A VFRoe scheme for 1D shallow water flows : wetting and drying simulation

TL;DR

This work targets robust 1D shallow-water simulations with topography, addressing wetting/drying by recasting the Saint-Venant equations into a celerity–speed form and developing a positivity-preserving approximate Roe solver. The finite-volume scheme uses a Godunov-type update with a Roe-based Riemann solver to compute interface fluxes while maintaining nonnegative intermediate celerity, enabling reliable handling of dry and submerged states. The methodology extends to topographic variations and bank transitions through entropy corrections and Davidé–LeVeque-inspired wetting/drying treatment, validated by submerged and wetting/drying test cases on a 25 m domain. Together, these contributions provide a practical, stable framework for urban flood modeling and potential extension to two dimensions.

Abstract

A finite-volume method for the one-dimensional shallow-water equations including topographic source terms is presented. Exploiting an original idea by Leroux, the system of partial-differential equations is completed by a trivial equation for the bathymetry. By applying a change of variable, the system is given a celerity-speed formulation, and linearized. As a result, an approximate Riemann solver preserving the positivity of the celerity can be constructed, permitting wetting and drying flow simulations to be performed. Finally, the simulation of numerical test cases is presented.

Figures (15)

• Figure 1: Chenal de Cotonou
• Figure 2:
• Figure 3: Ecoulement torrentiel
• Figure 4: Ecoulement fluvial
• Figure 5: bancs-couvrants-découvrants

Theorems & Definitions (5)

• Proposition 4.1
• Proposition 4.2
• Proposition 4.3
• Proposition 4.4
• Proposition 4.5