Theoretical and numerical aspects of the interfacial coupling: The scalar Riemann problem and an application to multiphase flows

This note is devoted to the study of the one dimensional interfacial coupling of two PDE systems at a given fixed interface, say x = 0. Each system is posed on a half-space, namely x < 0 and x > 0. As an interfacial model, a coupling condition whose objective is to enforce the continuity (in a weak sense) of a prescribed variable is imposed at x = 0. We focus on the coupling of two scalar conservation laws and state an existence result for the coupled Riemann problem. Numerical experiments are also proposed. In the long version of this note we will then consider, both from a theoretical and a numerical point of view, the coupling of two-phase flow models namely a drift-flux model and a two-fluid model. In particular, the link between both models will be addressed using asymptotic expansions.