arXiv subjects

Francis Filbet

Publications and source records attributed to Francis Filbet.

At least 19 recordsLinked to original sources

arXiv2026

Discrete hypocoercive estimates for discontinuous Galerkin methods: application to the Vlasov-Poisson-Fokker-Planck system

We develop and analyze a class of structure-preserving discontinuous Galerkin schemes for the nonlinear Vlasov-Poisson-Fokker-Planck model, reformulated as a hyperbolic system through a Hermite expansion in the velocity variable. We discretize the Vlasov-Fokker-Planck equation with the discontinuous Galerkin method, while the Poisson equation is approximated with either a discontinuous Galerkin method or a Raviart-Thomas mixed finite element method. We prove the exponential relaxation to equilibrium for suitable initial data, uniformly with respect to the discretization parameters thanks to discrete hypocoercivity arguments. Moreover, we check that the resulting semi-discrete schemes preserve the physical invariants along with the L 2 variational structure of the linearized model. Numerical simulations verify the accuracy and the long-time behavior of the scheme.

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arXiv2025

Convergence analysis of a Crank-Nicolson scheme for strongly magnetized plasmas

The present paper is devoted to the convergence analysis of an asymptotic preserving particle scheme designed to serve as a particle pusher in a Particle-In-Cell (PIC) method for the Vlasov equation with a strong inhomogeneous magnetic field. The asymptotic preserving scheme that we study removes classical strong restrictive stability constraints on discretization steps while capturing the large-scale dynamics, even when the discretization is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization and stiffness parameters and match sharply numerical tests. The present analysis is expected to be representative of the general analysis of a class of schemes, developed by the authors, conceived as implicit-explicit schemes on augmented formulations.

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arXiv2025

Numerical analysis of the homogeneous Landau equation: approximation, error estimates and simulation

We construct a numerical solution to the spatially homogeneous Landau equation with Coulomb potential on a domain $D_L$ with N retained Fourier modes. By deriving an explicit error estimate in terms of $L$ and $N$, we demonstrate that for any prescribed error tolerance and fixed time interval $[0, T ]$, there exist choices of $D_L$ and $N$ satisfying explicit conditions such that the error between the numerical and exact solutions is below the tolerance. Specifically, the estimate shows that sufficiently large $L$ and $N$ (depending on initial data parameters and $T$) can reduce the error to any desired level. Numerical simulations based on this construction are also presented. The results in particular demonstrate the mathematical validity of the spectral method proposed in the referenced literature.

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arXiv2025

A modified Crank-Nicolson scheme for the Vlasov-Poisson system with a strong external magnetic field

We propose and study a Particle-In-Cell (PIC) method based on the Crank-Nicolson time discretization for the Vlasov-Poisson system with a strong and inhomogeneous external magnetic field with fixed direction, where we focus on the motion of particles in the plane orthogonal to the magnetic field. In this regime, the time step can be subject to stability constraints related to the smallness of Larmor radius and plasma frequency [21]. To avoid this limitation, our approach is based on numerical schemes [9, 10, 12], providing a consistent PIC discretization of the guiding-center system taking into account variations of the magnetic field. We carry out some theoretical proofs and perform several numerical experiments to validate the method and its underlying concepts.

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arXiv2025

On the approximation of the von Neumann equation in the semiclassical limit. Part II : numerical analysis

This paper is devoted to the numerical analysis of the Hermite spectral method proposed in [14], which provides, in the semiclassical limit, an asymptotic preserving approximation of the von Neumann equation. More precisely, it relies on the use of so-called Weyl's variables to effectively address the stiffness associated to the equation. Then by employing a truncated Hermite expansion of the density operator, we successfully manage this stiffness and provide error estimates by leveraging the propagation of regularity in the exact solution.

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arXiv2025

A structure and asymptotic preserving scheme for the quasineutral limit of the Vlasov-Poisson system

In this work, we propose a new numerical method for the Vlasov-Poisson system that is both asymptotically consistent and stable in the quasineutral regime, i.e. when the Debye length is small compared to the characteristic spatial scale of the physical domain. Our approach consists in reformulating the Vlasov-Poisson system as a hyperbolic problem by applying a spectral expansion in the basis of Hermite functions in the velocity space and in designing a structure-preserving scheme for the time and spatial variables. Through this Hermite formulation, we establish a convergence result for the electric field toward its quasineutral limit together with optimal error estimates. Following this path, we then propose a fully discrete numerical method for the Vlasov-Poisson system, inspired by the approach in arXiv:2306.14605 , and rigorously prove that it is uniformly consistent in the quasineutral limit regime. Finally, we present several numerical simulations to illustrate the behavior of the proposed scheme. These results demonstrate the capability of our method to describe quasineutral plasmas and confirm the theoretical findings: stability and asymptotic preservation.

