Flooding and drying in finite-element discretizations of shallow-water equations. Part 2: Two dimensions

``Flooding'' and ``drying'' in the shallow-water equations consists of the time-dependent motion of the water line or free boundary over topography. A novel numerical algorithm is developed to deal with flooding and drying in a second-order space discontinuous finite-element method. The mesh has fixed Eulerian elements and mixed Eulerian-Lagrangian elements at free boundaries. The algorithm contains the following aspects. The weak formulation is adapted to include the movement of the nodes on the free boundary. A time-step restriction is derived by analyzing the HLLC numerical flux used in order to ensure positivity of the mean water depth. The mesh quality is maintained by updating the mesh topology by locally deforming an underlying basic mesh of rectangles. A good triangular mesh is found by locally swapping or inserting diagonals in this (locally deformed) rectangular mesh. The water depth and the velocity are approximated using mean and slope information. Finally, even although the mean water depth is ensured to be positive, the water depth is adjusted when it falls below zero by limiting the slope information of the depth. The algorithm devised can also be applied or extended to other mathematical models with a two-dimensional continuity equation, such as the two-dimensional compressible equations of motion.