A semi-implicit two dimensional solver for a covariant formulation of the shallow water equations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917530989232128 |
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| author | Tavelli, Maurizio Zanotti, Olindo |
| author_facet | Tavelli, Maurizio Zanotti, Olindo |
| contents | In this paper we combine a flexible covariant formulation of the shallow water equations with the semi-implicit numerical scheme developed over the years by Casulli and collaborators. After adopting an orthogonal, but non-orthonormal, coordinate basis on two dimensional manifolds, and by writing the divergence of symmetric tensors in a way that avoids the introduction of Christoffel symbols, the shallow water equations preserve a very close resemblance to the usual one expressed in Cartesian coordinates. In this way, a stable semi-implicit scheme can be derived by using an implicit discretization for the gradient of surface elevation in the momentum equations and for the velocity in the continuity equation, with stability properties that are independent of the celerity. We have tested the new method over a variety of challenging benchmarks, including, among the others, the smooth wave propagation over a water globe and the deformation of an artery branch. Two appealing additional features make the method particularly powerful with respect to oceanographic applications: firstly, thanks to the wetting and drying ability of our semi-implicit approach, no pathological behaviors occur at the poles; secondly, the scheme is naturally well-balanced, and it is able to preserve perfect stationarity, up to machined precision, of the entire ocean configuration of the earth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A semi-implicit two dimensional solver for a covariant formulation of the shallow water equations Tavelli, Maurizio Zanotti, Olindo Fluid Dynamics Numerical Analysis In this paper we combine a flexible covariant formulation of the shallow water equations with the semi-implicit numerical scheme developed over the years by Casulli and collaborators. After adopting an orthogonal, but non-orthonormal, coordinate basis on two dimensional manifolds, and by writing the divergence of symmetric tensors in a way that avoids the introduction of Christoffel symbols, the shallow water equations preserve a very close resemblance to the usual one expressed in Cartesian coordinates. In this way, a stable semi-implicit scheme can be derived by using an implicit discretization for the gradient of surface elevation in the momentum equations and for the velocity in the continuity equation, with stability properties that are independent of the celerity. We have tested the new method over a variety of challenging benchmarks, including, among the others, the smooth wave propagation over a water globe and the deformation of an artery branch. Two appealing additional features make the method particularly powerful with respect to oceanographic applications: firstly, thanks to the wetting and drying ability of our semi-implicit approach, no pathological behaviors occur at the poles; secondly, the scheme is naturally well-balanced, and it is able to preserve perfect stationarity, up to machined precision, of the entire ocean configuration of the earth. |
| title | A semi-implicit two dimensional solver for a covariant formulation of the shallow water equations |
| topic | Fluid Dynamics Numerical Analysis |
| url | https://arxiv.org/abs/2605.25544 |