Implicit time schemes for compressible fluid models based on relaxation methods

In this work, we are interested in the implicit discretization of compressible fluid models. When applied to this type of models, standard implicit schemes suffer from a lack of efficiency due to the ill-conditioning of the associated matrices. In this paper we propose an alternative strategy based on a relaxation method coupled with an implicit scheme. This method allows us to approximate the full complex problem by a collection of simpler ones that can be solved with standard methods. We obtain eventually a second order implicit method with some additional parallelism able to treat hyperbolic systems with small diffusion.