An efficient box-scheme for convection-diffusion equations with sharp contrast in the diffusion coefficients

In this paper, we introduce a box-scheme for time-dependent convection-diffusion equations, following principles previously introduced by Courbet in [Rech. Aérospatiale 4 (1990) 21] for hyperbolic problems. This scheme belongs to the category of mixed finite-volume schemes. This means that it works on irregular meshes (finite-volume scheme) and computes simultaneously the principal unknown and its gradient in all Peclet regimes, ranging from pure diffusion (Pe=0) to pure convection (Pe=+∞). The present paper focuses mainly on the design of the scheme, which is non-standard, in the case of the 1D convection-diffusion equation. The version of the scheme presented here is of first or second order depending on the local Peclet number. We extend the 1D scheme afterwards in 2D by an ADI like technique. Several numerical results on 1D and 2D test-cases of interest for flow simulation in porous media are presented, some of them exhibiting sharp contrasts in diffusion coefficients.