Linearized implicit time advancing and defect correction applied to sediment transport simulations

The numerical simulation of sediment transport problems is considered in this paper. The physical problem is modeled through the shallow-water equations coupled with the Exner equation to describe the time evolution of the bed profile. The spatial discretization of the governing equations is carried out by a finite-volume method and a modified Roe scheme designed for non-conservative systems. As for the time advancing, starting from an explicit method, a linearized implicit scheme is generated, in which the flux Jacobian is computed through automatic differentiation. Second-order accuracy in space is then obtained through MUSCL reconstruction and in time through a backward differentiation formula associated with a defect-correction approach. The implicit time advancing is compared in terms of accuracy and computational time with the explicit approach for one-dimensional and two-dimensional sediment transport problems, characterized by different time scales for the evolution of the bed and of the water flow. It is shown that, whenever the use of large time steps is compatible with the capture of the water flow dynamics and of the bedload evolution, the implicit scheme is far more efficient than its explicit counterpart with a CPU reduction up to more than two orders of magnitude. This makes implicit time differencing an attractive option for complex real life applications in this area.