On steady-state preserving spectral methods for homogeneous Boltzmann equations

My collaborators Francis Filbet, Lorenzo Pareschi and myself just wrote a note entitled On steady-state preserving spectral methods for homogeneous Boltzmann equations.

In this note, we present a general way to construct spectral methods for the collision operator of the Boltzmann equation which preserves exactly the Maxwellian steady-state of the system.
We show that the resulting method is able to approximate with spectral accuracy the solution uniformly in time.

  Our approach is based on a micro-macro decomposition of the solution to the kinetic equation considered, and is in particular not limited to the Boltzmann equation. Its numerical complexity is the same than the one of a classical spectral method.

A Hierarchy of Hybrid Numerical Methods for Multi-Scale Kinetic Equations

My collaborator Francis Filbet and myself just submitted a new paper, entitled A Hierarchy of Hybrid Numerical Methods for Multi-Scale Kinetic Equations.

In this paper, we construct a hierarchy of hybrid numerical methods for multi-scale kinetic equations based on moment realizability matrices, a concept introduced by Levermore, Morokoff and Nadiga. Following such a criterion, one can consider hybrid scheme where the hydrodynamic part is given either by the compressible Euler or Navier-Stokes equations, or even with more general models, such as the Burnett or super-Burnett systems.

We present applications of this method to the Boltmann equation for rarefied gases, in one dimension of space and three dimensions of velocity, for both Euler and Navier-Stokes fluid description. We prove numerically that our hierarchy of hybrid fluid-kinetic solvers can provide different numerical methods able to achieve the accuracy of a pure kinetic one, with efficiency sometimes almost comparable with the one of a fluid model.

Spectral study of the granular gases operator

I have just submitted a new paper, entitled A Spectral Study of the Linearized Boltzmann Equation for Diffusively Excited Granular Media.

In this work, we are interested in the spectrum of the diffusively excited granular gases equation, in a space inhomogeneous setting, linearized around an homogeneous equilibrium.
We perform a study which generalizes to a non-hilbertian setting and to the inelastic case the seminal work of Ellis and Pinsky about the spectrum of the linearized Boltzmann operator. We first give a precise localization of the spectrum, which consists in an essential part lying on the left of the imaginary axis and a discrete spectrum, which is also of nonnegative real part for small values of the inelasticity parameter. We then give the so-called inelastic “dispersion relations”, and compute an expansion of the branches of eigenvalues of the linear operator, for small Fourier (in space) frequencies and small inelasticity.
One of the main novelty in this work, apart from the study of the inelastic case, is that we consider an exponentially weighted L1(m−1) Banach setting instead of the classical L2(M1,0,1-1) Hilbertian case, endorsed with Gaussian weights. We prove in particular that the results of Ellis and Pinsky hold also in this space.

KI-Net Young Researcher Workshop 2013

I am co-organizing the 2013 KI-Net Young Researcher Workshop, Kinetic and macroscopic models for complex systems, which will take place at the Center for Scientific Computing and Mathematical Modelling (CSCAMM) from October 14 to October 18, 2013. Some spots are still available to apply. Here is the abstract of this meeting:

Complex systems have emerged as a dominant topic in modern science. From animal behaviors to social networks, the impact of complex systems are noticeable on a daily basis. New mathematical models have been developed to apprehend those phenomena. For instance, kinetic models constitute a valuable resource to model complex systems with detailed interactions. On the other hand, macroscopic models focus on the evolution of average quantities which offer a better understanding of systems at larger scales. Both approaches provide new mathematical difficulty (e.g. pattern formation, stability analysis, numerical validation) and ultimately challenge our understanding of complex systems.

Rosenau Approximations

The research article Large-time behavior of the solutions to Rosenau type approximations to the heat equation, written in collaboration with Giuseppe Toscani, has been accepted for publication in SIAM J. App. Math. Super!

In this article we study the large-time behavior of the solution to a general Rosenau type approximation to the heat equation, by showing that the solution to this approximation approaches the fundamental solution of the heat equation (the heat kernel), {but at a slower rate than the usual heat equation}.
This result is valid in particular for the central differences scheme approximation of the heat equation, a property which to our knowledge has never been observed before.

Rescaling Velocity Method

The article A Rescaling Velocity Method for Dissipative Kinetic Equations – Applications to Granular Media, written in collaboration with F. Filbet has been accepted for publication in the Journal of Computational Physics. Neat!

This paper introduces a new numerical algorithm based on a relative energy scaling for collisional kinetic equations allowing to study numerically their long time behavior, without the usual problems related to the change of scales in velocity variables. It is based on the knowledge of the hydrodynamic limit of the model considered, but is able to compute solutions for either dilute or dense regimes. Several applications are presented for Boltzmann-like equations. This method is particularly efficient for numerical simulations of the granular gases equation with dissipative energy: it allows to study accurately the long time behavior of this equation and is very well suited for the study of clustering phenomena.

Fast DVM Methods

The article Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation, written in collaboration with C. Mouhot and L. Pareschi, has been accepted for publication in ESAIM: Mathematical Modeling and Numerical Analysis. Yay!
Here is a short abstract of this work:

This paper deals with a class of algorithms for computing the Boltzmann collision operator for hard spheres, and therefore computing the solutions to the Boltzmann equation. This class of algorithms is known as discrete-velocity methods, and is highly stable, due to the nice positivity and conservation properties of the schemes. The direct numerical evaluation of such methods involve a prohibitive cost, typically O(N2d+1) where d is the dimension of the velocity space. In this paper, we derive fast summation techniques for the evaluation of discrete-velocity schemes which permits to reduce the computational cost from O(N2d+1) to O(Ndlog N), with almost no loss of accuracy. It involves discrete (and fast) Fourier transforms and Farey series in number theory. We show some detailed numerical simulations supporting the interest of these numerical methods.

Dissertation and New Affiliation

The 21st of September, I defended my thesis dissertation, entitled

Quelques contributions à l’analyse mathématique et numérique d’équations cinétiques collisionnelles

in the University of Lyon. The manuscript can be found here.

Some new adventures begin now, as I am starting a post-doc position at the CSCAMM, in the University of Maryland, College Park!

Rosenau type approximations to the heat equation

My collaborator Giuseppe Toscani and myself submitted a new paper, entitled Large-time behavior of the solutions to Rosenau type approximations to the heat equation.
We study in this paper the validity of the approximation to the linear diffusion equation proposed by Rosenau as a regularized version of the Chapman-Enskog expansion of hydrodynamics. This approximation essentially is realized by substituting the heat equation with a linear kinetic equation of Boltzmann type, describing collisions of particles with a fixed background. This remark allows to consider the Rosenau approximation as a particular realization of a model Boltzmann equation, in which the background distribution is a general probability density with bounded variance. In addition to the Rosenau distribution, we considered also a point masses background, which furnishes the central difference scheme to solve numerically the linear diffusion equation.