The paper entitle Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations has been accepted for publication in the SMAI Journal of Computational Mathematics. Yay!

# 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:

# 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.

# Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations

Together with my collaborators Ward Melis and Giovanni Samaey, we submitted a new paper, titled: Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations :

We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions.

# An efficient numerical method for solving the Boltzmann equation in multidimensions

The paper An efficient numerical method for solving the Boltzmann equation in multidimensions, written with Giacomo Dimarco, Raphaël Loubère, Jacek Narski and myself has been published in the Journal of Computational Physics. Great!

# Residual equilibrium schemes for time dependent partial differential equations

The paper about Residual equilibrium schemes for time dependent partial differential equations written in collaboration with Lorenzo Pareschi has been accepted in Computers and Fluids. Yay!

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.

# An efficient numerical method for solving the Boltzmann equation in multidimensions

Giacomo Dimarco, Raphaël Loubère, Jacek Narski and myself just submitted a preprint about An efficient numerical method for solving the Boltzmann equation in multidimensions. Yay!

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J. Comput. Phys., Vol. 255, 2013, pp 680-698] originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3D×3D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.

# ABPDE 2

From June 15th to 17th, we are organizing in Lille the second edition of the workshop Asymptotic Behavior of systems of PDE arising in physics and biology: theoretical and numerical points of view. Registration are still open, and there are still spots for posters and contributed talks: don’t hesitate to apply by sending me an email.

# Hydrodynamic limit of granular gases to pressureless Euler in dimension 1

Together with my collaborator Pierre-Emmanuel Jabin, we’ve just submitted a paper entitled Hydrodynamic limit of granular gases to pressureless Euler in dimension 1. This works aims to provide the first rigorous proof of the hydrodynamic limit of the kinetic, granular gases equation with strong inelasticity towards the fluid, pressureless Euler system, in dimension 1 of space and velocity. More precisely,

We investigate the behavior of granular gases in the limit of small Knudsen number, that is very frequent collisions. We deal with the strongly inelastic case, in one dimension of space and velocity. We are able to prove the convergence toward the pressureless Euler system. The proof relies on dispersive relations at the kinetic level, which leads to the so-called Oleinik property at the limit.

# Residual equilibrium schemes for time dependent partial differential equations

Together with my collaborator Lorenzo Pareschi, we have just submitted a new preprint entitled Residual equilibrium schemes for time dependent partial differential equations. This work extends the note about steady state preserving spectral methods to a large class of dissipative PDE’s. Here is the abstract:

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.