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.

Discrete Diffusion Limit and Hypocoercivity

Marianne Bessemoulin-Chatard, Maxime Herda and myself just submitted a new preprint, entitled Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations :

In this article, we are interested in the asymptotic analysis of a finite volume scheme for one dimensional linear kinetic equations, with either Fokker-Planck or linearized BGK collision operator. Thanks to appropriate uniform estimates, we establish that the proposed scheme is Asymptotic-Preserving in the diffusive limit. Moreover, we adapt to the discrete framework the hypocoercivity method proposed by [J. Dolbeault, C. Mouhot and C. Schmeiser, Trans. Amed. Math. Soc., 367, 6 (2015)] to prove the exponential return to equilibrium of the approximate solution. We obtain decay estimates that are uniform in the diffusive limit. Finally, we present an efficient implementation of the proposed numerical schemes, and perform numerous numerical simulations assessing their accuracy and efficiency in capturing the correct asymptotic behaviors of the models.

The numerical method developed in this preprint, as well as all the numerical tests, are available on the following jupyter notebook:
MyBinder

ABPDE 3

From August 28 to August 31, 2018, the third edition of the Asymptotic Behavior of systems of PDE arising in physics and biology: theoretical and numerical points of view (ABPDE III) will take place in LILLIAD, Villeneuve d’Ascq.

The main goals of this workshop are the theoretical study of asymptotic behaviors (in large time or with respect to some parameters) of problems arising in physics and biology and the development of asymptotic preserving numerical methods.

The third edition of this workshop features nine plenary speakers. In addition, several contributed talks and a poster session will completethe programs. Submissions are open until July 8th.

ABPDE 3 website