Single-phase test cases
Test cases presented in this section are dealing with single-phase compressible problems. In this part, ECOGEN solves the Euler equations [Eul57]:
\[
\label{eqEuler}
\begin{array}{l}
\displaystyle \frac{\partial\rho}{\partial t}+ div(\rho {\mathbf{u}} )=0\\
\displaystyle\frac{\partial\rho {\mathbf{u}}}{\partial t}+ div(\rho {\mathbf{u}} \otimes {\mathbf{u}} +p \mathbf{I})=\mathbf{0} \\
\displaystyle \frac{\partial\rho E}{\partial t}+ div((\rho E+p){\mathbf{u}})=0
\end{array}
\]
where \(\rho\) represents the density, \(\mathbf{u}\) the velocity vector, \(p\) the pressure and \(E = e +\frac{\mathbf{u}^2}{2}\) the total energy, with \(e\) the internal energy. The closure relation for this model is ensured by any convex equation of state (EOS) \(p = p(\rho,e)\) (see section Materials for details about implemented EOS in ECOGEN). Euler equations are solved thanks to an explicit finite-volume Godunov-like scheme [GZI+79] that is coupled with HLLC approximate Riemann solver [Tor13] for flux computation.