/**
# Benchmarks
## Friction test cases
Here we test the friction source term in the case of
fluvial/torrential flows for three differents source terms. The
original test case is the one of MacDonald[1] and you can find more
similar test cases in Delestre & al.[2]. The principle is to set the
topography in order to have an analytical solution for h(x). The
source term is treated semi-implicitly at the same order than the
Saint-Venant function, which is at order 2, except for the Darcy 1
order case.
### Case description
We consider a 1D topography $z_b(x)$ which is 1000 metres long.
At steady state, the momentum equation of Saint-Venant can be written as :
$$
\partial_x z_b(x) = (\frac{q^2}{g h^3}-1) \partial_x h(x) - S_f \ ,
$$
where $S_f$ is the friction term. By chosing an analytical function $hex(x)$
for $h(x)$, we can find the resulting topography with a r-k method.
### Fluvial
For the sub-critical case, the analytical water height at steady state is :
$$
hex(x) = \left( \frac{4}{g} \right)^{1/3} \left( 1 + 0.5\times exp\left( -16\times (\frac{x}{1000}-0.5)^2 \right) \right).
$$
The slope is initially dry ($h(x) = u(x) = 0$) and we chose as boundaries conditions :
$$
q[left] = 1.5 m^2 .s^{-1} \ and \ h[left] = hex(0) ,
$$
$$
q[right] = 1.5 m^2 .s^{-1} \ and \ h[right] = hex(1000).
$$
* [Manning](fricfluv_man.c) with $n = 0.033$
* [Poiseuille](fricfluv_poiseuille.c) with $\nu = 1e-6$
* [Darcy](fricfluv_darcy.c) with $f = 0.093$
* [Darcy with a 1 order treatment for the source term](fricfluv_darcy_0.c) with $f = 0.093$
We plot an example of the resulting topography for the Darcy friction :
![Resulting topography](/topofluvial_darcy.png)
### Torrential
For the super-critical case, the analytical water height at steady state is :
$$
hex(x) = \left( \frac{4}{g} \right)^{1/3} \left( 1 - 0.2\times exp\left( -36\times (\frac{x}{1000}-0.5)^2 \right) \right).
$$
The slope is initially dry ($h(x) = u(x) = 0$) and we chose as boundaries conditions :
$$
q[left] = 2.5 m^2 .s^{-1} \ and \ h[left] = hex(0).
$$
* [Manning](frictor_man.c) with $n = 0.04$
* [Poiseuille](frictor_poiseuille.c) with $\nu = 1e-6$
* [Darcy](frictor_darcy.c) with $f = 0.065$
* [Darcy with a 1 order treatment for the source term](frictor_darcy_O.c) with $f = 0.065$
We plot an example of the resulting topography for the Darcy friction :
![Resulting topography](/topotorrential_darcy.png)
### Order treatment
We compare the convergence of $|h|_1$ for the different source treatment
(order 1 and 2), for the torrential case with the darcy friction term :
![Error Convergence](/tor_darcy10_error.png)
Note that the convergence depends strongly on the chosen source term.
### Subcritical to super-critical
The flow is subcritical upstream and super-critical downstream. The water
height is :
$$
h_{ex}(x) = \left( \frac{4}{g} \right)^{1/3} \left( 1 - 3\times tanh\left( -3\times (\frac{x}{1000}-0.5)^2 \right) \right).
$$
for $0 < x < 500$ and :
$$
h_{ex}(x) = \left( \frac{4}{g} \right)^{1/6} \left( 1 - 6\times tanh\left( -3\times (\frac{x}{1000}-0.5)^2 \right) \right).
$$
for $500 < x < 1000$.
* [Darcy](fricsubsup_darcy.c) with $q0 = 2\ m. s^{-1}$ and $f = 0.042$
## Rain test cases
We add a rain term to the MacDonald test case, as seen in Delestre et
al.[2]. Write R the constant rain intensity, the flux along the
channel is written :
$$
q(x) = q0 + x*R.
$$
So the topography is now :
$$
\partial_x z_b(x) = (\frac{q(x)^2}{g h^3}-1) \partial_x h(x) -\frac{2q(x) R}{g h(x)^2}- S_f
$$
We take the same function for hex(x) seen with the friction test cases in both torrential
and fluvial case and we use the Darcy friction term :
* [Rain in a fluvial flow with Darcy friction](fricfluv_rain_darcy.c)
* [Rain in a torrential case with Darcy friction](frictor_rain_darcy.c)
## References
[1] I. MacDonald, M. Baines, N. Nichols, and P. G. Samuels, “Analytic
Benchmark Solutions for Open-Channel Flows,” no. November,
pp. 1041–1045, 1997.
[2] O. Delestre, C. Lucas, P. A. Ksinant, F. Darboux, C. Laguerre,
T. N. T. Vo, F. James, and S. Cordier, “SWASHES: A compilation of
shallow water analytic solutions for hydraulic and environmental
studies,” Int. J. Numer. Methods Fluids, vol. 72, pp. 269–300, 2013.
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