One-Phase and Two-Phase Flow Simulation Using High-Order HDG and High-Order Diagonally Implicit Time Integration Schemes

We present two high-order hybridizable discontinuous Galerkin (HDG) formulations combined with high-order diagonally implicit Runge-Kutta schemes to solve one-phase and two-phase flow problems through porous media. The HDG method is locally conservative and allows reducing the size of the global systems due to the hybridization procedure, and the pressure, the saturation and the velocity converge with order P + 1 in L2-norm, with P being the polynomial degree. In addition, an element-wise post-process can be applied to obtain a convergence rate of P + 2 in L2-norm for the pressure and saturation. To achieve these rates of convergence the temporal errors should be small enough. For this purpose we combine HDG with high-order diagonally implicit Runge-Kutta (DIRK) temporal schemes. Finally, we present four examples dealing with 2D and 3D problems, and high-order structured and unstructured meshes.