| Issue |
ESAIM: M2AN
Volume 57, Number 2, March-April 2023
|
|
|---|---|---|
| Page(s) | 1087 - 1110 | |
| DOI | https://doi.org/10.1051/m2an/2023009 | |
| Published online | 12 April 2023 | |
A flux globalization based well-balanced path-conservative central-upwind scheme for the shallow water flows in channels
1
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
2
Department of Mathematics, SUSTech International Center for Mathematics and Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen 518055, P.R. China
3
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, P.R. China
* Corresponding author: 12131231@mail.sustech.edu.cn
Received:
7
May
2022
Accepted:
24
January
2023
We develop a flux globalization based well-balanced (WB) path-conservative central-upwind (PCCU) scheme for the one-dimensional shallow water flows in channels. Challenges in developing numerical methods for the studied system are mainly related to the presence of nonconservative terms modeling the flow when the channel width and bottom topography are discontinuous. We use the path-conservative technique to treat these nonconservative product terms and implement this technique within the flux globalization framework, for which the friction and aforementioned nonconservative terms are incorporated into the global flux: This results in a quasi-conservative system, which is numerically solved using the Riemann-problem-solver-free central-upwind scheme. The WB property of the resulting scheme (that is, its ability to exactly preserve both still- and moving-water equilibria at the discrete level) is ensured by performing piecewise linear reconstruction for the equilibrium variables rather than the conservative variables, and then evaluating the global flux using the obtained point values of the equilibrium quantities. The robustness and excellent performance of the proposed flux globalization based WB PCCU scheme are demonstrated in several numerical examples with both continuous and discontinuous channel width and bottom topography. In these examples, we clearly demonstrate the advantage of the proposed scheme over its simpler counterparts.
Mathematics Subject Classification: 76M12 / 65M08 / 86-08 / 35L65 / 35L67
Key words: Shallow water flows in channels / steady-state solutions / well-balanced methods / semi-discrete central-upwind schemes / flux globalization / path-conservative central-upwind schemes
© The authors. Published by EDP Sciences, SMAI 2023
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