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Volume 18, Issue 1
The Generalized Riemann Problem for Two-Layer Shallow Water Equations with Two-Velocities

Shuzhi Liu & Qinglong Zhang

DOI: 10.4208/aamm.OA-2024-0001

Adv. Appl. Math. Mech., 18 (2026), pp. 242-266.

Published online: 2025-10

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  • Abstract

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.

  • Keywords

Generalized Riemann problem (GRP), two-layer shallow water equations, layer-wise discretization, Riemann invariants, Rankine-Hugoniot condition.

  • AMS Subject Headings

35L40, 76L05, 76M12, 76B70, 65M08

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-18-242, author = {Liu , Shuzhi and Zhang , Qinglong}, title = {The Generalized Riemann Problem for Two-Layer Shallow Water Equations with Two-Velocities}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2025}, volume = {18}, number = {1}, pages = {242--266}, abstract = {

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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2024-0001}, url = {http://global-sci.org/intro/article_detail/aamm/24526.html} }
TY - JOUR T1 - The Generalized Riemann Problem for Two-Layer Shallow Water Equations with Two-Velocities AU - Liu , Shuzhi AU - Zhang , Qinglong JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 242 EP - 266 PY - 2025 DA - 2025/10 SN - 18 DO - http://doi.org/10.4208/aamm.OA-2024-0001 UR - https://global-sci.org/intro/article_detail/aamm/24526.html KW - Generalized Riemann problem (GRP), two-layer shallow water equations, layer-wise discretization, Riemann invariants, Rankine-Hugoniot condition. AB -

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.

Liu , Shuzhi and Zhang , Qinglong. (2025). The Generalized Riemann Problem for Two-Layer Shallow Water Equations with Two-Velocities. Advances in Applied Mathematics and Mechanics. 18 (1). 242-266. doi:10.4208/aamm.OA-2024-0001
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