Entropy Method and Asymptotic Behaviours of Finite Volume Schemes

When deriving a numerical scheme for a system of PDEs coming for instance from physics or engineering, it is crucial to propose a scheme which preserves the asymptotic behaviour of the continuous system, with respect to time as with respect to some parameters. In this paper, we want to show how the entropy method can be applied to some finite volume schemes and permits to show that some schemes are asymptotic preserving. We focus on two problems: the nonlinear diffusion equation (long time behaviour) and the drift-diffusion system (long time behaviour and quasi-neutral limit). Some results have been obtained in collaboration with jungel and Schuchnigg [10] and the others with Bessemoulin-Chatard and Vignal [4].