Thèse soutenue

Analyse mathématique de schémas volume finis pour la simulation des écoulements quasi-géostrophiques à bas nombre de Froude

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Auteur / Autrice : Minh Hieu Do
Direction : Pascal Omnes
Type : Thèse de doctorat
Discipline(s) : Mathématiques
Date : Soutenance le 19/12/2017
Etablissement(s) : Sorbonne Paris Cité
Ecole(s) doctorale(s) : École doctorale Galilée (Villetaneuse, Seine-Saint-Denis)
Partenaire(s) de recherche : Laboratoire : Laboratoire Analyse, géométrie et applications (LAGA) (Villetaneuse, Seine-Saint-Denis)
Etablissement de préparation : Université Sorbonne Paris Nord (Bobigny, Villetaneuse, Seine-Saint-Denis ; 1970-....)
Jury : Président / Présidente : Fayssal Benkhaldoun
Examinateurs / Examinatrices : Emmanuel Audusse, Yohan Penel, Vladimir Zeitlin, Carine Lucas
Rapporteur / Rapporteuse : François Bouchut, Christophe Berthon

Résumé

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The shallow water system plays an important role in the numerical simulation of oceanic models, coastal flows and dam-break floods. Several kinds of source terms can be taken into account in this model, such as the influence of bottom topography, Manning friction effects and Coriolis force. For large scale oceanic phenomena, the Coriolis force due to the Earth’s rotation plays a central role since the atmospheric or oceanic circulations are frequently observed around the so-called geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis source term. The ability of numerical schemes to well capture the lake at rest, has been widely studied. However, the geostrophic equilibrium issue, including the divergence free constraint on the velocity, is much more complex and only few works have been devoted to its preservation. In this manuscript, we design finite volume schemes that preserve the discrete geostrophic equilibriuminordertoimprovesignificantlytheaccuracyofnumericalsimulationsofperturbations around this equilibrium. We first develop collocated and staggered schemes on rectangular and triangular meshes for a linearized model of the original shallow water system. The crucial common point of the various methods is to adapt and combine several strategies known as the Apparent Topography, the Low Mach and the Divergence Penalisation methods, in order to handle correctly the numerical diffusions involved in the schemes on different cell geometries, so that they do not destroy geostrophic equilibria. Finally, we extend these strategies to the non-linear case and show convincing numerical results.