High-Order Entropy-Conservative Schemes and Kinetic Relations for van der Waals Fluids

We consider the mixed (hyperbolic–elliptic) system of two conservation laws modeling the dynamics of van der Waals fluids. Viscosity and capillarity effects are taken into account. We introduce a new class of semidiscrete high-order schemes which are entropy conservative (in the sense of Tadmor) when the viscosity is neglected and, otherwise, dissipate the associated mathematical entropy. Our numerical schemes generalize the works by E. Tadmor (1987, Math. Comput.49, 91) and P. G. LeFloch and C. Rohde (2000, SIAM J. Numer. Anal.37, 2023) who proposed second-order and third-order entropy-conservative schemes, respectively. ing B. T. Hayes and P. G. LeFloch (1998, SIAM J. Numer. Anal.35, 2169), we demonstrate numerically that balanced viscosity and capillarity terms in van der Waals fluids may generate nonclassical shock waves or subsonic propagating phase transitions. Such waves are undercompressive and do not satisfy standard entropy criteria. They must be characterized by a kinetic function, which we determine numerically in this paper from vanishing viscosity and capillarity. The kinetic relation is an efficient tool to discuss the interplay among the viscosity, capillarity, and discretization parameters in van der Waals fluids.