Design of asymptotic preserving schemes for the hyperbolic heat equation on unstructured meshes

The transport equation, in highly scattering regimes, has a limit in which the dominant behavior is given by the solution of a di usion equation. Angular discretization like the discrete ordinate method (SN ), the truncated spherical harmonic expansion (PN ) or also nonlinear moment models have the same property. For such systems it would be interesting to construct nite volume schemes on unstructured meshes which have the same dominant behavior even if the meshes are coarse. Such schemes are generally called di usion asymptotic preserving (AP) schemes and are designed presently at most on Cartesian meshes. In this work we give some answers for unstructured meshes, when considering the lowest order possible angular discretization of the transport equation that is the P! model also refereed to as the hyperbolic heat equation, the Cattaneo's equation or the rst order formulation of the telegraph equation. We start from the modi ed upwind AP scheme proposed by Jin and Levermore [JL96] for this equation in 1-D. We show that extended in 2-D on unstructured meshes, the classical edge formulation of this scheme (and also for other AP schemes) is no longer asymptotic preserving. To solve this problem, we propose new schemes built on a node formulation of the Jin and Levermore's scheme which use the analogy between P1 model and acoustic equations for which schemes with corner's uxes have been built in the context of gas dynamics [Maz07, MABO07].