A neural delta-f method for low-noise PIC simulations

• Martin Campos Pinto

Room: Petri-Turing The delta-f method is a powerful tool to reduce statistical errors in particle simulations of kinetic problems. In its traditional form where it amounts of using an equilibrium state as a control variate, it is essentially limited to regimes where the distribution does not strongly deviate from this equilibrium. In general regimes, several methods have been proposed to extend the approach but the problem is still considered open by many experts in the field. In this talk I will review these approaches and present a new method where the control variate is evolved using neural networks, with promising numerical results both in low and high dimensions.

Damping around inhomogeneous stationary states of the Vlasov-HMF model

• Erwan Faou (INRIA)

Room: Petri-Turing We study the dynamics of perturbations around inhomogeneous stationary states of the Vlasov-HMF (Hamiltonian Mean-Field) model. These stationary solutions are built with compact support and satisfy a linearized stability criterion (Penrose criterion). We show a scattering behavior to a modified state over long (but finite) time depending on the size of the perturbation. This implies a damping effect with an algebraic rate. The key ingredients are based on the analysis of echoes in the dynamics generated by the action-angle variables of the inhomogenous stationary state. This is a joint work with Torryanand Seetohul and Frédéric Rousset.

TBA

• Michel Mehrenberger (Marseille)

Room: Petri-Turing

Numerical analysis of a kinetic equation with a non-local Hamilton-Jacobi limit

• Hélène HIVERT (Inria - Géosciences Rennes)

Room: Petri-Turing In this talk, I will consider the kinetic equation studied in [Bouin, Calvez, Grenier, Nadin, 2023]. I will present the design and analysis of a numerical scheme adapted to the asymptotic behavior of the equation when considered in a large deviations regime. In this regime, the equation degenerates into a nonlocal Hamilton–Jacobi equation, which falls outside the standard frameworks of numerical analysis for Hamilton–Jacobi equations. I will show how a semi-Lagrangian scheme for the original kinetic equation allows one to derive a numerical method suited to the asymptotic regime, and how the convergence of this scheme can be established by relying on the analysis of the well-posedness of the continuous problem.

Time splitting for nonlinear Schrödinger equations: Strichartz estimates and consequences

• Remi Carles

Room: Petri-Turing Strichartz estimates have allowed spectacular progress in the analytical study of nonlinear dispersive equations such as nonlinear Schrödinger equations. We show two consequences regarding error estimates for time splitting method: a global in time error estimate, obtained with Chunmei Su, and low regularity error estimates in the presence of an harmonic potential, using in addition Weyl-Hörmander pseudodifferential calculus.

A micro-macro model for self-propelled particle systems of Vicsek type

• Anais Crestetto (Nantes)

Room: Petri-Turing From the Vicsek model, describing at the individual scale the motion of particles that tend to align with each others, a kinetic equation of Fokker-Planck type can be written. Works of Degond, Frouvelle, Motsch and Navoret investigate the large-scale ("hydrodynamic") limit, leading to the SOH model, and propose numerical scheme to approach its solution. In this talk, I will first present a shock-preserving numerical scheme for the 1D SOH model. In our approach, the SOH model is composed of 2 equations: for the density and the velocity angle unknowns. We will then discuss a micro-macro model which aims to add a kinetic contribution to the SOH model. Here is the difficulty: the collisional operator of the kinetic model only admits a one-dimensional set of collisional invariants. This let us recover the macro equation on the density but not the equation on the velocity angle. To get this one, we have to consider "generalized" collision invariants. The obtained micro-macro model is well adapted to the development of asymptotic preserving scheme. This work has been done with Marie Compain and Christophe Berthon.

Generalised UGK and UGKWP Scheme in the Diffusive Limit

• Julien Mathiaud (Rennes)

Room: Petri-Turing The unified gas kinetic scheme (UGKS) was initially designed to address multiscale challenges in rarefied gas dynamics and then extended to radiative transfert theory, as described by BGK like relaxation models. In this talk, we extend its application to linear kinetic models with non isotropic scattering collision operators, as well as Fokker-Planck models . These problems typically exhibit a fully diffusive nature in the optically thick limit (corresponding to a small Knudsen number). It still leads to an asymptotic preserving (AP) property not only in this diffusive regime but also in the free transport limit. A series of numerical experiments confirm the effectiveness of the approach.