Together with my collaborators Domenico Caparello and Lorenzo Pareschi, we submitted a new paper entitlled High-Order Asymptotic-Preserving IMEX schemes for an ES-BGK model for Gas Mixtures.

In this work we construct a high-order Asymptotic-Preserving (AP)
Implicit-Explicit (IMEX) scheme for the ES-BGK model for gas mixtures
introduced in [Brull, Commun. Math. Sci., 2015]. The time discretization
is based on the IMEX strategy proposed in [Filbet, Jin, J. Sci.
Comput., 2011] for the single-species BGK model and is here extended to
the multi-species ES-BGK setting. The resulting method is fully
explicit, uniformly stable with respect to the Knudsen number and, in
the fluid regime, it reduces to a consistent and high-order accurate
solver for the limiting macroscopic equations of the mixture. The IMEX
structure removes the stiffness associated with the relaxation term so
that the time step is constrained only by a hyperbolic CFL condition.
The full solver couples a high-order space and velocity discretization
that includes third-order time integration, a CWENO3 finite-volume
reconstruction in space, exact conservation of macroscopic moments in
the discrete velocity space, and a multithreaded implementation. The
proposed approach can handle an arbitrary number of species. Its
accuracy and robustness are demonstrated on a set of multidimensional
kinetic tests for gas mixtures, where the AP property and the correct
asymptotics are numerically verified across different regimes.

Together with my collaborators Marianne Bessemoulin-Chatard and Tino Laidin, we have submitted a new paper entitled Discrete H-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity.

In this article, we study the long-time behavior of a finite-volume discretization for a nonlinear kinetic reaction model involving two interacting species. Building upon the seminal work of [Favre, Pirner, Schmeiser, ARMA, 2023], we extend the discrete exponential convergence to equilibrium result established in [Bessemoulin-Chatard, Laidin, Rey, IMAJNA, 2025], which was obtained in a perturbative framework using weighted L2 estimates. The analysis applies to a broader class of exponentially decaying initial data, without requiring proximity to equilibrium, by exploiting the properties of the Boltzmann entropy. The proof relies on the propagation of the initial L infinity bounds, derived from monotonicity properties of the scheme, allowing controlled linearizations within the nonlinear entropy estimates. Moreover, we show that the time-discrete dissipation inherent to the numerical scheme plays a crucial stabilizing role, providing control over the nonlinear terms.

Together with Gabriella Puppo and Tommaso Tenna, we’ve just submitted a new preprint entitled Formal derivation of an isentropic two-phase flow model from the multi-species Boltzmann equation:

Starting from the multi-species Boltzmann equation for a gas mixture, we propose the formal derivation of the isentropic two-phase flow model introduced in [Romenski, E., and Toro, E. F., Comput. Fluid Dyn. J., 13 (2004)]. We examine the asymptotic limit as the Knudsen numbers approach zero, in a regime characterized by resonant intra-species collisions, where interactions between particles of the same species dominate. This specific regime leads to a multi-velocity and multi-pressure hydrodynamic model, enabling the explicit computation of the coefficients for the two-phase macroscopic model. Our derivation also accounts for the inclusion of the evolution of the volume fraction, which is a key variable in many macroscopic multiphase models

Together with my PhD student Domenico Caparello and my collaborator Lorenzo Pareschi, we have submitted a new work entitled Hierarchical dynamic domain decomposition for the multiscale Boltzmann equation:

In this work, we present a hierarchical domain decomposition method for the multi-scale Boltzmann equation based on moment realizability matrices, a concept introduced by Levermore, Morokoff, and Nadiga. This criterion is used to dynamically partition the two-dimensional spatial domain into three regimes: the Euler regime, an intermediate kinetic regime governed by the ES-BGK model, and the full Boltzmann regime. The key advantage of this approach lies in the use of Euler equations in regions where the flow is near hydrodynamic equilibrium, the ES-BGK model in moderately non-equilibrium regions where a fluid description is insufficient but full kinetic resolution is not yet necessary, and the full Boltzmann solver where strong non-equilibrium effects dominate, such as near shocks and
boundary layers. This allows for both high accuracy and significant computational savings, as the Euler solver and the ES-BGK models are considerably cheaper than the full kinetic Boltzmann model.To ensure accurate and efficient coupling between regimes, we employ asymptotic-preserving (AP) numerical schemes and fast spectral solvers for evaluating the Boltzmann collision operator. Among the main novelties of this work are the use of a full 2D spatial and 3D velocity decomposition, the integration of three distinct physical regimes within a unified solver framework, and a parallelized implementation exploiting CPU multithreading. This combination enables robust and scalable simulation of multiscale kinetic flows with complex geometries.

