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

This paper 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 generally imposed at $x=0$.
   We first focus on the coupling of two scalar conservation laws and state an existence result for the coupled Riemann problem. Numerical experiments are also proposed. We 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.