A new relaxation method for the compressible Navier-Stokes equations

We consider the compressible Navier-Stokes equations for gas flows endowed with general pressure and temperature laws as long as they are compatible with the existence of an entropy and Gibbs relations. We extend the relaxation method introduced for the Euler equations by Coquel and Perthame. Keeping the same "sub-characteristic" conditions for the hyperbolic fluxes and using a consistent splitting of the diffusive fluxes based on a global temperature, we prove the stability of the relaxation system via the sign of the production of a suitable entropy. A first order asymptotic analysis around equilibrium states confirms the stability result. Finally, we present a numerical implementation of the method.