Together with my former PhD student Tino Laidin, we’ve just submitted a paper entitled A Parareal in time numerical method for the collisional Vlasov equation in the hyperbolic scaling:

We present the design of a multiscale parareal method for
kinetic equations in the fluid dynamic regime. The goal is to reduce the
cost of a fully kinetic simulation using a parallel in time procedure.
Using the multiscale property of kinetic models, the cheap, coarse
propagator consists in a fluid solver and the fine (expensive)
propagation is achieved through a kinetic solver for a collisional
Vlasov equation. To validate our approach, we present simulations in the
1D in space, 3D in velocity settings over a wide range of initial data
and kinetic regimes, showcasing the accuracy, efficiency, and the
speedup capabilities of our method.

Together with my PhD student Tommaso Tenna, we have just submitted a new preprint, entitled The Boltzmann equation for a multi-species inelastic mixture. A granular gas is a collection of macroscopic particles that interact through energy-dissipating collisions, also known as inelastic collisions. This inelasticity is characterized by a collision mechanics in which mass and momentum are conserved and kinetic energy is dissipated. Such a system can be described by a kinetic equation of the Boltzmann type. Nevertheless, due to the macroscopic aspect of the particles, any realistic description of a granular gas should be written as a mixture model composed of M different species, each with its own mass. We propose in this work such a granular multi-species model and analyse it, providing Povzner-type inequalities, and a Cauchy theory in general Orlicz spaces. We also analyse its large time behavior, showing that it exhibits a mixture analogue of the seminal Haff’s Law.

Together with my collaborators Marianne Bessemoulin-Chatard and Tino Laidin, we have just submitted a new paper, entitled Discrete hypocoercivity for a nonlinear kinetic reaction model. In this work, we propose a finite volume discretization of a one dimensional nonlinear reaction kinetic model proposed in [Neumann, Schmeiser, Kint. Rel. Mod. 2016], which describes a 2-species recombination-generation process. Specifically, we establish the long-time convergence of approximate solutions towards equilibrium, at exponential rate. The study is based on an adaptation for a discretization of the linearized problem of the L2 hypocoercivity method introduced in [Dolbeault, Mouhot, Schmeiser, 2015]. From this, we can deduce a local result for the discrete nonlinear problem. As in the continuous framework, this result requires the establishment of a maximum principle, which necessitates the use of monotone numerical fluxes.

I have the pleasure to start the academic year 2023 as a newly appointed Full Professor of Applied Mathematics at Université Côte d’Azur, in Nice, France. Farewell, Lille!

DATAHYKING is a Marie Curie Doctoral Network funded by the European commission under Grant Agreement No. 101072546. The DATAHYKING Doctoral Network aims at training a new generation of modeling and simulation experts to develop virtual experimentation tools and workflows that can reliably and efficiently exploit the potential of mathematical modeling and simulation of interacting particle systems.

To this end, we create a data-driven simulation framework for kinetic models of interacting particle systems, and define a common methodology for these future modeling and simulation experts. DATAHYKING will focus on:

• Developing reliable and efficient simulation methods;
• Designing robust consensus-based optimisation, also for machine learning;
• Developing multifidelity methods for uncertainty quantification and data assimilation;
• Applications in traffic flow, finance and granular flow, also in collaboration with industry.

Applications are still open for the 13 PhD theses funded by the program, including 3 in Lille. JOIN US!

The 5th edition of the conference Asymptotic Behavior of systems of PDEs arising in physics and biology (ABPDE 5) will take place in Lille on June 5-9. People interested in attending and possibly giving a presentation can register here.