Speakers

High order schemes for the Boltzmann model on unstructured meshes

In this talk, we present recent advances in the development of high-order numerical methods for the solution of the multidimensional Boltzmann equation on unstructured polygonal meshes. The proposed approaches combine discrete velocity formulations with fast spectral methods for the collision operator and high-order spatial discretizations based on finite volume and discontinuous Galerkin techniques. Time integration relies on Implicit–Explicit schemes, including Runge–Kutta and linear multistep methods, ensuring stability, high accuracy, and correct asymptotic behavior from the kinetic to the fluid regimes. Particular emphasis is placed on compact reconstructions, shock-capturing strategies, and computational efficiency on parallel architectures. The performance of the methods is demonstrated through convergence studies, comparisons with simplified kinetic models and DSMC results, and a range of two-dimensional benchmark problems, including subsonic and supersonic flows around a NACA 0012 airfoil.

The talk also introduces a new Boltzmann-type model for binary mixtures of inert gases, preserving the structure of the original equations while accounting for inter-species interactions.

Numerical schemes for Euler-Poisson equations in the quasi-neutral limit  

In this talk, I will discuss new numerical schemes for Euler-Poisson (EP) and Euler-Poisson-Boltzmann (EPB) equation that are able to deal with quasineutral regime. In such regimes, standard numerical methods face severe stability constraints, rendering them practically unusable. For the EP system, we introduce and analyse a new class of finite-volume penalized IMEX Runge-Kutta methods. For the EPB system, a fully implicit finite volume scheme is constructed, for which unconditional energy decay is proven. In addition, a discrete analogue of the modulated energy estimate (around constant states) are established. 

This talk is based on joint works with G. Dimarco (University of Ferrara, Italy) and S. Samantaray (University of Mainz, Germany), and with M. Badsi (University of Nantes, France). 

PDE-Constrained Optimization in Kinetic Equations

PDE-constrained optimization has become a powerful framework for addressing inverse and control problems in systems governed by partial differential equations (PDEs). Kinetic theory encompasses a broad class of equations that describe the non-equilibrium dynamics of interacting particles, and applying PDE-constrained optimization to these systems reveals a variety of unique and intriguing behaviors.

In this talk, I will discuss two representative cases. In the first, we explore the use of PDE-constrained optimization for stabilizing plasma instabilities—a challenge central to achieving controlled fusion. Dispersion-relation-based linear analysis and landscape analysis are employed to identify optimal stabilization strategies. In the second case, we highlight an essential yet often overlooked adjustment required for gradient computation when PDEs are solved using particle methods.

Neural semi-Lagrangian method for high-dimensional advection-diffusion problems

We are interested in numerically solving high-dimensional advection-diffusion equations, such as kinetic equations or parametric problems. Traditional numerical methods suffer from the curse of dimensionality, as the number of degrees of freedom grows exponentially with dimension. Recently, methods based on neural networks have proven effective in reducing the number of degrees of freedom by enriching classical approximation spaces. In this presentation, we will introduce a semi-Lagrangian neural method: at each time step, it consists of advecting the solution exactly, following the characteristic curves of the equation, and projecting it onto the neural approximation space. We provide rough error estimates and present several high-dimensional numerical experiments to evaluate the performance of the method. This is a joint work with Emmanuel Franck, Victor Michel-Dansac and Vincent Vigon.

Uncertainty Quantification for Kinetic Equations: A Journey from Monte Carlo to Deep Neural Surrogates through Multi-Fidelity Methods

Kinetic equations are widely used to model complex systems in areas such as gas dynamics, transport theory, and socio-economic modeling. In many applications these models involve uncertain parameters or data, making uncertainty quantification an essential component of reliable simulations. However, the high dimensionality of kinetic equations often makes standard approaches computationally expensive.

This mini-course presents several modern strategies to tackle uncertainty quantification in kinetic equations. We first review classical Monte Carlo methods, and then introduce multi-fidelity techniques that exploit simplified models to accelerate stochastic simulations. Finally, we discuss the use of deep neural network surrogates as data-driven approximations to further reduce computational costs.

The goal of the course is to provide a unified overview of these approaches and illustrate how their combination can lead to scalable and efficient algorithms for uncertainty propagation in kinetic models.

Dispersive effects in quasilinear hyperbolic equations

Lecture 1 - Dispersive equations

1. • Linear dispersive equations - dispersion relation

   • KdV equation - conservation properties, phase space portrait, solitons

   • Other dispersive equations: Boussinesq equation, BBM approach

   • Numerical solution of dispersive systems: pseudo-spectral methods

Lecture 2 - Quasilinear systems: shocks or dispersion?

• • Dissipative vs dispersive regularisation

  • The Euler equations of gas dynamics

  • Waves on a periodic background

  • A systematic approach: the method of multiple scales

  • Dispersive behaviour of quasilinear systems on periodic background