Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures

Together with Rafael Bailo, we have just submitted a new preprint, entitled Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures:

We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and BGK equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the schemes on a range of scenarios, demonstrating its prowess in dealing with extreme mass ratios, fluid instabilities, and other complex phenomena.

Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation

Together with Lorenzo Pareschi, we’ve just submitted a new preprint, entitled Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation :

Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. We then apply the new spectral method to the evaluation of the Boltzmann collision term, and prove spectral consistency and stability of the resulting Fourier-Galerkin approximation scheme. Various numerical experiments illustrate the theoretical findings.

On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation

Together with Lorenzo Pareschi, we’ve just submitted a new preprint, entitled On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation.

Spectral methods, thanks to the high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the Boltzmann collision operator. On the other hand, the loss of some local invariants leads to the wrong long time behavior. A way to overcome this drawback, without sacrificing spectral accuracy, has been proposed recently with the construction of equilibrium preserving spectral methods. Despite the ability to capture the steady state with arbitrary accuracy, the theoretical properties of the method have never been studied in details. In this paper, using the perturbation argument developed by Filbet and Mouhot for the homogeneous Boltzmann equation, we prove stability, convergence and spectrally accurate long time behavior of the equilibrium preserving approach.

Convergence of knowledge in a stochastic cultural evolution model with population structure, social learning and credibility biases

Our paper with Sylvain Billiard, Maxime Derex and Ludovic Maisonneuve has been accepted for publication in M3AS. YAY \o/

Understanding how knowledge is created and propagates within groups is crucial to explain how human populations have evolved through time. Anthropologists have relied on different theoretical models to address this question. In this work, we introduce a mathematically oriented model that shares properties with individual based approaches, inhomogeneous Markov chains and learning algorithms, such as those introduced in [F. Cucker, S. Smale, Bull. Amer. Math. Soc., 39 (1), 2002] and [F. Cucker, S. Smale and D.~X Zhou, Found. Comput. Math., 2004]. After deriving the model, we study some of its mathematical properties, and establish theoretical and quantitative results in a simplified case. Finally, we run numerical simulations to illustrate some properties of the model.

Our main result is that, as time goes to infinity, individuals' knowledge can converge to a common shared knowledge that was not present in the convex combination of initial individuals' knowledge.

Recent development in kinetic theory of granular materials: analysis and numerical methods

Together with my collaborators José Antonio Carrillo, Jingwei Hu and Zheng Ma, we have just submitted a new preprint, entitled Recent development in kinetic theory of granular materials: analysis and numerical methods.

Over the past decades, kinetic description of granular materials has received a lot of attention in mathematical community and applied fields such as physics and engineering. This article aims to review recent mathematical results in kinetic granular materials, especially for those which arose since the last review by Villani on the same subject. We will discuss both theoretical and numerical developments. We will finally showcase some important open problems and conjectures by means of numerical experiments based on spectral methods.

Finite Volume Method for a System of Continuity Equations Driven by Nonlocal Interactions

With my collaborator Anissa El Keurti, we’ve just submitted a short note on a Finite Volume Method for a System of Continuity Equations Driven by Nonlocal Interactions .

We present a new finite volume method for computing numerical approximations of a system of nonlocal transport equation modeling interacting species. This method is based on the work [F. Delarue, F. Lagoutire, N. Vauchelet, Convergence analysis of upwind type schemes for the aggregation equation with pointy potential, Ann. Henri. Lebesgue 2019], where the nonlocal continuity equations are 10 treated as conservative transport equations with a nonlocal, nonlinear, rough velocity field. We analyze some properties of the method, and illustrate the results with numerical simulations.

Convergence of knowledge in a cultural evolution model with population structure, random social learning and credibility biases

Together with Sylvain Billiard, Maxime Derex and Ludovic Maisonneuve, we have just submitted a new preprint, entitled Convergence of knowledge in a cultural evolution model with population structure, random social learning and credibility biases:

Understanding how knowledge is created and propagates within groups is crucial to explain how human populations have evolved through time. Anthropologists have relied on different theoretical models to address this question. In this work, we introduce a mathematically oriented model that shares properties with individual based approaches, inhomogeneous Markov chains and learning algorithms, such as those introduced in [F. Cucker, S. Smale, Bull. Amer. Math. Soc., 39 (1), 2002] and [F. Cucker, S. Smale and D.~X Zhou, Found. Comput. Math., 2004]. After deriving the model, we study some of its mathematical properties, and establish theoretical and quantitative results in a simplified case. Finally, we run numerical simulations to illustrate some properties of the model.