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?


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