/** # 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 }