High‐order hybridizable discontinuous Galerkin formulation with fully implicit temporal schemes for the simulation of two‐phase flow through porous media

We present a memory‐efficient high‐order hybridizable discontinuous Galerkin (HDG) formulation coupled with high‐order fully implicit Runge‐Kutta schemes for immiscible and incompressible two‐phase flow through porous media. To obtain the same high‐order accuracy in space and time, we propose using high‐order temporal schemes that allow using large time steps. Therefore, we require unconditionally stable temporal schemes for any combination of element size, polynomial degree, and time step. Specifically, we use the Radau IIA and Gauss‐Legendre schemes, which are unconditionally stable, achieve high‐order accuracy with few stages, and do not suffer order reduction in this problem. To reduce the memory footprint of coupling these spatial and temporal high‐order schemes, we rewrite the nonlinear system. In this way, we achieve a better sparsity pattern of the Jacobian matrix and less coupling between stages. Furthermore, we propose a fix‐point iterative method to further reduce the memory consumption. The saturation solution may present sharp fronts. Thus, the high‐order approximation may contain spurious oscillations. To reduce them, we introduce artificial viscosity. We detect the elements with high‐oscillations using a computationally efficient shock sensor obtained from the saturation solution and the post‐processed saturation of HDG. Finally, we present several examples to assess the capabilities of our formulation.