Comparison of High-order Numerical Schemes for Flows in Heterogeneous and Anisotropic Porous Media

We consider unstationnary flows in porous media through Richards equation. Backward Differentiation Formulas are used to discretize the unstationnary term and we propose to compare DG (Discontinuous Galerkin) and DDFV (Discrete Duality Finite Volume) schemes for the discretization of the diffusive term. On the one hand, the flexibility of DG methods and their ample theoretical foundation make them a reliable choice for a number of computational problems. Two DG methods are used for Richards equation; we consider here the Symmetric Interior Penalty Galerkin (SIPG) method because it preserves the natural symmetry of the discrete diffusion operator, and the Local Discontinuous Galerkin (LDG) method which ensures convergence with a positive penalty parameter. On the other hand, the e fficiency of DDFV methods has also been proved to approximate the diffusive fl ux; besides, as finite voume methods, they ensure good preservation of physical properties and offer superconvergence in the L2-norm on a regular basis. Accuracy and robustness of theses schemes are tested and compared on relevant test cases, specially in heterogeneous and anisotropic medium.