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Séminaire Equations aux dérivées partielles

organisé par l'équipe Modélisation et contrôle

  • Killian Vuillemot

    A new unfitted finite element method: $\varphi$-FEM

    23 septembre 2025 - 14:00Salle de conférences IRMA

    $\varphi$-FEM is a new finite element method, proposed to solve partial differential equations on complex domains, using simple non-conforming meshes. The method relies on the use of a level-set function $\varphi$, which defines the domain and its boundary. In this presentation, I will introduce the method in the simple case of the resolution of the Poisson equation with Dirichlet boundary conditions. Then I will present the extension to the case of mixed Dirichlet/Neumann boundary conditions. I will also present results for the resolution of the Heat equation with Dirichlet boundary conditions or linear and non-linear elasticity problems. I will finally present different evolutions of the method, including its combination with Neural Operators or the use of the finite difference method. I will also discuss perspectives and future challenges for $\varphi$-FEM.

  • Jordan Berthoumieu

    Stability of soliton for nonlinear non-integrable Schrödinger equations with non-trivial far field

    30 septembre 2025 - 14:00Salle de conférences IRMA

    Many innovative technologies in areas such as optical fibers, superfluids and Bose-Einstein condensates are based on quantum physical models described by nonlinear Schrödinger equations. Our main focus is put on the nonlinear Schrödinger equation with non-zero limit at infinity that is a generalization of the well-known Gross-Pitaevskii equation (GP) with a non-vanishing condition at infinity. The presentation will relate to defocusing nonlinearities for which we are specially interested and on the properties of travelling waves. Many different behaviors for these travelling waves have been highlighted, according to the shape of the nonlinearity. Nevertheless, we have been able to prove the existence of travelling waves with small momentum. Moreover, we shall dwell on the existence and uniqueness travelling waves with speed close to the speed of sound, the orbital stability of a well-prepared chain of such travelling waves, as well as the asymptotic stability of these special solutions.

  • Nikita Afanasev

    Conservative-characteristic Schemes: Recent Developments in Active Flux and CABARET Methods

    7 octobre 2025 - 14:00Salle de conférences IRMA

    In recent years, conservative-characteristic methods have been extensively used to numerically solve different hyperbolic PDEs. These methods use a combinaton of a finite volume method to approximate the cell-averages in mesh cells using conservative form of equations, and an arbitrary method to approximate the point values in edges/faces of the mesh using the non-conservative form of equations. Therefore, the overall method remains conservative (for the averages), retaining a lot of flexibility in how to deal with the point values. In this talk, we will describe 2 of such methods: CABARET and Active Flux, including some applications for both schemes.

    CABARET, first introduced by V. Goloviznin and later refined by S. Karabasov [1], is a second-order explicit conservative-characteristic method. Its special feature is the extrapolation of Riemann invariants along the linearized characteristics to evolve the point values. We will discuss this method in detail and introduce various applications for problems in oceanology [2], fluid-structure interaction [3], transonic flows [4] and thermoacoustic instability [5].

    Active Flux method, first introduced by T. Eymann and P. Roe [6], has been adapted to solve many problems for hyperbolic systems of PDEs on orthogonal and polygonal meshes. There are many versions of this method, and we will concentrate on the work of R. Abgrall and his group. This version of Active Flux is a third-order scheme [7], which works on general polygonal meshes (for 2D) and uses the method of lines to approximate the point values on edges and nodes of the mesh. We will describe the base algorithm for two-dimensional problems on a plane, and also we introduce the generalization of Active Flux method on triangular meshes to hyperbolic problems on a sphere [8].

    References:

    [1] S. Karabasov and V. Goloviznin “Compact Accurately Boundary-Adjusting High-REsolution Technique for Fluid Dynamics”, Journal of Computational Physics, 228(19), pp. 7426–7451, 2009.

    [2] V.M. Goloviznin, P.A. Maiorov, P.A. Maiorov and A.V. Solovjev “Validation of the Low Dissipation Computational Algorithm CABARET-MFSH for Multilayer Hydrostatic Flows with a Free Surface on the Lock-release Experiments”, Journal of Computational Physics, 463, p. 111239, 2023.

    [3] N. Afanasiev, V. Goloviznin, P. Maiorov and A. Solovjev “Simulating the dynamics of a fluid with a free surface in a gravitational field by a CABARET method”, Mathematical notes of NEFU, 29(4), pp. 77–94, 2022.

    [4] N. Afanasiev and V. Goloviznin, “A Locally Implicit Time-Reversible Sonic Point Processing Algorithm for One-Dimensional Shallow-Water Equations”, Journal of Computational Physics, 434, p. 110220, 2021.

    [5] N. A. Afanasiev, V. M. Goloviznin, V. N. Semenov et al. “Direct simulation of thermoacoustic instability in gas generators using the cabaret scheme”, Mathematical Models and Computer Simulations, 13(5), pp. 820–830, 2021.

    [6] T.A. Eymann and P.L. Roe. “Active flux schemes”, AIAA, 382(19), 2011.

    [7] R. Abgrall, J. Lin and Y. Liu “Active flux for triangular meshes for compressible flows problems”, Beijing Journal of Pure & Applied Mathematics, 2(1), pp. 1–33, 2025.

    [8] N. Afanasev and R.Abgrall “Active Flux Method on a Sphere”, Submitted, 2025.

  • Nilo Schwencke

    TBA

    4 novembre 2025 - 14:00A confirmer

  • León Avila León

    TBA

    18 novembre 2025 - 14:00A confirmer

  • Boris Gnamah

    Problème inverse de sources dans deux EDPs paraboliques couplées de type advection-dispersion-réaction

    2 décembre 2025 - 14:00Salle de conférences IRMA

    On cherche à résoudre un problème inverse non linéaire de source dans un système de deux équations aux dérivées partielles paraboliques 2D couplées d'advection-dispersion-réaction. Dans ce système, nous abordons l'identification de plusieurs sources inconnues, mélangées et distribuées, définissant le membre de droite de sa première équation en utilisant certaines observations locales liées à l'état de la solution de sa deuxième équation couplée. Nous développons des fonctions adjointes appropriées permettant d'établir des écarts de réciprocité remplis par les éléments inconnus définissant les sources recherchées. Ces fonctions adjointes sont définies par des potentiels scalaires dérivés de champs colinéaires aux directions orthogonales indiquées par les vecteurs propres du tenseur de dispersion symétrique. À partir de certaines interfaces de mesure mises en place dans le domaine surveillé, nous établissons un résultat qui permet de faire la détection et l'identification de la source.

  • Florian De Vuyst

    TBA

    20 janvier 2026 - 14:00Salle de conférences IRMA