A NUMERICAL STUDY OF RIEMANN PROBLEM SOLUTIONS AND STABILITY FOR A SYSTEM OF VISCOUS CONSERVATION-LAWS OF MIXED TYPE

A numerical study of the isothermal fluid equations with a nonmonotone equation of state (like that of van der Waals) and with viscosity and capillarity terms is presented. This system is ill-posed (i.e., elliptic in x vs. t) in some regions of state space and well-posed (i.e., hyperbolic) in other regions. Thus, it may serve as a model for describing dynamic phase transitions. Numerical computations of phase jumps, shock waves, and rarefaction waves for this system are presented. Although the solution of the Riemann problem is not unique, all of these waves are found to be stable to infinitesimal initial perturbations. Criteria are found for instability after O(1) initial perturbations. An analytic argument is made for stability of phase transitions.