src/compressible.h
Compressible gas dynamics
The Euler system of conservation laws for a compressible gas can be written
\displaystyle \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
\displaystyle 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
}