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

Positivity-Preserving and Well-Balanced Adaptive Surface Reconstruction Schemes for Shallow Water... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Scientific Computing Springer Journals

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

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References (34)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
0885-7474
eISSN
1573-7691
DOI
10.1007/s10915-022-01943-3
Publisher site
See Article on Publisher Site

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.

Journal

Journal of Scientific Computing – Springer Journals

Published: Sep 1, 2022

Keywords: Well-balanced; Positivity-preserving; Irregular quadrangles; Surface reconstructions; Adaptive moving mesh; Two-dimensional shallow water equations; 76M12; 35L65; 65M08

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