Novel numerical approach for 1D variable density shallow flows over uneven rigid and erodible beds

The numerical modelling of hyperconcentrated shallow flows is a challenging task because they exhibit special features, such as propagation over dry bed, profound bed elevation modifications due to erosion or deposition phenomena, and flow discontinuities. In this paper, a novel depthpositivity preserving HLLC Riemann solver is devised in order to approximate the solution of the Riemann problem for the one-dimensional Hyper-concentrated Shallow Flows equations over horizontal bed. The solver is used as a building block for the construction of HCSF, a wellbalanced Finite Volume scheme for the solution of the Hyperconcentrated Shallow Flows equations with variable elevation. HCSF is able to handle the case of dry bed, to take into account the variability of the topography also in presence of bed discontinuities, considering the flow resistance and the mass exchange between the flowing mixture and the mobile bed. The numerical tests carried out confirm the well-balancing property of the scheme proposed, the robustness in presence of dry bed, the ability to approximate the analytic solution of problems with smooth or discontinuous bed, and the ability to reproduce reasonably the results of a laboratory experiment.