Papers by emmanuel frenod
arXiv (Cornell University), Nov 3, 2010
In this paper, we present some new results about the approximation of the Vlasov-Poisson system w... more In this paper, we present some new results about the approximation of the Vlasov-Poisson system with a strong external magnetic field by the 2D finite Larmor radius model. The proofs within the present work are built by using two-scale convergence tools, and can be viewed as a new slant on previous works of Frénod & Sonnendrücker and Bostan on the 2D finite Larmor Radius model. In a first part, we recall the physical and mathematical contexts. We also recall two main results from previous papers of Frénod & Sonnendrücker and Bostan. Then, we introduce a set of variables which are so-called canonical gyrokinetic coordinates, and we write the Vlasov equation in these new variables. Then, we establish some two-scale convergence and weak-* convergence results.
Transactions on Engineering Technologies, 2018
In this paper we study the two-scale behavior of the electromagnetic field in 3D in a composite m... more In this paper we study the two-scale behavior of the electromagnetic field in 3D in a composite material. It is the continuation of the paper (Canot and Frenod Method of homogenization for the study of the propagation of electromagnetic waves in a composite 2017) [7] in which we obtain existence and uniqueness results for the problem, we performed an estimate that allows us to approach homogenization. Techniques of asymptotic expansion and two-scale convergence are used to obtain the homogenized problem. We justify the two-scale expansion numerically in the second part of the paper.
The purpose of this article is to study behavior of the electromagnetic field in 3D in and near c... more The purpose of this article is to study behavior of the electromagnetic field in 3D in and near composite material. For this, time-harmonic Maxwell equations, for a conducting two-phase composite and the air above, are considered.
Communications in Computational Physics, 2015
With the aim of solving in a four dimensional phase space a multi-scale Vlasov-Poisson system, we... more With the aim of solving in a four dimensional phase space a multi-scale Vlasov-Poisson system, we propose in a Particle-In-Cell framework a robust time-stepping method that works uniformly when the small parameter vanishes. As an exponential integrator, the scheme is able to use large time steps with respect to the typical size of the solution’s fast oscillations. In addition, we show numerically that the method has accurate long time behaviour and that it is asymptotic preserving with respect to the limiting Guiding Center system.
This Proceeding presents the method that allows us to get the Gyro-Kinetic Approximation of the D... more This Proceeding presents the method that allows us to get the Gyro-Kinetic Approximation of the Dynamical System satisfied by the trajectory of a particle submitted to a Strong Magnetic Field. The goal of the method is to build a change of coordinates in order to make the dynamic of two components of the trajectory to disappear. This change of coordinates
Comptes Rendus Mécanique, 2014
We change a previous time-stepping algorithm for solving a multi-scale Vlasov-Poisson system with... more We change a previous time-stepping algorithm for solving a multi-scale Vlasov-Poisson system within a Particle-In-Cell method, in order to do accurate long time simulations. As an exponential integrator, the new scheme allows to use large time steps compared to the size of oscillations in the solution.
Discrete & Continuous Dynamical Systems - S, 2014
In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a sm... more In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a small parameter, we propose a time-stepping method which is numerically uniformly accurate when the parameter goes to zero. Based on an exponential time differencing approach, the scheme is able to use large time steps with respect to the typical size of the fast oscillations of the solution.
Multiscale Modeling & Simulation, 2006
In this paper, we propose a new method to forecast the drift of objects in near coastal ocean on ... more In this paper, we propose a new method to forecast the drift of objects in near coastal ocean on a period of several weeks. The proposed approach consists in estimating the probability of events linked to the drift using Monte Carlo simulations. It couples an averaging method which permits to decrease the computational cost and a statistical method in order to take into account the variability of meteorological loading factors.
SIAM Journal on Mathematical Analysis, 2001
Kinetic & Related Models, 2014
Considering a Hamiltonian Dynamical System describing the motion of charged particle in a Tokamak... more Considering a Hamiltonian Dynamical System describing the motion of charged particle in a Tokamak or a Stellarator, we build a change of coordinates to reduce its dimension. This change of coordinates is in fact an intricate succession of mappings that are built using Hyperbolic Partial Differential Equations, Differential Geometry, Hamiltonian Dynamical System Theory and Symplectic Geometry, Lie Transforms and a new tool which is here introduced : Partial Lie Sums.
