Well-balanced physics-based finite volume schemes for Saint-Venant-Exner-type models of sediment transport

Raimund Bürger; Enrique D. Fernández Nieto; José Garres Díaz; Jorge Moya

doi:10.29393/ppudec-20bpbr40020

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Raimund Bürger Universidad de Concepción, Facultad de Ciencias Físicas y Matemática, Departamento de Ingeniería Matemática. Concepción, Chile.
Enrique D. Fernández Nieto Departamento de Matem´atica Aplicada I & IMUS, ETS Arquitectura, Universidad de Sevilla, Avda. Reina Mercedes No. 2, 41012 Sevilla, España.
José Garres Díaz Departamento de Matem´atica Aplicada II & IMUS, ETS Ingenier´ıa, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, España.
Jorge Moya Departamento de Matem´atica Aplicada I & IMUS, ETS Arquitectura, Universidad de Sevilla, Avda. Reina Mercedes No. 2, 41012 Sevilla, España.

DOI:

https://doi.org/10.29393/ppudec-20bpbr40020

Keywords:

Finite volume method, depth-averaged model, well-balanced methods, sediment transport

Resumen

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.

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