| Issue |
ESAIM: M2AN
Volume 59, Number 4, July-August 2025
|
|
|---|---|---|
| Page(s) | 2253 - 2277 | |
| DOI | https://doi.org/10.1051/m2an/2025053 | |
| Published online | 31 July 2025 | |
Flux gobalization based well-balanced path-conservative central-upwind schemes for the shallow water flows in channels with arbitrary cross-section
1
Department of Mathematics and Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, P.R. China
2
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, P.R.China
* Corresponding author: 12131231@mail.sustech.edu.cn
Received:
17
May
2024
Accepted:
18
June
2025
A second-order flux globalization based well-balanced (WB) path-conservative central-upwind (PCCU) scheme for the shallow water flows in rectangular cross-section channels has been recently proposed in Chen et al. [ESAIM Math. Model. Numer. Anal. 57 (2023) 1087–1110]. Here, we extend this method to a more general case of arbitrary cross-section channels and to a higher (fifth) order of accuracy. One of the challenges in extending the numerical methods for the arbitrary cross-section channel system is related to converting the equilibrium variables (need to evaluate the global fluxes in a well-balanced manner) to the conservative ones. Unlike the case of rectangular cross-section channels, this conversion process requires the use of carefully designed nonlinear equation solver. Challenges in extending the scheme to higher-order are mainly related to the development of higher-order WB and non-oscillatory reconstructions (interpolations). The introduced WB schemes are capable of exactly preserving a wide variety of steady-state solutions including both the still- and moving-water ones. The proposed schemes are tested on a number of numerical examples with both continuous and discontinuous channel breadth.
Mathematics Subject Classification: 76M12 / 65M08 / 86-08 / 35L65 / 35L67
Key words: Shallow water flows in arbitrary cross-section channels / steady-state solutions / well-balanced methods / flux globalization / path-conservative central-upwind schemes
© The authors. Published by EDP Sciences, SMAI 2025
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