A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles

We propose in this work an original finite volume scheme for the system of gas dynamics in a nozzle. Our numerical method is based on a piecewise constant discretization of the cross-section and on an approximate Riemann solver in the sense of Harten, Lax and van Leer. The solver is obtained by the use of a relaxation approximation that leads to a positive and entropy satisfying numerical scheme for all variation of section, even discontinuous sections with arbitrary large jumps. To do so, we introduce, in the first step of the relaxation solver, a singular dissipation measure superposed on the standing wave, which enables us to control the approximate speeds of sound and thus the time step, even for extreme initial data.