Derivation of the Nonlinear Shallow Water Equations Over Unsaturated Porous Media: A Discontinuous Galerkin Method for Coarse-Grained Beaches

This work examines the physics of free-surface flow and groundwater flow within a coupled model. Coupled models for such phenomena are not clearly justified, and there is a lack of precision in the derivation of such models. The primary objective of this work is to derive a coupled model of the shallow water equations (SWE) and Richards’ equation (RE) using asymptotic considerations. The numerical coupling approach chosen for the unified model will be described as a parallel coupling. Additionally, numerical considerations regarding how to solve this model using the discontinuous Galerkin (DG) methods will be provided. Furthermore, the exchange of information between the two models, which are time-synchronized, will be explained. The solution of RE coupled with SWE, following the described procedure and implemented using the DG formulation, is integrated into RIVAGE (an in-house numerical code based on the DG method). This implementation is then tested on a numerical problem and validated against an experimental benchmark.