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Moving Water Equilibria Preserving Discontinuous Galerkin Method for the Shallow Water Equations

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Abstract

In this paper, a new well-balanced discontinuous Galerkin (DG) method is proposed for the shallow water equations in the sense of moving-water equilibria preservation. The significant ingredient for the scheme is the approximation of the equilibrium variables in the DG piecewise polynomial space, rather than the conservative variables in most existing well-balanced schemes. Thus, the conservative variables are non-polynomial functions of the equilibrium variables. This approach is easily accomplished without the reference equilibrium state recovery and a special source term approximation. To obtain the moving water equilibrium preservation of the scheme, the strategy pursued is to modify the numerical fluxes based on the generalized hydrostatic reconstruction. A rigorous theoretical analysis proves the exact equilibrium preservation of the scheme. Numerical tests demonstrate that this DG method is high-order accurate, successfully maintains the still water and the generalized moving water equilibria states, and captures small perturbations of the hydrostatic state without numerical oscillations near shocks even on the discontinuous bottom topography.

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Data Availability

The datasets generated during the current study are available from the corresponding author on reasonable request. They support our published claims and comply with field standards.

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Funding

Research of Yinhua Xia was supported by NSFC grant No. 12271498, and National Key R &D Program of China No. 2022YFA1005202/2022YFA1005200. Research of Yan Xu was supported by NSFC Grant No. 12071455.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

    Jiahui Zhang, Yinhua Xia & Yan Xu

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  1. Jiahui Zhang
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  2. Yinhua Xia
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  3. Yan Xu
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Contributions

All authors contributed to the study conception and design. Algorithm implementation and data collection were performed by Jiahui Zhang. The first draft of the manuscript was written by Jiahui Zhang and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Yinhua Xia.

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Zhang, J., Xia, Y. & Xu, Y. Moving Water Equilibria Preserving Discontinuous Galerkin Method for the Shallow Water Equations. J Sci Comput 95, 48 (2023). https://doi.org/10.1007/s10915-023-02174-w

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