Institut de recherche mathématique avancée

Résumé : In particle methods, also known as Lagrangian methods, the discretization of the simulation domain is realized on a set of numerical particles that follow the dynamics of the system and are displaced with respect to the flow velocity. Vortex methods belong to particle methods and allow to discretize in a Lagrangian way the vorticity-velocity formulation of the Navier-Stokes equations, for the simulation of incompressible flows. This work is focused on a semi-Lagrangian variant of the Vortex Methods, the so-called « Remeshed Vortex Methods » (RVM). In this approach, the Lagrangian particles are regularly re-located on a Cartesian grid (through a « particle-remeshing » procedure) in order to avoid one of the major drawback of purely mesh-free Lagrangian schemes, namely the distorsion of particle distribution, leading to convergence loss. These methods constitute an extension of the « Particle-In-Cell » methods and can be seen as a hybrid Lagrangian-Eulerian approach; they take advantage from both the low numerical dissipation brought by the Lagrangian treatment of the flow advection and the efficiency of Eulerian schemes to handle the Poisson equation of the problem, the viscous term or the external forces. The present talk will present the numerical characteristics of the RVM, through a comparison of Direct Numerical Simulations (DNS) obtained with other numerical methods (like a Lattice Boltzmann method (LBM), i.e. an Eulerian method based on a mesoscopic flow description). We will then highlight the interest of RVM to afford Large Eddy Simulations (LES) and present the sub-grid scale models suited to RVM for the LES of homogeneous isototropic turbulents flows and of turbulent flows over solid walls.

Résumé : La majoration des sommes de Kloosterman obtenue par Deligne est un résultat crucial, exploité notamment en théorie analytique des nombres. La borne qu'il obtient est de l'ordre de la racine carrée des bornes "naïves", ce qui est comparable aux gains obtenus en admettant l'hypothèse de Riemann. L'objectif de cet exposé sera d'expliquer comment ces sommes d'exponentielles s'expriment dans un langage géométrique/algébrique, puis d'expliquer les avantages d'une telle traduction.

Résumé : Toric varieties are a special kind of algebraic varieties whose geometry is governed by combinatorial data: cones, fans and lattice polytopes. This makes them both explicitly computable and surprisingly rich. In this talk, I will give an introduction to toric varieties, emphasizing the dictionary between geometry and combinatorics and illustrating the theory through concrete examples. If time allows, I will also discuss their broad applications.

Résumé : TBA