# Benjamin GrailleUniversité Paris-Saclay · Mathematics

Benjamin Graille

Assistant Professor

## About

51

Publications

5,007

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367

Citations

Citations since 2016

Introduction

Additional affiliations

October 2005 - present

September 2001 - September 2005

## Publications

Publications (51)

Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fi...

Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of cons...

Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of cons...

Multiresolution provides a fundamental tool based on the wavelet theory to build adaptive numerical schemes for Partial Differential Equations and time-adaptive meshes, allowing for error control. We have introduced this strategy before to construct adaptive lattice Boltzmann methods with this interesting feature.Furthermore, these schemes allow fo...

We consider an adaptive multiresolution-based lattice Boltzmann scheme, which we have recently introduced and studied from the perspective of the error control and the theory of the equivalent equations. This numerical strategy leads to high compression rates, error control and its high accuracy has been explained on uniform and dynamically adaptiv...

Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fi...

Lattice Boltzmann Methods (LBM) stand out for their simplicity and computational efficiency while offering the possibility of simulating complex phenomena. While they are optimal for Cartesian meshes, adapted meshes have traditionally been a stumbling block since it is difficult to predict the right physics through various levels of meshes. In this...

In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times when the time step is proportional to the space step and tends to zero. We do this by a combination of theory and numerical experiment. The classical formal analysis when all the relaxation parameters are fixed and the time...

In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times when the time step is proportional to the space step and tends to zero. We do this by a combination of theory and numerical experiment. The classical formal analysis when all the relaxation parameters are fixed and the time...

In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative vel...

This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic reconnections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model to investigate the influence on shock solutions of a nonconserva...

In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative vel...

We consider a mono-dimensional two-velocities scheme used to approximate the solutions of a scalar hyperbolic conservative partial differential equation. We prove the convergence of the discrete solution toward the unique entropy solution by first estimating the supremum norm and the total variation of the discrete solution, and second by construct...

This contribution deals with the fluid modeling of multicomponent magnetized plasmas in thermo-chemical non-equilibrium from the partially- to fully-ionized collisional regimes, aiming at the predictive simulation of magnetic reconnection in Sun chromosphere conditions. Such fluid models are required for large-scale simulations by relying on high p...

The present contribution starts from a two-temperature single-momentum multicom-ponent diffusion model coupled to Maxwell's equations obtained in Wargnier et al. (2018a). It is the asymptotic limit of small Debye length of the model rigorously derived from kinetic theory by Graille et al. (2009) using a multi-scale Chapman Enskog expansion. The mod...

This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic reconnections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model in order to properly investigate the influence on shock solution...

This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic re-connections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model in order to properly investigate the influence on shock solutio...

In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times when the time step is proportional to the space step and tends to zero. We do this by a combination of theory and numerical experiment. The classical formal analysis when all the relaxation parameters are fixed and the time...

We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the simulation of diffusive processes. When the mesh is refined while holding the diffusivity constant, we first obtain asymptotic convergence. When the mesh size tends to zero, however, this convergence breaks down in a curious fashion, and we observe qualitative discrepancie...

We consider multi relaxation times lattice Boltzmann scheme with two particle distributions for the thermal Navier Stokes equations formulated with conservation of mass and momentum and dissipation of volumic entropy. Linear stability is taken into consideration to determine a coupling between two coefficients of dissipation. We present interesting...

In this contribution, we study the theoretical and numerical stability of a
bidimensional relative velocity lattice Boltzmann scheme. These relative
velocity schemes introduce a velocity field parameter called "relative
velocity" function of space and time. They generalize the d'Humi\`eres multiple
relaxation times scheme and the cascaded automaton...

This paper studies the stability properties of a two dimensional relative
velocity scheme for the Navier-Stokes equations. This scheme inspired by the
cascaded scheme has the particularity to relax in a frame moving with a
velocity field function of space and time. Its stability is studied first in a
linear context then on the non linear test case...

We study the formal precision of the relative velocity lattice Boltzmann
schemes. They differ from the d'Humi\`eres schemes by their relaxation phase:
it occurs for a set of moments parametrized by a velocity field function of
space and time. We deal with the asymptotics of the relative velocity schemes
for one conservation law: the third order equ...

We focus on mono-dimensional hyperbolic systems approximated by a particular lattice Boltz- mann scheme. The scheme is described in the framework of the multiple relaxation times method and stability conditions are given. An analysis is done to link the scheme with an explicit finite differences approximation of the relaxation method proposed by Ji...

We propose the derivation of acoustic-type isotropic partial differential
equations that are equivalent to linear lattice Boltzmann schemes with a
density scalar field and a momentum vector field as conserved moments. The
corresponding linear equivalent partial differential equations are generated
with a new "Berliner version" of the Taylor expansi...

In this contribution, a new class of lattice Boltzmann schemes is introduced
and studied. These schemes are presented in a framework that generalizes the
multiple relaxation times method of d'Humi\`eres. They extend also the Geier's
cascaded method. The relaxation phase takes place in a moving frame involving a
set of moments depending on a given r...

