Schémas préservant l'asymptotique hyperpolique-parabolique

In this work we focus on explicit finite volume schemes for systems of conservations laws in two dimensions with stiff source terms. Such systems may degenerate into diffusion equations. It is a major numerical challenge to follow this degeneracy. We propose a general framework to design an asymptotic preserving scheme, that is stable and consistent under a classical hyperbolic CFL condition in both hyperbolic and diffusive regime, for any two-dimensional unstructured mesh. Moreover, the scheme developed also preserves the set of admissible states, which is mandatory to keep physical solutions in stiff configurations. This construction is achieved by using a non-linear scheme as a target scheme for the diffusive equation, which gives the form of the global scheme for the complete system of conservation laws. Numerical results are provided to validate the scheme in both regimes.