Raviart-Thomas and Brezzi-Douglas-Marini finite element approximations of the shallow-water equations.

*(English)*Zbl 1140.76022Summary: We present an analysis of discrete shallow-water equations using Raviart-Thomas and Brezzi-Douglas-Marini finite elements. For inertia-gravity waves, the discrete formulations are obtained and the dispersion relations are computed in order to quantify the dispersive nature of the schemes on two meshes made up of equilateral and biased triangles. A linear algebra approach is also used to ascertain the possible presence of spurious modes arising from the discretization. The geostrophic balance is examined, and the smallest representable vortices are characterized on both structured and unstructured meshes. Numerical solutions of two test problems to simulate gravity and Rossby modes are in good agreement with analytical results.

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76B65 | Rossby waves (MSC2010) |

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\textit{V. Rostand} and \textit{D. Y. Le Roux}, Int. J. Numer. Methods Fluids 57, No. 8, 951--976 (2008; Zbl 1140.76022)

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