# src/compressible.h

# Compressible gas dynamics

The Euler system of conservation laws for a compressible gas can be written

$${\partial}_{t}\left(\begin{array}{c}\hfill \rho \hfill \\ \hfill E\hfill \\ \hfill {w}_{x}\hfill \\ \hfill {w}_{y}\hfill \end{array}\right)+{\nabla}_{x}\cdot \left(\begin{array}{c}\hfill {w}_{x}\hfill \\ \hfill \frac{{w}_{x}}{\rho}(E+p)\hfill \\ \hfill \frac{{w}_{x}^{2}}{\rho}+p\hfill \\ \hfill \frac{{w}_{y}{w}_{x}}{\rho}\hfill \end{array}\right)+{\nabla}_{y}\cdot \left(\begin{array}{c}\hfill {w}_{y}\hfill \\ \hfill \frac{{w}_{y}}{\rho}(E+p)\hfill \\ \hfill \frac{{w}_{y}{w}_{x}}{\rho}\hfill \\ \hfill \frac{{w}_{y}^{2}}{\rho}+p\hfill \end{array}\right)=0$$

with $\rho $ the gas density, $E$ the total energy, $\mathbf{\text{w}}$ the gas momentum and $p$ the pressure given by the equation of state

$$p=(\gamma -1)(E-\rho {\mathbf{\text{u}}}^{2}/2)$$

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 $\mathbf{\text{w}}$. The constant $\gamma $ is represented by *gammao* here, with a default value of 1.4.

```
scalar ρ[], E[];
vector w[];
scalar * scalars = {ρ, 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 ρ = 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/ρ, p = (gammao - 1.)*(E - 0.5*w2/ρ);
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/ρ);
e[0] = un - c; // min
e[1] = un + c; // max
}
```