Flooding and drying in finite-element discretizations of shallow-water equations. Part 1: One dimension

Free boundaries in shallow-water equations demarcate the time-dependent water line between ``flooded'' and ``dry'' topography. A novel numerical algorithm to treat flooding and drying in a formally second-order explicit space discontinuous finite element discretization of the one-dimensional or symmetric shallow-water equations is presented. The algorithm uses fixed Eulerian flooded elements and one mixed Eulerian-Lagrangian element at each free boundary. The positivity of the mean water depth is ensured via a time step restriction based on analysis of a maximum principle for the discretized continuity equation while using an HLLC flux. The algorithm and its implementation are tested in comparison with a large and relevant suite of known exact solutions. The essence of the flooding and drying algorithm pivots around the analysis of a continuity equation with a fluid velocity and a pseudo density (in the shallow water case the depth). It therefore also applies, for example, to space discontinuous finite-element discretizations of the compressible Euler equations in which vacuum regions emerge, in analogy of the above dry regions. The approach is hypothesized to extend to finite-volume discretizations with similar mean level and slope reconstruction.