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Positivity-Preserving and Well-Balanced Adaptive Surface Reconstruction Schemes for Shallow Water Equations with Wet-Dry Fronts

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Abstract

We aim to propose a robust and efficient surface reconstruction (SR) scheme for two-dimensional shallow water equations with wet-dry fronts together with adaptive moving mesh methods on irregular quadrangles. The key ingredient of the surface reconstruction is to define Riemann states based on smoothing the water surface or the bottom topography on the cell boundary. The main difficulties in using adaptive moving mesh methods for shallow water equations are to guarantee the positivity of the water depth and the stationary solution near wet-dry fronts. We use a geometrical conservative method to recover the numerical solutions from the mesh of the previous time level and prove positivity-preserving and well-balanced properties. It is a challenging work to preserve stationary solutions for the adaptive moving mesh method when the computational domain contains wet-dry fronts. To overcome this issue, we propose three steps, which consist of redefining the bottom topography on the new meshes, fixing the mesh vertex of the partially flooded cells, and avoiding the extrema of the solutions on the new meshes. The current adaptive SR schemes can maintain the still-water steady state even if the computational domain contains wet-dry fronts and guarantee the water depth to be nonnegative. We illustrate the performance of the current adaptive SR scheme using several classic experiments of two-dimensional shallow water equations with wet-dry fronts.

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (No. 11971481,11901577,12071481).

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Authors and Affiliations

  1. Department of Mathematics, College of Liberal Arts and Science, National University of Defense Technology, Changsha, Hunan, 410073, China

    Xu Qian, Jian Dong & Songhe Song

  2. State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, 410073, China

    Songhe Song

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  1. Xu Qian
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  2. Jian Dong
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  3. Songhe Song
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Correspondence to Jian Dong.

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Qian, X., Dong, J. & Song, S. Positivity-Preserving and Well-Balanced Adaptive Surface Reconstruction Schemes for Shallow Water Equations with Wet-Dry Fronts. J Sci Comput 92, 111 (2022). https://doi.org/10.1007/s10915-022-01943-3

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  • DOI: https://doi.org/10.1007/s10915-022-01943-3

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