Implicit Discontinuous Galerkin Scheme for Discontinuous Bathymetry in Shallow Water Equations

One important issue in the approach with shallow water equations, which is not restricted to discontinuous Galerkin formulation, is the limitation to step geometries (discontinuous bathymetry), due to the hydrostatic assumption employed for the derivation of shallow water equations from the Navier-Stokes equations. In addition, the explicit Runge-Kutta time-stepping schemes do not come without any drawbacks even though the majority of discontinuous Galerkin applications have employed explicit ones due to simplicity. In this study, the recently developed implicit discontinuous Galerkin scheme is combined with the surface gradient method for steps (SGMS). The developed scheme is verified with flows over discontinuous bathymetry, i.e., vertical steps and weirs. For one-dimensional problems, the flows over a step and over a rectangular weir are simulated. As for two-dimensional problems, the flow over a weir and the dam-break flow over a step followed by the L-shaped and 45°-bend channels are simulated. The numerical solutions are compared with the experimental data. In all cases, good agreements were observed and the effectiveness of the developed scheme was verified.