Numerical Approximation of the One-Dimensional Vlasov–Poisson System with Periodic Boundary Conditions

Wollman, Stephen; Ozizmir, Ercument

doi:10.1137/s0036142993233585

Cited by 19 publications

(29 citation statements)

References 23 publications

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“…These figures point out that the approximated solution is very sensitive to the choice of α. It has to be noted, however, that the parameter α that well performs for the initial datum (59) does not necessarily behave optimally as time increases. For example, the value α = 1.8 looks the best choice for the approximation of the initial distribution (see Figure 2 on bottom), nevertheless, keeping α constantly equal to such a value brings to instability before arriving at time T = 30.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Semi-Lagrangian methods have been proposed in different frameworks such as the Finite Volume method [26,4], Discontinuous Galerkin method [2,3,39], finite difference methods based on ENO and WENO polynomial reconstructions [21], as well as in the propagation of solutions along the characteristics in an operator splitting context [1,17,20,28,27,52,23]. These methods offer an alternative to Particle-in-Cell (PIC) methods [22,58,59,12,18,19,42,43,47,53] and to the so called Transform methods based on spectral approximations [45,48,41,57,56]. PIC methods are very popular in the plasma physics community and are the most widely used methods because of their robustness and relative simplicity [6].…”

Section: Introductionmentioning

confidence: 99%

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A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials

Fatone

Funaro

Manzini

2019

Commun. Appl. Math. Comput.

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In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF time-stepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on the standard two-stream instability benchmark problem. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.

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Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials

Fatone

Funaro

Manzini

2019

Commun. Appl. Math. Comput.

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“…The right-hand side of the Poisson equation depends on the charges carried by the macro-particles. The convergence of the PIC method for the Vlasov-Poisson system was proved in [21,49,50]. The PIC method has been successfully used to simulate the behavior of collisionless laboratory and space plasmas and provides excellent results for the modeling of large scale phenomena in one, two or three space dimensions [8].…”

Section: Introductionmentioning

confidence: 99%

Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov–Poisson system

Fatone

Funaro

Manzini

2019

Journal of Computational Physics

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“…Cottet and Raviart [5] present a precise mathematical analysis of the particle method for solving the one-dimensional Vlasov-Poisson system. We also mention the papers of Wollman and Ozizmir [19] and Wollman [20] on the topic. Ganguly and Victory give a convergence result for the Vlasov-Maxwell system [10].…”

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confidence: 99%

Convergence of a Finite Volume Scheme for the Vlasov--Poisson System

Filbet¹

2001

SIAM J. Numer. Anal.

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We propose a finite volume scheme to discretize the one-dimensional Vlasov-Poisson system. We prove that, if the initial data are positive, bounded, continuous, and have their first moment bounded, then the numerical approximation converges to the weak solution of the system for the weak topology of L ∞ . Moreover, if the initial data belong to BV , the convergence is strong in C 0 (0, T ; L 1 loc ). To prove the convergence of the discrete electric field, we obtain an estimation in W 1,∞ (Ω T ). Then we havewhere (E, f ) is the unique weak solution of the Vlasov-Poisson system. Introduction. The Vlasov-Poisson system is a model for the motion of a collisionless plasma of electrons in a uniform background of ions and describes the evolution of the distribution function of electrons (solution of the Vlasov equation)under the effects of the transport and self-consistent electric field (solution of the Poisson equation). The coupling between both equations gives a nonlinear problem.The numerical resolution of the Vlasov equation is most often performed using particle methods (PIC), which consist of approximating the plasma by a finite number of particles. The trajectories of these particles are computed from the characteristic curves given by the Vlasov equation. The interactions with self-consistent and external fields are computed by a numerical method using a mesh of the physical space (see, e.g., Birdsall and Langdon [2] or Cottet and Raviart [5]). This method enables us to get satisfying results with few particles.Methods relying on a discretization of the phase space have also been proposed (see, e.g., Shoucri and Knorr [15] and Klimas and Farrell [11]) and seem to be more efficient in some cases, for example, when particles in the tail of the distribution play an important physical role, or when the numerical noise due to the finite number of particles becomes too important. Among them, the semi-Lagrangian method [16] consists of directly computing the distribution function on a grid of the phase space. This computation is done by following the characteristic curves at each time step and interpolating the value at the origin of the characteristics by a cubic spline method.This interpolation method works well for simple geometries of the physical space but does not seem to be well suited to more complex geometries.To remedy this problem a possible approach is to use the finite volume method which is known to be a robust and computationally cheap method for the discretization of conservation laws (see, e.g., Eymard, Gallouet, and Herbin [9] and the references

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Numerical Approximation of the One-Dimensional Vlasov–Poisson System with Periodic Boundary Conditions

Cited by 19 publications

References 23 publications

A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials

A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials

Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov–Poisson system

Convergence of a Finite Volume Scheme for the Vlasov--Poisson System

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