Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916133923192832 |
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| author | Kivva, Sergii |
| author_facet | Kivva, Sergii |
| contents | First-order hydrostatic reconstruction (HR) schemes for shallow water equations are highly diffusive whereas high-order schemes can produce entropy-violating solutions. Our goal is to develop a flux correction with maximum antidiffusive fluxes to obtain entropy solutions of shallow water equations with variable bottom topography. For this purpose, we consider a hybrid explicit HR scheme whose flux is a convex combination of first-order Rusanov flux and high-order flux. The conditions under which the explicit first-order HR scheme for shallow water equations satisfies the fully discrete entropy inequality have been studied. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the first-order HR scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution to the shallow water equations. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. Numerical results of testing various HR schemes on different benchmarks are presented. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_17774 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows Kivva, Sergii Numerical Analysis 65M08, MSC 65M22, 76M12 First-order hydrostatic reconstruction (HR) schemes for shallow water equations are highly diffusive whereas high-order schemes can produce entropy-violating solutions. Our goal is to develop a flux correction with maximum antidiffusive fluxes to obtain entropy solutions of shallow water equations with variable bottom topography. For this purpose, we consider a hybrid explicit HR scheme whose flux is a convex combination of first-order Rusanov flux and high-order flux. The conditions under which the explicit first-order HR scheme for shallow water equations satisfies the fully discrete entropy inequality have been studied. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the first-order HR scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution to the shallow water equations. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. Numerical results of testing various HR schemes on different benchmarks are presented. |
| title | Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows |
| topic | Numerical Analysis 65M08, MSC 65M22, 76M12 |
| url | https://arxiv.org/abs/2305.17774 |