compressible.h

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/** @file compressible.h
 */
/**
# Compressible gas dynamics
 
The Euler system of conservation laws for a compressible gas can be
written
 
\f[
\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
\f]
 
with \f$\rho\f$ the gas density, \f$E\f$ the total energy, \f$\mathbf{w}\f$ the
gas momentum and \f$p\f$ the pressure given by the equation of state
 
\f[
p = (\gamma - 1)(E - \rho\mathbf{u}^2/2)
\f]
 
with \f$\gamma\f$ 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 \f$\rho\f$ and the total energy
\f$E\f$. The only conserved vector is the momentum \f$\mathbf{w}\f$.  The constant
\f$\gamma\f$ 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)
{
 
  /**
  We first recover each value (\f$\rho\f$, \f$E\f$, \f$w_x\f$ and \f$w_y\f$) 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 \f$u \pm \sqrt(\gamma p / \rho)\f$. */
 
  double c = sqrt(gammao*p/rho);
  e[0] = un - c; // min
  e[1] = un + c; // max
}