A robust and well-balanced scheme for the 2D Saint-Venant system on unstructured meshes with friction source term

• Published in: International journal for numerical methods in fluids Vol. 78; no. 2; pp. 89 - 121
• Main Author: Duran, A.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 20-05-2015; Wiley Subscription Services, Inc; Wiley
• Subjects: Analysis of PDEs; Computer simulation; Drying; Finite element method; finite volume; Fluid mechanics; Friction; Mathematical models; Mathematics; Mechanics; nonlinear shallow-water equations; Numerical Analysis; Physics; Preserves; Robustness; Topography; Two dimensional; unstructured mesh; well-balanced schemes
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4011

SummaryIn the following lines, we propose a numerical scheme for the shallow‐water system supplemented by topography and friction source terms, in a 2D unstructured context. This work proposes an improved version of the well‐balanced and robust numerical model recently introduced by Duran et al. (J. Comp. Phys., 235, 565–586, 2013) for the pre‐balanced shallow‐water equations, accounting for varying topography. The present work aims at relaxing the robustness condition and includes a friction term. To this purpose, the scheme is modified using a recent method, entirely based on a modified Riemann solver. This approach preserves the robustness and well‐balanced properties of the original scheme and prevents unstable computations in the presence of low water depths. A series of numerical experiments are devoted to highlighting the performances of the resulting scheme. Simulations involving dry areas, complex geometry and topography are proposed to validate the stability of the numerical model in the neighbourhood of wet/dry transitions. Copyright © 2015 John Wiley & Sons, Ltd.
The present work concerns the extension of a 1D friction approach in the context of 2D unstructured meshes. The method is positive preserving, stratifies the C‐property and allows the cure of the problem of vanishing water heights. Accurate and stable results are reached, notably for flooding and drying events in the presence of complex geometry, varying topography and nonlinear friction terms.