A Kinetic Method for Hyperbolic–Elliptic Equations and Its Application in Two-Phase Flow

A kinetic method for hyperbolic–elliptic equations is presented in this paper. In the mixed type system, the coexistence of liquid and gas and the phase transition between them are described by the van der Waals-type equation of state (EOS). Because the fluid is unstable in the elliptic region, the interface between liquid and gas can be captured naturally through condensation and evaporation processes, which continuously remove any “averaged” numerical fluid away from the elliptic region at the interfaces. As a result, a sharp liquid–gas interface can be constructed from the competition between the numerical diffusion and phase transition. The numerical examples presented in this paper include both phase transition and multifluid interface problems.