Entropy stable hydrostatic reconstruction schemes for shallow water systems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917699687284736 |
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| author | Ersing, Patrick Goldberg, Sven Winters, Andrew R. |
| author_facet | Ersing, Patrick Goldberg, Sven Winters, Andrew R. |
| contents | In this work, we develop a new hydrostatic reconstruction procedure to construct well-balanced schemes for one and multilayer shallow water flows, including wetting and drying. Initially, we derive the method for a path-conservative finite volume scheme and combine it with entropy conservative fluxes and suitable numerical dissipation to preserve an entropy inequality in the semi-discrete case. We then combine the novel hydrostatic reconstruction with a collocated nodal split-form discontinuous Galerkin spectral element method, extending the method to high-order and curvilinear meshes. The high-order method incorporates an additional positivity-limiter and is blended with a compatible subcell finite volume method to maintain well-balancedness at wet/dry fronts. We prove entropy stability, well-balancedness, and positivity-preservation for both methods. Numerical results for the high-order method validate the theoretical findings and demonstrate the robustness of the scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14119 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropy stable hydrostatic reconstruction schemes for shallow water systems Ersing, Patrick Goldberg, Sven Winters, Andrew R. Numerical Analysis 65M12, 65M20, 65M70, 76M22, 35L60, 86A05 In this work, we develop a new hydrostatic reconstruction procedure to construct well-balanced schemes for one and multilayer shallow water flows, including wetting and drying. Initially, we derive the method for a path-conservative finite volume scheme and combine it with entropy conservative fluxes and suitable numerical dissipation to preserve an entropy inequality in the semi-discrete case. We then combine the novel hydrostatic reconstruction with a collocated nodal split-form discontinuous Galerkin spectral element method, extending the method to high-order and curvilinear meshes. The high-order method incorporates an additional positivity-limiter and is blended with a compatible subcell finite volume method to maintain well-balancedness at wet/dry fronts. We prove entropy stability, well-balancedness, and positivity-preservation for both methods. Numerical results for the high-order method validate the theoretical findings and demonstrate the robustness of the scheme. |
| title | Entropy stable hydrostatic reconstruction schemes for shallow water systems |
| topic | Numerical Analysis 65M12, 65M20, 65M70, 76M22, 35L60, 86A05 |
| url | https://arxiv.org/abs/2406.14119 |