The Riemann Problem for Hyperbolic Equations under a Nonconvex Flux with   Two Inflection Points
  This report addresses the solution of Riemann problems for hyperbolic equations when the nonlinear characteristic fields loose their genuine nonlinearity. In this context, exact solvers for nonconvex 1D Riemann problems are developed. First a scalar conservation law for a nonconvex flux with two inflection points is studied. Then the P-system for an isothermal version of the van der Waals gas model is examined in a range of temperatures allowing for a nonconvex pressure function. Eventually the system of the Euler equations of gasdynamics is considered for the polytropic van der Waals gas. In this case, a suitably large specific heat is considered such that the isentropes display a local loss of convexity near the saturation curve and the critical point. Such a nonconvex physical model allows for nonclassical waves to appear as a result of the change of sign of the fundamental derivative of gasdynamics. The solution of the Riemann problem for the considered real gas model reduces to a system of two nonlinear equations for the values of the density on the two sides of the contact discontinuity, much in the same manner of a recently proposed solution method for gases admitting nonlinear wavefields only fully genuine. Solutions containing nonclassical and mixed waves are presented for the three mathematical models. Vacuum formation is described analytically including the presence of composite rarefaction waves, and to the best of authors knowledge, produced numerically here for the first time. 
Marco Fossati and Luigi Quartapelle
physics.flu-dyn