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.

Semi-Lagrangian SAV method for the Vlasov-Maxwell equations

In this talk, a new numerical method to approximate the solution of the Vlasov-Maxwell equations will be presented.

The method uses a phase space discretization and its main properties are: energy and charge conservation thanks to a semi-implicit treatment of the Maxwell equations, but allowing for an explicit and efficient update of the unknown. One of the main ingredients lies in the introduction of an auxiliary scalar variable inspired from the Scalar Auxiliary Variable (SAV) approach together with a suitable splitting which enables the use of a semi-Lagrangian method. 

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.