The Generalized Riemann Problem for Two-Layer Shallow Water Equations with Two-Velocities (Advances in Applied Mathematics and Mechanics, 2025)
This paper proposes a direct Eulerian generalized Riemann problem (GRP) scheme for two-layer shallow water equations. The model takes into account the distinctions between different densities and velocities, and is obtained by taking the vertical averaging across the layer depth. The source terms generated from the mass and momentum exchange prevent us from solving the Riemann problem analytically. We consider an equivalent conservative two-layer model which describes the horizontal velocity with two degrees of freedom. The rarefaction wave and the shock wave are analytically resolved by using the Riemann invariants and Rankine-Hugoniot condition, respectively. Numerical simulations are also given on some typical problems in order to verify the good performance of the GRP method.