A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system

Manzini, Gianmarco; Delzanno, Gian Luca; Vencels, Juris; Markidis, Stefano

doi:10.1016/j.jcp.2016.03.069

A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system

Journal Article · Fri Apr 22 00:00:00 EDT 2016 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2016.03.069· OSTI ID:1331266

Manzini, Gianmarco ^[1]; Delzanno, Gian Luca ^[2]; Vencels, Juris ^[2]; Markidis, Stefano ^[3]

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)

In this study, we present the design and implementation of an L²-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 second-order accurate Crank–Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a Jacobian-Free 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 semi-discrete and discrete setting. The L²-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 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.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: LDRD; USDOE; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-06NA25396

OSTI ID:: 1331266

Alternate ID(s):: OSTI ID: 1347624

Report Number(s):: LA-UR--15-27359

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 317; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials Fatone, Lorella; Funaro, Daniele; Manzini, Gianmarco Communications on Applied Mathematics and Computation, Vol. 1, Issue 3 https://doi.org/10.1007/s42967-019-00027-8	journal	May 2019
Convergence of spectral discretizations of the Vlasov-Poisson system Manzini, Gianmarco; Funaro, Daniele; Delzanno, Gian Luca arXiv https://doi.org/10.48550/arxiv.1612.08013	text	January 2016
Annotations on the virtual element method for second-order elliptic problems Manzini, Gianmarco arXiv https://doi.org/10.48550/arxiv.1612.09144	preprint	January 2016
Quadratic conservative scheme for relativistic Vlasov--Maxwell system Shiroto, Takashi; Ohnishi, Naofumi; Sentoku, Yasuhiko arXiv https://doi.org/10.48550/arxiv.1802.07238	text	January 2018
Conservative discontinuous Galerkin/Hermite Spectral Method for the Vlasov-Poisson System Filbet, Francis; Xiong, Tao arXiv https://doi.org/10.48550/arxiv.2004.02685	preprint	January 2020
On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system Bessemoulin-Chatard, Marianne; Filbet, Francis arXiv https://doi.org/10.48550/arxiv.2106.07468	text	January 2021

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