Existence result for the coupling problem of two scalar conservation laws with Riemann initial data

This paper is devoted to the coupling problem of two scalar conservation laws through a fixed interface located for instance at x=0. Each scalar conservation law is associated with its own (smooth) flux function and is posed on a half-space, namely x < 0 or x > 0. At interface x = 0 we impose a coupling condition whose objective is to enforce in a weak sense the continuity of a prescribed variable, which may differ from the conservative unknown (and the flux functions as well). We prove existence of a solution to the coupled Riemann problem using a constructive approach. The latter allows in particular to highlight interesting features like non uniqueness of both continuous and discontinuous (at interface x = 0) solutions. The behavior of some numerical scheme is also investigated.