Discrete & Continuous Dynamical Systems - S, 2014
In this paper we consider the model built in [3] for short term dynamics of dunes in tidal area. ... more In this paper we consider the model built in [3] for short term dynamics of dunes in tidal area. We construct a Two-Scale Numerical Method based on the fact that the solution of the equation which has oscillations Two-Scale converges to the solution of a well-posed problem. This numerical method uses on Fourier series. 1. Introduction. This paper deals with numerical simulations of sand transport problems. Its goal is to build a Two-Scale Numerical Method to simulate dynamics of dunes in tidal area. This paper enters a work program concerning the development of Two-Scale Numerical Methods to solve PDEs with oscillatory singular perturbations linked with physical phenomena. In Ailliot, Frénod and Monbet [2], such a method is used to manage the tide oscillation for long term drift forecast of objects in coastal ocean waters. Frénod, Mouton and Sonnendrücker [5] made simulations of the 1D Euler equation using a Two-Scale Numerical Method. In Frénod, Salvarani and Sonnendrücker [6], such a method is used to simulate a charged particle beam in a periodic focusing channel. Mouton [9, 10] developped a Two-Scale Semi Lagrangian Method for beam and plasma applications. We consider the following model, valid for short-term dynamics of dunes, built and studied in [3]:
Numerische Mathematik, 2007
Mathematical Models and Methods in Applied Sciences, 2013
In this paper, we build a two-scale macro–micro decomposition of the Vlasov equation with a stron... more In this paper, we build a two-scale macro–micro decomposition of the Vlasov equation with a strong magnetic field. This consists in writing the solution of this equation as a sum of two oscillating functions with circumscribed oscillations. The first of these functions has a shape which is close to the shape of the two-scale limit of the solution and the second one is a correction built to offset this imposed shape. The aim of such a decomposition is to be the starting point for the construction of two-scale asymptotic-preserving schemes.
Mathematical Models and Methods in Applied Sciences, 2009
We study the two-scale asymptotics for a charged beam under the action of a rapidly oscillating e... more We study the two-scale asymptotics for a charged beam under the action of a rapidly oscillating external electric field. After proving the convergence to the correct asymptotic state, we develop a numerical method for solving the limit model involving two time scales and validate its efficiency for the simulation of long time beam evolution.
Journal of Differential Equations, 2010
From a scale analysis of hydrodynamic phenomena having a significant action on the drift of an ob... more From a scale analysis of hydrodynamic phenomena having a significant action on the drift of an object in coastal ocean waters, we deduce equations modeling the associated hydrodynamic fields over a time period of several weeks. These models are essentially non linear hyperbolic systems of PDE involving a small parameter. Then from the models we extract a simplified and nevertheless typical one for which we prove that its classical solution exists on a time interval which is independent of the small parameter. We then show that the solution weak− * converges as the small parameter goes to zero and we characterize the equation satisfied by the weak− * limit.
Discrete & Continuous Dynamical Systems - A, 2010
In this paper we build models for short-term, mean-term and long-term dynamics of dune and megari... more In this paper we build models for short-term, mean-term and long-term dynamics of dune and megariple morphodynamics. They are models that are degenerated parabolic equations which are, moreover, singularly perturbed. We, then give an existence and uniqueness result for the short-term and mean-term models. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the short-term model is homogenized.
Discrete and Continuous Dynamical Systems - Series S, 2014
In this note, a classification of Homogenization-Based Numerical Methods and (in particular) of N... more In this note, a classification of Homogenization-Based Numerical Methods and (in particular) of Numerical Methods that are based on the Two-Scale Convergence is done. In this classification stand: Direct Homogenization-Based Numerical Methods; H-Measure-Based Numerical Methods; Two-Scale Numerical Methods and TSAPS: Two-Scale Asymptotic Preserving Schemes.
Discrete and Continuous Dynamical Systems - Series S, 2016
This note recalls what are "Homogenization-Based Numerical Methods". Then it introduces the paper... more This note recalls what are "Homogenization-Based Numerical Methods". Then it introduces the papers of this Special Issue. In a third section it advocates for building a project in order to build "Homogenization-Based Software for Simulation of Multi-Scale Complex Systems".
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Papers by emmanuel frenod