Hyperbolicity of a semi-Lagrangian formulation of the hydrostatic free-surface Euler system
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2023
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| _version_ | 1866929642849435648 |
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| author | Di Martino, Bernard Hassanieh, Chourouk El Godlewski, Edwige Guillod, Julien Sainte-Marie, Jacques |
| author_facet | Di Martino, Bernard Hassanieh, Chourouk El Godlewski, Edwige Guillod, Julien Sainte-Marie, Jacques |
| contents | By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. The system one obtains can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multilayer $\mathbb{P}_0$-discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15083 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hyperbolicity of a semi-Lagrangian formulation of the hydrostatic free-surface Euler system Di Martino, Bernard Hassanieh, Chourouk El Godlewski, Edwige Guillod, Julien Sainte-Marie, Jacques Analysis of PDEs Numerical Analysis 35Q31, 35F50, 35L02, 45K05, 35P05, 35P15, 76B99 By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. The system one obtains can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multilayer $\mathbb{P}_0$-discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system. |
| title | Hyperbolicity of a semi-Lagrangian formulation of the hydrostatic free-surface Euler system |
| topic | Analysis of PDEs Numerical Analysis 35Q31, 35F50, 35L02, 45K05, 35P05, 35P15, 76B99 |
| url | https://arxiv.org/abs/2308.15083 |