Flooding and drying in discontinuous Galerkin discretizations of shallow water equations

Flooding and drying in space or space-time discontinuous Galerkin (DG) discretizations provides an accurate and efficient numerical scheme. Moreover, the space-time DG method is particularly suitable for moving or deforming meshes. The shallow water equations, which can exhibit flooding and drying due to the movement of water front, are considered and discretized with linear polynomial approximations. In the finite elements connected to the water front, the means (zeroth order term) are used to conserve the mass and momentum, and the slopes (first order term) are used to follow the front movement accurately, which contrasts finite volume schemes where the slopes have to be reconstructed. The front movement can be governed by a front equation in a restricted situation in which the front is single valued in either one of the horizontal coordinate directions. In general, however, the water line can be complicated and the topology of the domain can change as islands become flooded or fall dry. We are developing the following two approaches for this general case. In an Eulerian-Lagrangian space DG method, the free boundary nodes at the water line are allowed to move, while the time step is restricted and the mesh is locally updated to keep the dry and wet mesh around the water line well-behaved. In the space-time DG method a level set is introduced to demarcate the water line while the water depth and level set are matched properly. As a preliminary step, we have obtained numerical results for two exact solutions by prescribing the exact front movement. These are second order accurate in space and time