Unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations

• Published in: Journal of computational physics Vol. 231; no. 8; pp. 3317 - 3339
• Main Authors: Cea, L., Vázquez-Cendón, M.E.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.04.2012; Elsevier
• Subjects: Bed friction; Computational techniques; Depth averaged models; Exact sciences and technology; Finite volume method; Flux; Friction; High order schemes; Hydrodynamic equations; Mathematical analysis; Mathematical methods in physics; Mathematical models; Physics; Scalar transport; Scalars; Shallow water equations; Transport; Unstructured mesh; Shallow water equations; Bed friction; Unstructured mesh; High order schemes; Depth averaged models; Scalar transport; Finite volume method; Transport equation; Velocity fields; Finite volume methods; Calculation methods; Hydrodynamic model; Uniform flow; Calculation; Shallow-water equations; Gravity
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2012.01.007

The finite volume discretisation of the shallow water equations has been the subject of many previous studies, most of which deal with a well-balanced conservative discretisation of the convective flux and bathymetry. However, the bed friction discretisation has not been so profusely analysed in previous works, while it may play a leading role in certain applications of shallow water models. In this paper we analyse the numerical discretisation of the bed friction term in the two-dimensional shallow water equations, and we propose a new unstructured upwind finite volume discretisation for this term. The new discretisation proposed improves the accuracy of the model in problems in which the bed friction is a relevant force in the momentum equation, and it guarantees a perfect balance between gravity and bed friction under uniform flow conditions. The relation between the numerical scheme used to solve the hydrodynamic equations and the scheme used to solve a scalar transport model linked to the shallow water equations, is also analysed in the paper. It is shown that the scheme used in the scalar transport model must take into consideration the scheme used to solve the hydrodynamic equations. The most important implication is that a well-balanced and conservative scheme for the scalar transport equation cannot be formulated just from the water depth and velocity fields, but has to consider also the way in which the hydrodynamic equations have been solved.