Bienvenue sur la page web de François DUBOIS !
                               

Enseignant-chercheur associé au Laboratoire de Mathématiques d'Orsay, équipe "Analyse Numérique et Equations aux Dérivées Partielles".
Professeur des Universités (mathématiques appliquées) au Conservatoire National des Arts et Métiers à Paris et membre du laboratoire de mécanique des structures et des systèmes couplés.
Membre de la Société Mathématique de France.


La vidéo de l'exposé du 9 avril "An asymptotic expansion approach to the lattice Boltzmann method with applications to compressible Navier Stokes equations" au Seminar: Hyperbolic equations ; structure preserving numerical methods and other topics, coordonné par Christian Klingenberg (Würzburg Universität), est disponible en ligne.

Groupe de travail   "Schémas de Boltzmann sur réseau"
Prochaine séance : mercredi 5 mai 2021 à 15h  (9h du matin à Montréal)
                                                  (heure Française, GMT+1) dans une salle "big blue button"
Thomas Bellotti (Étudiant en thèse au Centre de Mathématiques Appliquées de l'École Polytechnique, Palaiseau)
"Adaptive multiresolution-based lattice Boltzmann schemes and their accuracy analysis via the equivalent equations"
Résumé
Multiresolution provides a fundamental tool based on the wavelet theory to build adaptive numerical schemes for Partial Differential Equations and time-adaptive meshes, allowing for precise error control [4]. This strategy is used to build adaptive lattice Boltzmann methods with this desirable feature [3, 2]. Furthermore, these schemes allow for an effective memory compression of the solution when spatially localized phenomena – such as shocks – are involved. Nevertheless, the peculiar way of modeling the desired physical phenomena in the lattice Boltzmann schemes calls, besides the possibility of controlling the error introduced by the mesh adaptation, for a deeper understanding of how this new scheme could alter the physics approximated by the numerical strategy. This issue is studied by writing the equivalent equations [5] of the adaptative method after having put the scheme under an adapted formalism [1]. It provides an essential tool to master the perturbations introduced by the adaptive numerical strategy, which can thus be devised in such a way as to preserve the desired features of the reference scheme. The theoretical considerations are corroborated by numerical experiments in both the 1D and 2D context, showing the relevance of the asymptotic analysis. In particular, we show that our numerical method outperforms traditional approaches, whether or not the solution of the reference scheme converges to the solution of the target equation. Furthermore, we discuss the influence of different collision strategies for non-linear problems, showing that they have only a marginal impact on the quality of the solution.
Références
[1] T. Bellotti, L. Gouarin, B. Graille, and M. Massot. Accuracy analysis of adaptive multiresolution- based lattice boltzmann schemes via the equivalent equations. In preparation. SMAI Journal of Computational Mathematics, 2021.
[2] T. Bellotti, L. Gouarin, B. Graille, and M. Massot. Multidimensional fully adaptive lattice boltzmann methods with error control based on multiresolution analysis. Journal of Computational Physics, 2021. Submitted, - available on HAL : https://hal.archives-ouvertes.fr/hal-03158073.
[3] T. Bellotti, L. Gouarin, B. Graille, and M. Massot. Multiresolution-based mesh adaptation and error control for lattice boltzmann methods with applications to hyperbolic conservation laws. SIAM J. Scientific Computing, 2021. Submitted - available on HAL : https://hal.archives-ouvertes.fr/ hal-03148621 and ArXiv : https://arxiv.org/abs/2102.12163.
[4] Albert Cohen, Sidi Kaber, Siegfried Müller, and Marie Postel. Fully adaptive multiresolution finite volume schemes for conservation laws. Mathematics of Computation, 72(241):183-225, 2003.
[5] François Dubois. Equivalent partial differential equations of a lattice boltzmann scheme. Computers & Mathematics with Applications, 55(7):1441-1449, 2008.

Groupe de travail "Modélisation quantique", ISC-PIF, 113 rue Nationale, 75013 Paris.
Travaux de recherche en modélisation mathématique et numérique et modélisation quantique.
La recherche scientifique est faite aussi de rencontres, colloques et séminaires.
L'enseignement conduit à rédiger des cours, parfois à les publier, proposer des exercices et mettre au point des travaux pratiques. Mais rien n'interdit de faire des maths pour le plaisir !
Quelques documents rares, même sur internet.
Cette page est en évolution dynamique. Merci de me faire part de vos remarques !
There is no english version of this sheet. Nevertheless, most of the papers are written in scientific english. Please enter !
mise à jour : 21 avril 2021