Volume 55, Number 5, September-October 2021
|Page(s)||2185 - 2210|
|Published online||13 October 2021|
A central-upwind scheme for two-layer shallow-water flows with friction and entrainment along channels
Institute of Mathematics, National University of Mexico, Blvd. Juriquilla, 3001 Queretaro, Mexico
2 Department of Mathematics, California State University Northridge, 18111 Nordhoff St, Northridge, CA 91330-8313, USA
* Corresponding author: firstname.lastname@example.org
Accepted: 24 August 2021
We present a new high-resolution, non-oscillatory semi-discrete central-upwind scheme for one-dimensional two-layer shallow-water flows with friction and entrainment along channels with arbitrary cross sections and bottom topography. These flows are described by a conditionally hyperbolic balance law with non-conservative products. A detailed description of the properties of the model is provided, including entropy inequalities and asymptotic approximations of the eigenvalues of the corresponding coefficient matrix. The scheme extends existing central-upwind semi-discrete numerical methods for hyperbolic conservation and balance laws and it satisfies two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water depth for each layer, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. Along with the description of the scheme and proofs of these two properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.
Mathematics Subject Classification: 76M12 / 35L65
Key words: Hyperbolic systems of conservation and balance laws / semi-discrete schemes / Saint-Venant system of shallow-water equations with friction / non-oscillatory reconstructions / channels with irregular geometry
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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