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arXiv2024

On the approximation of the von Neumann equation in the semi-classical limit. Part I : numerical algorithm

We propose a new approach to discretize the von Neumann equation, which is efficient in the semi-classical limit. This method is first based on the so called Weyl's variables to address the stiffness associated with the equation. Then, by applying a truncated Hermite expansion of the density operator, we successfully handle this stiffness. Additionally, we develop a finite volume approximation for practical implementation and conduct numerical simulations to illustrate the efficiency of our approach. This asymptotic preserving numerical approximation, combined with the use of Hermite polynomials, provides an efficient tool for solving the von Neumann equation in all regimes, near classical or not.

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arXiv2023

Trend to equilibrium and diffusion limit for the inertial Kuramoto-Sakaguchi equation

In this paper, we study the inertial Kuramoto-Sakaguchi equation for interacting oscillatory systems. On the one hand, we prove the convergence toward corresponding phase-homogeneous stationary states in weighted Lebesgue norm sense when the coupling strength is small enough. In [10], it is proved that when the noise intensity is sufficiently large, equilibrium of the inertial Kuramoto-Sakaguchi equation is asymptotically stable. For generic initial data, every solutions converges to equilibrium in weighted Sobolev norm sense. We improve this previous result by showing the convergence for a larger class of functions and by providing a simpler proof. On the other hand, we investigate the diffusion limit when all oscillators are identical. In [19], authors studied the same problem using an energy estimate on renormalized solutions and a compactness method, through which error estimates could not be discussed. Here we provide error estimates for the diffusion limit with respect to the mass m $\ll$ 1 using a simple proof by imposing slightly more regularity on the solution.

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arXiv2023

A structure and asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck model

We propose a numerical method for the Vlasov-Poisson-Fokker-Planck model written as an hyperbolic system thanks to a spectral decomposition in the basis of Hermite functions with respect to the velocity variable and a structure preserving finite volume scheme for the space variable. On the one hand, we show that this scheme naturally preserves both stationary solutions and linearized free-energy estimate. On the other hand, we adapt previous arguments based on hypocoercivity methods to get quantitative estimates ensuring the exponential relaxation to equilibrium of the discrete solution for the linearized Vlasov-Poisson-Fokker-Planck system, uniformly with respect to both scaling and discretization parameters. Finally, we perform substantial numerical simulations for the nonlinear system to illustrate the efficiency of this approach for a large variety of collisional regimes (plasma echos for weakly collisional regimes and trend to equilibrium for collisional plasmas) and to highlight its robustness (unconditional stability, asymptotic preserving properties).

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arXiv2022

On a discrete framework of hypocoercivity for kinetic equations

We propose and study a fully discrete finite volume scheme for the Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally preserves the stationary solution and the weighted L 2 relative entropy. Then, we adapt the arguments developed in [12] based the hypocoercivity method to get quantitative estimates on the convergence to equilibrium of the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with respect to both the time variable and the scaling parameter at play.

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arXiv2022

Asymptotically preserving particle methods for strongly magnetizedplasmas in a torus

We propose and analyze a class of particle methods for the Vlasov equation with a strong external magnetic field in a torus configuration. In this regime, the time step can be subject to stability constraints related to the smallness of Larmor radius. To avoid this limitation, our approach is based on higher-order semi-implicit numerical schemes already validated on dissipative systems [3] and for magnetic fields pointing in a fixed direction [9, 10, 12]. It hinges on asymptotic insights gained in [11] at the continuous level. Thus, when the magnitude of the external magnetic field is large, this scheme provides a consistent approximation of the guiding-center system taking into account curvature and variation of the magnetic field. Finally, we carry out a theoretical proof of consistency and perform several numerical experiments that establish a solid validation of the method and its underlying concepts.

