only uses a SAT for $$h$$, but no further treatment of the other variables is needed, right?

[cwittens]

Thanks! This looks good to me. Just for clarification: Do you have a formal proof of entropy conservation for this semidiscretization?

Yes. I dont have a TeXed version of the proof yet, but I did calculate it and energy is conserved for both flat and variable bottom topography.

It only uses a SAT for h , but no further treatment of the other variables is needed, right?

In the conservative form you also have additional terms for $u$, $w$ and $\eta$, but when going to primitive variables they all cancle out and only the term for $h$ is left.

[JoshuaLampert]

Thanks! This looks good to me. Just for clarification: Do you have a formal proof of entropy conservation for this semidiscretization?

Yes. I dont have a TeXed version of the proof yet, but I did calculate it and energy is conserved for both flat and variable bottom topography.

It only uses a SAT for h , but no further treatment of the other variables is needed, right?

In the conservative form you also have additional terms for u , w and η , but when going to primitive variables they all cancle out and only the term for h is left.

Ah, that's interesting. Thanks for the explanation!