A new relaxation method for the compressible Navier-Stokes equations

We derive a new relaxation method for the compressible Navier-Stokes equations endowed with general pressure and temperature laws compatible with the existence of an entropy functional and Gibbs relations. Our method is an extension of the energy relaxation method introduced by Coquel and Perthame for the Euler equations. We first introduce a consistent splitting of the diffusion fluxes as well as a global temperature for the relaxation system. We then prove that under the same subcharacteristic conditions as for the relaxed Euler equations and for a specific form of the global temperature and the heat flux splitting, the stability of the relaxation system may be obtained from the non-negativity of a suitable entropy production. A first-order asymptotic analysis around equilibrium states confirms the stability result. Finally, we present a numerical implementation of the method allowing for a straightforward use of Navier-Stokes solvers designed for ideal gases as well as numerical results illustrating the accuracy of the proposed algorithm.