A numerical model for simulation of bedload transport in unsteady trans-critical river flow

We present a numerical model for the simulation of unsteady trans-critical flow with bedload transport in rivers. The approach uses second-order well-balanced finite volume methods to solve the Saint-Venant-Exner equations with rectangular cross-section in 1 dimension. The equations are treated as a non-conservative hyperbolic system with distinct wave speeds, influenced by sediment transport processes. Two finite volume approximate Riemann solvers are implemented, based on an augmented Roe solver and an adapted HLL solver that both deal with the Saint-Venant-Exner equations as a coupled system. The three source terms of the model (bottom, friction and width) are discretized in a way that preserves of lake at rest equilibrium and positivity of the water depth. Model stability is ensured by a Courant-Friedrichs-Lewy (CFL) condition which depends on the wave speeds of the coupled system. Finally second-order schemes are proposed, based on a modified Heun time scheme and linear state reconstructions at cell interfaces and slope limiters. The paper highlights the challenges of computing the Jacobian matrix for various sediment transport models. We propose using different approximations of the wave speeds and present exact solid flux derivatives for many classical sediment transport laws. We also propose approximate solid flux derivatives which allow a simple generalization of the numerical model to any law, enabling real life industrial applications. The model's performance was validated against analytical and experimental data, proving its sturdiness and precision. We also compare our approach to other numerical methods and to the Mascaret industrial code and its 1-dimensional sediment transport module Courlis.