A positive and asymptotic preserving scheme for the linear transport equation on 2D unstructured meshes

In this paper, we propose a finite volume scheme for the linear transport equation in two space dimensions. This scheme is based on a second order upwind flux where the velocity is modified so as to recover the correct diffusion limit. A partially implicit time discretization is used. This allows to have good properties while keeping the computational cost per iteration very low. The resulting scheme is asymptotic preserving , positive under a classical CFL condition, conservative and second order consistent in all the regimes. These properties are valid on general unstructured meshes and the computational cost is similar to an explicit scheme. Eventually, the extension of this scheme to 3 D unstructured meshes is straightforward and its properties remain valid.