/**
# Compressible gas dynamics
The Euler system of conservation laws for a compressible gas can be
written
$$
\partial_t\left(\begin{array}{c}
\rho \\
E \\
w_x \\
w_y \\
\end{array}\right)
+
\nabla_x \cdot\left(\begin{array}{c}
w_x \\
\frac{w_x}{\rho} ( E + p ) \\
\frac{w_x^2}{\rho} + p \\
\frac{w_y w_x}{\rho} \\
\end{array}\right)
+
\nabla_y \cdot\left(\begin{array}{c}
w_y \\
\frac{w_y}{\rho} ( E + p ) \\
\frac{w_y w_x}{\rho} \\
\frac{w_y^2}{\rho} + p \\
\end{array}\right)
= 0
$$
with $\rho$ the gas density, $E$ the total energy, $\mathbf{w}$ the
gas momentum and $p$ the pressure given by the equation of state
$$
p = (\gamma - 1)(E - \rho\mathbf{u}^2/2)
$$
with $\gamma$ the polytropic exponent. This system can be solved using
the generic solver for systems of conservation laws. */
#include "conservation.h"
/**
The conserved scalars are the gas density $\rho$ and the total energy
$E$. The only conserved vector is the momentum $\mathbf{w}$. The constant
$\gamma$ is represented by *gammao* here, with a default value of 1.4. */
scalar rho[], E[];
vector w[];
scalar * scalars = {rho, E};
vector * vectors = {w};
double gammao = 1.4 ;
/**
The system is entirely defined by the `flux()` function called by the
generic solver for conservation laws. The parameter passed to the
function is the array `s` which contains the state variables for each
conserved field, in the order of their definition above (i.e. scalars
then vectors). */
void flux (const double * s, double * f, double e[2])
{
/**
We first recover each value ($\rho$, $E$, $w_x$ and $w_y$) and then
compute the corresponding fluxes (`f[0]`, `f[1]`, `f[2]` and
`f[3]`). */
double rho = s[0], E = s[1], wn = s[2], w2 = 0.;
for (int i = 2; i < 2 + dimension; i++)
w2 += sq(s[i]);
double un = wn/rho, p = (gammao - 1.)*(E - 0.5*w2/rho);
f[0] = wn;
f[1] = un*(E + p);
f[2] = un*wn + p;
for (int i = 3; i < 2 + dimension; i++)
f[i] = un*s[i];
/**
The minimum and maximum eigenvalues for the Euler system are the
characteristic speeds $u \pm \sqrt(\gamma p / \rho)$. */
double c = sqrt(gammao*p/rho);
e[0] = un - c; // min
e[1] = un + c; // max
}