Implicit time advancing applied to shallow water problems coupled with different models of sediment transport

The numerical simulation of sediment transport problems is considered. The physical problem is modeled through the shallow-water equations coupled with the Exner equation to describe the time evolution of the bed profile. Different models of solid transport discharge of increasing complexity are considered. The spatial discretisation of the governing equations is carried out by a finite-volume method and a modified Roe scheme designed for non-conservative systems. Linearized implicit schemes for time advancing are built through a recently proposed strategy, based on automatic differentiation to compute the flux Jacobians and on the defect correction approach to reach second-order accuracy. Explicit schemes for time advancing are compared with implicit ones in one-dimensional sediment transport problems, characterized by different time scales for the evolution of the bed. It is shown that, independently of the model used for the solid transport discharge, for slow and intermediate speeds of interaction bewteen the bedload and the water flow, for which the use of large time steps is compatible with the capture of the bed evolution, implicit time advancing is far more efficient than explicit one with a computational cost reduction up to more than three orders of magnitude.