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Séminaire Equations aux dérivées partielles

organisé par l'équipe Modélisation et contrôle

  • Yvonne Alama Bronsard

    Numerical approximations to nonlinear dispersive equations, from short to long times

    16 janvier 2025 - 14:00Salle de conférences IRMA

    The first part of this talk deals with the numerical approximation to nonlinear dispersive equations, such as the prototypical nonlinear Schrödinger equation. We introduce novel integration techniques allowing for the construction of schemes which perform well both in smooth and non-smooth settings. We obtain symmetric low-regularity schemes with very good structure preserving properties over long times. Higher order extensions will be presented, following new techniques based on decorated trees series inspired by singular stochastic PDEs via the theory of regularity structures. In the second part, we introduce a new approach for designing and analyzing schemes for some nonlinear and nonlocal integrable PDEs, including the well-known Benjamin-Ono equation. This work is heavily inspired by recent theoretical breakthroughs in the field of nonlinear integrable equations, and opens the way to numerical approximations which are far more accurate and efficient for simulating integrable PDEs, from short up to long times.