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arXiv2022

On the convergence of discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system

We prove the convergence of discontinuous Galerkin approximations for the Vlasov-Poisson system written as an hyperbolic system using Hermite polynomials in velocity. To obtain stability properties, we introduce a suitable weighted L 2 space, with a time dependent weight, and first prove global stability for the weighted L 2 norm and propagation of regularity. Then we prove error estimates between the numerical solution and the smooth solution to the Vlasov-Poisson system.

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arXiv2022

Fokker-planck Multi-species Equations In The Adiabatic Asymptotics

The main concern of the present paper is the study of the multi-scale dynamics of thermonuclear fusion plasmas via a multi-species Fokker-Planck kinetic model. One of the goals is the generalization of the standard Fokker-Planck collision operator to a multispecies one, conserving mass, total momentum and energy, as well as satisfying Boltzmann's H-theorem. Secondly, the paper investigates in more details the reduced model used for the electron description in present simulations, and which considers the electrons in a thermodynamic equilibrium (adiabatic regime), whereas the ions are kept kinetic. On the one hand, we perform some mathematical asymptotic limits to obtain in the electron/ion low mass ratio limit the above-mentioned electron adiabatic regime. On the other hand, we develop a numerical scheme , based on a Hermite spectral method, and perfom numerical simulations to illustrate and investigate in more details this asymptotics.

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arXiv2022

Concentration phenomena in Fitzhugh-Nagumo's equations: A mesoscopic approach

We consider a spatially extended mesoscopic FitzHugh-Nagumo model with strong local interactions and prove that its asymptotic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. As the local interactions strongly dominate, the weak solution to the mesoscopic equation under consideration converges to the local equilibrium, which has the form of Dirac distribution concentrated to an averaged membrane potential. Our approach is based on techniques widely developed in kinetic theory (Wasserstein distance, relative entropy method), where macroscopic quantities of the mesoscopic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigorous link between microscopic and reaction-diffusion models.

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arXiv2021

On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system

We study a class of spatial discretizations for the Vlasov-Poisson system written as an hyperbolic system using Hermite polynomials. In particular, we focus on spectral methods and discontinuous Galerkin approximations. To obtain L 2 stability properties, we introduce a new L 2 weighted space, with a time dependent weight. For the Hermite spectral form of the Vlasov-Poisson system, we prove conservation of mass, momentum and total energy, as well as global stability for the weighted L 2 norm. These properties are then discussed for several spatial discretizations. Finally, numerical simulations are performed with the proposed DG/Hermite spectral method to highlight its stability and conservation features.

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arXiv2020

Asymptotic preserving schemes for the FitzHugh-Nagumo transport equation with strong local interactions

This paper is devoted to the numerical approximation of the spatially-extended FitzHugh-Nagumo transport equation with strong local interactions based on a particle method. In this regime, the time step can be subject to stability constraints related to the interaction kernel. To avoid this limitation, our approach is based on higher-order implicit-explicit numerical schemes. Thus, when the magnitude of the interactions becomes large, this method provides a consistent discretization of the macroscopic reaction-diffusion FitzHugh-Nagumo system. We carry out some theoretical proofs and perform several numerical experiments that establish a solid validation of the method and its underlying concepts.

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arXiv2020

A spectral collocation method for the Landau equation in plasma physics

In this paper we present a spectral collocation method for the fast evaluation of the Landau collision operator for plasma physics, which allows us to obtain spectrally accurate numerical solutions. The method is inspired by the seminal work [36], but it is specifically designed for Coulombian interactions, taking into account the particular structure of the operator. It allows us to reduce the number of discrete convolutions to provide an approximation of the Landau operator. Then, we show that the method preserves the total mass whereas momentum and energy are approximated with spectral accuracy. Numerical results for the Landau equation in three dimensions in velocity space are presented to illustrate the efficiency of the present approach.

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arXiv2020

Convergence of a Finite Volume Scheme for a System of Interacting Species with Cross-Diffusion

In this work we present the convergence of a positivity preserving semi-discrete finite volume scheme for a coupled system of two non-local partial differential equations with cross-diffusion. The key to proving the convergence result is to establish positivity in order to obtain a discrete energy estimate to obtain compactness. We numerically observe the convergence to reference solutions with a first order accuracy in space. Moreover we recover segregated stationary states in spite of the regularising effect of the self-diffusion. However, if the self-diffusion or the cross-diffusion is strong enough, mixing occurs while both densities remain continuous.

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