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An Oscillation-free Discontinuous Galerkin Method for Shallow Water Equations

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Abstract

In this paper, we develop an oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. Due to the nonlinear hyperbolic nature of the shallow water equation, the exact solution may contain shock discontinuities; thus, the numerical solution may generate spurious oscillations if no special treatment is taken near the discontinuities. Besides, it is desirable to construct a numerical scheme that possesses the well-balanced property, which preserves exactly the hydrostatic equilibrium solutions up to the machine error. Based on the existing well-balanced DG schemes proposed in Xing (J Comput Phys 257:536–553, 2014), Xing and Shu (Commun Comput Phys 1: 100–134, 2006), we add an extra damping term into these schemes to control the spurious oscillations. One of the advantages of the damping term in suppressing the oscillations is that the damping technique seems to be more convenient for the theoretical analysis, at least in the semi-discrete analysis. With the careful construction of the damping term, the proposed method not only achieves non-oscillatory property without compromising any order of accuracy, but also inherits the the well-balanced property from original schemes. We also present a detailed procedure for constructing the damping term and a theoretical analysis for the preservation of the well-balanced property. Extensive numerical experiments, including one- and two-dimensional space, demonstrate that the proposed method has desired properties without sacrificing any order of accuracy.

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Funding

Y. Liu: Research is partially supported by the fellowship of China Postdoctoral Science Foundation No. 2020TQ0343, and NSFC grant No. 11688101. J. Lu: Research is partially supported by NSFC grant No. 11901213. Q. Tao: Research is partially supported by NSFC grant No. U1930402 and the fellowship of China Postdoctoral Science Foundation No. 2020TQ0030, 2021M700357. Y. Xia: Research is partially supported by NSFC grant No. 11871449.

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics, Hua Loo-Keng Center for Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Yong Liu

  2. School of Mathematics, South China University of Technology, Canton, 510641, Guangdong, China

    Jianfang Lu

  3. Beijing Computational Science Research Center, Beijing, 100193, China

    Qi Tao

  4. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 233026, Anhui, People’s Republic of China

    Yinhua Xia

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  1. Yong Liu
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Correspondence to Qi Tao.

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Liu, Y., Lu, J., Tao, Q. et al. An Oscillation-free Discontinuous Galerkin Method for Shallow Water Equations. J Sci Comput 92, 109 (2022). https://doi.org/10.1007/s10915-022-01893-w

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