Application of Fully Implicit Discontinuous Galerkin Wet–Dry Scheme for Shallow Water Equations

Moving boundaries (often called free boundaries) in shallow-water equations mark the time-dependent water line between ‘(flooded) wet’ and ‘dry’ regions and this feature is of utmost practical importance and should be adequately incorporated into the simulation. An efficient and conceptually simple implicit scheme is proposed for the one- and two-dimensional shallow-water equation systems with moving boundary treatment (i.e., wet–dry scheme). For the implementation of a wet–dry scheme in which very low water depth is present, a fully implicit backward-Euler scheme was developed for the fixed Eulerian mesh domain and a positivity-preserving limiter was employed. For the approximate Riemann problem, the local Lax–Friedrichs flux is used. The Jacobians are estimated by centered finite differences in an approximate way rather than exact formulas, which, in principle, greatly reduce the coding efforts compared to the analytical formulation and will make extensions to higher dimensional schemes. Some benchmark problems with exact solutions and/or reference experimental data are presented for the domain where the initially dry bed presents. Overall, satisfactory results are obtained.