The Riemann problem for two-layer shallow water equations with bottom topography

Abstract

In this article, we propose a model describing laminar shallow water flows with two velocities to handle an uneven bottom. The model is established by taking the different velocities into account. The source terms generated from the bottom topography prevent us from solving the Riemann problem directly. We first derive the elementary waves including the stationary wave where the global entropy condition is used to ensure uniqueness. Then, we analyze the resonance phenomenon and coalescence of waves by classifying the initial data into different regions. Finally, the Riemann problem is resolved explicitly on a case-by-case basis.

Keywords

• the riemann problem
• laminar shallow water equations
• riemann invariants
• the stationary wave
• rankine-hugoniot condition
• primary: 35l40
• 76l05
• 76m12
• 35l65
• secondary: 35f55