# Riemann solvers for metastable phases

Most Riemann solvers I've come across can solve the Riemann problem only under certain conditions such as convexity of the equation of state. But what happens if the fluid enters a metastable state or the spinodal region or the fluid state follows a nonconvex isentrope in the p-v diagram? All the approximate solvers described in the well known text books such as Toro break down here and I guess the same is true for exact solvers which only assume the three possible wave configurations of rarefaction,shock and contact.

But how to simulate for example a v.d.W - liquid including phase transitions? What to do if the pressure of the unstable/metastable state becomes negative? I've seen people doing the Maxwell construction in these cases. However, this results in in constant pressure-derivatives $$\left(\frac{\partial p}{\partial \rho}\right)_S$$ within these regions and as a result the adiabatic speed of sound becomes 0 which in turn makes the system of PDEs non-hyperbolic.

For the last couple of hours I was searching for a more general Riemann solver. However, I wasn't able to find one.

Did I miss something or is this an unsolved problem?