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A new well-balanced spectral volume method for solving shallow water equations over variable bed topography with wetting and drying

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Abstract

The shallow-water equations are a hyperbolic conservation law system with source terms, which can be used in various engineering applications. Designing a high-order numerical method to preserve exactly steady-state solutions is a challenging task. Another difficulty is the appearance of dry regions in the computational domain, where no water or very little water is present. Special attention needs to be paid; otherwise, numerical methods may fail in these regions creating unphysical negative water depths. In this paper, a new high-order well-balanced Chebyshev spectral volume with a new hydrostatic reconstruction (HR) scheme is presented to preserve the steady-state solutions, and at the same time, deal with wetting and drying without loss of mass conservation. In addition, the shallow water equations may have some discontinuous solutions, even for smooth initial conditions. We modify the C-WENO limiter to reconstruct the numerical approximation on target cells that have numerical oscillations. One of the significant advantages of the modified C-WENO limiter compared to other limiters is that it only depends on the numerical approximation of the target cell and immediate neighbors. With the modified C-WENO limiter, we can achieve a high order of accuracy and non-oscillatory properties and maintain the proposed method’s well-balanced and positivity-preserving properties. To restrict the time step to the Courant–Friedrichs–Lewy condition and ensure stability and accurate results, we introduce a semi-implicit discretization of the friction source term, which does not need an iteration method. Various numerical tests are presented to evaluate the proposed method’s performance in terms of high-order accuracy, well-balanced, positivity-preserving, non-oscillatory, and mass conservation properties.

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Authors and Affiliations

  1. Faculty of Civil, Water and Environmental Engineering, Shahid Beheshti University, Tehran, Iran

    Mohsen Hadadian Nejad Yousefi & Seyed Hossein Ghoreishi Najafabadi

  2. Department of Mathematics, Kosar University of Bojnord, PO Box 9415615458, Bojnord, Iran

    Emran Tohidi

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  1. Mohsen Hadadian Nejad Yousefi
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Hadadian Nejad Yousefi, M., Ghoreishi Najafabadi, S.H. & Tohidi, E. A new well-balanced spectral volume method for solving shallow water equations over variable bed topography with wetting and drying. Engineering with Computers 39, 3099–3130 (2023). https://doi.org/10.1007/s00366-022-01704-8

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