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Séminaire Equations aux dérivées partielles

organisé par l'équipe Modélisation et contrôle

  • Thomas Bellotti

    Boundary conditions analysis for a two-unknowns lattice Boltzmann scheme

    17 septembre 2024 - 14:00Salle de conférences IRMA

    In this talk, I will first present key results from our recent works on the numerical analysis of lattice Boltzmann schemes, which ignore boundary conditions. These schemes can be used to simulate systems of conservation laws. Next, I will theoretically examine boundary conditions in lattice Boltzmann methods, focusing on a simplified two-unknowns model. By mapping lattice Boltzmann schemes to finite difference schemes, we enable rigorous consistency and stability analyses. We propose kinetic boundary conditions for inflow and outflow scenarios, addressing the trade-off between accuracy and stability, which we successfully mitigate. Consistency is analyzed using modified equations, while stability is assessed via GKS (Gustafsson, Kreiss, and Sundström) theory. For coarse meshes, where GKS theory may falter, we employ spectral and pseudo-spectral analyses of the scheme's matrix to explain low-resolution effects. Lastly, I will discuss potential research directions related to boundary conditions in kinetic schemes.

  • Michel Duprez

    A finite difference scheme with an optimal convergence for elliptic PDEs on domains defined by a level-set function

    24 septembre 2024 - 14:00Salle de séminaires IRMA

    In this talk, we will present a new finite difference method, on regular grid, well suited for elliptic problems posed in a domain given by a level-set function. It is inspired by the phi-FEM paradigm which is a fictitious finite element method imposing the boundary conditions thanks to a level-set function describing the domain. We will consider here the Poisson equation with Dirichlet condition.We will prove the optimal convergence of our finite difference method in some Sobolev norms. Moreover, the discrete problem is proven to be well conditioned, i.e. the condition number of the associated matrix is of the same order as that of a standard method on a comparable mesh. We will then give some numerical results that confirm the optimal convergence in the considered Sobolev norms. An other advantage of our approach is that it uses standard libraries such as Numpy and Scipy in Python, and the implementation is very short (less than 100 lines in Python), making it a very low-cost numerical scheme in terms of computation time.

  • Yanfei Xiang

    TBA

    8 octobre 2024 - 14:00A confirmer

  • Lucas Ertzbischoff

    TBA

    19 novembre 2024 - 14:00Salle de conférences IRMA

    TBA

  • Yvonne Alama Bronsard

    TBA

    14 janvier 2025 - 14:00Salle de conférences IRMA

    TBA