Analyse mathématique de schémas volume finis pour la simulation des écoulements quasi-géostrophiques à bas nombre de Froude
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 |
Mots clés
Résumé
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.