A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system
Abstract
In this study, we present the design and implementation of an L ^{2}stable spectral method for the discretization of the Vlasov–Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit secondorder accurate Crank–Nicolson time discretization. The nonlinear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a JacobianFree Newton–Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre–Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semidiscrete and discrete setting. The L ^{2}stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affectmore »
 Authors:

 Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Istituto di Matematica Applicata e Tecnologie Informatiche, Pavia (Italy)
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 KTH Royal Institute of Technology, Stockholm (Sweden)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE Laboratory Directed Research and Development (LDRD) Program
 OSTI Identifier:
 1331266
 Alternate Identifier(s):
 OSTI ID: 1347624
 Report Number(s):
 LAUR1527359
Journal ID: ISSN 00219991
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 317; Journal Issue: C; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; mathematics; magnetic fusion energy; Vlasov–Poisson; Legendre–Fourier discretization; conservation laws stability
Citation Formats
Manzini, Gianmarco, Delzanno, Gian Luca, Vencels, Juris, and Markidis, Stefano. A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system. United States: N. p., 2016.
Web. doi:10.1016/j.jcp.2016.03.069.
Manzini, Gianmarco, Delzanno, Gian Luca, Vencels, Juris, & Markidis, Stefano. A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system. United States. doi:10.1016/j.jcp.2016.03.069.
Manzini, Gianmarco, Delzanno, Gian Luca, Vencels, Juris, and Markidis, Stefano. Fri .
"A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system". United States. doi:10.1016/j.jcp.2016.03.069. https://www.osti.gov/servlets/purl/1331266.
@article{osti_1331266,
title = {A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system},
author = {Manzini, Gianmarco and Delzanno, Gian Luca and Vencels, Juris and Markidis, Stefano},
abstractNote = {In this study, we present the design and implementation of an L2stable spectral method for the discretization of the Vlasov–Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit secondorder accurate Crank–Nicolson time discretization. The nonlinear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a JacobianFree Newton–Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre–Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semidiscrete and discrete setting. The L2stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.},
doi = {10.1016/j.jcp.2016.03.069},
journal = {Journal of Computational Physics},
number = C,
volume = 317,
place = {United States},
year = {2016},
month = {4}
}
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