In this paper we study the acoustic properties of a microstructured material such as glass wool or foam. In our model, the solid matrix is governed by linear elasticity and the surrounding fluid obeys Stokes equations. The microstructure is assumed to be periodic at some small scale ε and the viscosity coefficient of the fluid is assumed to be of o...

We derive a hydrodynamic model for the internal energy excitation of molecular gases in thermal nonequilibrium based on kinetic theory. The co-existence of fast and slow collisions in the system results in thermal nonequilibrium between the translational and internal energy modes. A proper scaling for the Boltzmann equation that accounts for the di...

In this paper, we investigate the numerous parameters choices for linear
lattice Boltzmann schemes according to the definition of the isotropic order
given in \cite{ADG11}. This property---written in a general framework including
all of the \ddqq schemes---can be read through a group operation. It implies
some relations on the parameters of the sch...

In this paper, we recall the linear version of the lattice Boltzmann schemes
in the framework proposed by d'Humi\'eres. According to the equivalent
equations we introduce a definition for a scheme to be isotropic at some order.
This definition is chosen such that the equivalent equations are preserved by
orthogonal transformations of the frame. The...

Multitemperature models are widely used in the aerospace community to model atmospheric entry flows. In this paper, we propose a general description of the internal energy excitation of a molecular gas in thermal nonequilibrium by distinguishing between slow and fast collisions. A multiscale Chapman-Enskog method is used to study thermalization and...

We examine both processes of ionization by electron and heavy‐particle impact in spatially uniform plasmas at rest in the absence of external forces. A singular perturbation analysis is used to study the following physical scenario, in which thermal relaxation becomes much slower than chemical reactions. First, electron‐impact ionization is investi...

We discuss transport coefficients in weakly ionized mixtures. We investigate the situations of weak and strong magnetic fields as well as electron temperature nonequilibrium. We present in each regime the Boltzmann equations, examples of transport fluxes, the structure of transport linear systems and discuss their solution by efficient iterative te...

In the present contribution, we derive from kinetic theory a unified fluid model for multicomponent plasmas by accounting for the electromagnetic field influence. We deal with a possible thermal nonequilibrium of the translational energy of the particles, neglecting their internal energy and the reactive collisions. Given the strong disparity of ma...

We investigate iterative algorithms for solving complex symmetric constrained singular systems arising in magnetized multicomponent transport. The matrices of the corresponding linear systems are symmetric with a positive semi-definite real part and an imaginary part with a compatible nullspace. We discuss well posedness, the symmetry of generalize...

We investigate the kinetic theory of partially ionized reactive gas mixtures in strong magnetic fields following Giovangigli et al (2003 Physica A 327 313-48). A new tensor basis is introduced for expanding the perturbed distribution functions associated with the viscous tensor. New symmetry properties of transport coefficients are established as w...

In the present contribution, we derive from kinetic theory a unified fluid model for multicomponent plasmas by accounting for the electromagnetic field influence. We deal with a possible thermal nonequilibrium of the translational energy of the particles, neglecting their internal energy and the reactive collisions. Given the strong disparity of ma...

We derive a model for reactive plasmas based on kinetic theory, accounting for an ionization mechanism and dealing with a possible thermal non-equilibrium of the trans- lational energy of the electrons and heavy particles, such as atoms and ions, given their strong disparity of mass. We conduct a dimensional analysis of the Boltzmann equation and u...

We investigate a system of partial differential equations modelling ionized magnetized reactive gas mixtures. In this model, dissipative fluxes are anisotropic linear combinations of fluid variable gradients and also include zeroth-order contributions modelling the direct effect of electromagnetic forces. There are also gradient dependent source te...

Reçu le 20 octobre 2004 ; accepté le 3 décembre 2004 Présenté par Pierre-Louis Lions Résumé On étudie un système d'équations aux dérivées partielles modélisant les plasmas réactifs dissipatifs. Les flux de transport comprennent des combinaisons linéaires anisotropes des gradients et des termes d'ordre zéro dus au champ électromagnétique et les term...

We investigate macroscopic models of PDE for ionised reactive gas
mixtures and we perform some mathematical studies and numerical
simulations. We obtain the macroscopic equations just like expressions for transport fluxes by using the Enskog expansion of a generalized Boltzmann equation. We then investigate symmetry properties given by the entropy...

We investigate a system of partial differential equations modeling ambipolar plasmas. The ambipolar — or zero current — model is obtained from general plasmas equations in the limit of vanishing Debye length. In this model, the electric field is expressed as a linear combination of macroscopic variable gradients. We establish that the governing equ...

We investigate partially ionized reactive gas mixtures in the presence of electric and magnetic fields. Our starting point is a generalized Boltzmann equation with a chemical source term valid for arbitrary reaction mechanism. We study the Enskog expansion and obtain macroscopic equations in the zeroth- and first-order regimes, together with transp...

## Projects

Projects (3)

My contributions to set the theoretical foundation for lattice Boltzmann methods from the standpoint of Numerical Analysis

Adaptive Multiresolution and Lattice Boltzmann schemes

Notions of stability, consistency, and convergence for the Lattice Boltzmann Schemes. Link with the relaxation method.