Well-balanced central-upwind scheme for a fully coupled shallow water system modeling flows over erodible bed

Intense sediment transport and rapid bed evolution are frequently observed under highly-energetic flows, and bed erosion sometimes is of the same magnitude as the flow itself. Simultaneous simulation of multiple physical processes requires a fully coupled system to achieve an accurate hydraulic and morphodynamical prediction. In this paper, we develop a high-order well-balanced finite-volume method for a new fully coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution.The nonequilibrium sediment transport equation is used to predict the sediment concentration variation. Since bed-load, sediment entrainment and deposition have significant effects on the bed evolution, an Exner-based equation is adopted together with the Grass bed-load formula and sediment entrainment and deposition models to calculate the morphological process. The resulting 5 × 5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-upwind scheme on a triangular grid. A computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. A special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed scheme. The proposed fully coupled model is verified on a number of numerical experiments.