Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes

This work is concerned with the numerical capture of stiff viscous shock solutions of Navier-Stokes equations for complex compressible materials, in the regime of large Reynolds numbers. After [2] and [6], a relevant numerical capture is known to require the satisfaction of an extended set of non classical Rankine-Hugoniot conditions due to the non conservation form of the governing PDE model. Here, we show how to enforce their validity at the discrete level without the need for solving local non linear algebraic problems. Non linearities are bypassed when introducing new averaging techniques which are proved to satisfy all the desirable stability properties when invoking suitable approximate Riemann solutions. A relaxation procedure is proposed to that purpose with the benefit of a fairly simple overall numerical method.