Preprint 2025-18
Raimund Bürger, Enrique D. Fernández Nieto, José Garres-Díaz, Jorge Moya:
Well-balanced physics-based finite volume schemes for Saint-Venant-Exner-type models of sediment transport
Abstract:
The Saint-Venant-Exner (SVE) model is widely used for the description of sediment transport including bedload, erosion, and deposition processes. A modified version of the SVE model, which includes sediment concentration incorporates exchange of sediment between the fluid and an erodible bed and a non-hydrostatic pressure for the fluid along with non-equilibrium entrainment and deposition velocities, is introduced. Gravitational effects on erosion are described by an effective shear stress formulation. This modified SVE model is derived from a general approach with density variations. It preserves the mass of both the sediment and the fluid, and satisfies a dissipative energy balance. On the other hand, well-balanced finite volume schemes adapted for SVE models are derived since standard well-balanced schemes for the Saint-Venant system with fixed bottom are in general no more well-balanced when applied to the SVE model. The latter property is due to the uncontrolled numerical diffusion associated with the bed evolution equation. Two novel techniques to achieve the well-balanced property for the modified SVE model are proposed. The first is a new polynomial-viscosity-matrix-based (PVM) scheme, denoted “PVM-2I”, that modifies the numerical approximation of the bed evolution equation according to its related characteristic speed. The second is a physically motivated correction of the numerical diffusion term for the Rusanov and Harten-Lax-van Leer (HLL) schemes. The proposed schemes are positivity-preserving for the water height. Numerical solutions are compared with exact solutions with gravitational effects, with a novel exact solution in non-equilibrium conditions, and with experimental data. It is illustrated how the use of standard non-well-balanced schemes leads to a large artificial (unphysical) erosion and completely degraded solutions. This undesirable behaviour is avoided by the proposed well-balanced schemes.
