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 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 conservationmore »
- 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 Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1331266
- Alternate Identifier(s):
- OSTI ID: 1347624
- Report Number(s):
- LA-UR-15-27359
Journal ID: ISSN 0021-9991
- Grant/Contract Number:
- AC52-06NA25396
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 317; Journal Issue: C; Journal ID: ISSN 0021-9991
- 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. https://doi.org/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. https://doi.org/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 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.},
doi = {10.1016/j.jcp.2016.03.069},
journal = {Journal of Computational Physics},
number = C,
volume = 317,
place = {United States},
year = {Fri Apr 22 00:00:00 EDT 2016},
month = {Fri Apr 22 00:00:00 EDT 2016}
}
Web of Science
Works referenced in this record:
On the velocity space discretization for the Vlasov–Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods
journal, January 2016
- Camporeale, E.; Delzanno, G. L.; Bergen, B. K.
- Computer Physics Communications, Vol. 198
Nonlinear Effects from Vlasov's Equation
journal, January 1963
- Engelmann, F.; Feix, M.; Minardi, E.
- Physics of Fluids, Vol. 6, Issue 2
Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form
journal, November 2015
- Delzanno, G. L.
- Journal of Computational Physics, Vol. 301
On the velocity space discretization for the Vlasov–Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods
journal, January 2016
- Camporeale, E.; Delzanno, G. L.; Bergen, B. K.
- Computer Physics Communications, Vol. 198
New approach for the study of linear Vlasov stability of inhomogeneous systems
journal, September 2006
- Camporeale, Enrico; Delzanno, Gian Luca; Lapenta, Giovanni
- Physics of Plasmas, Vol. 13, Issue 9
Time-Stable Boundary Conditions for Finite-Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-Order Compact Schemes
journal, April 1994
- Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul
- Journal of Computational Physics, Vol. 111, Issue 2
A Stable and Conservative Interface Treatment of Arbitrary Spatial Accuracy
journal, January 1999
- Carpenter, Mark H.; Nordström, Jan; Gottlieb, David
- Journal of Computational Physics, Vol. 148, Issue 2
The integration of the vlasov equation in configuration space
journal, November 1976
- Cheng, C. Z.; Knorr, Georg
- Journal of Computational Physics, Vol. 22, Issue 3
A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type
journal, January 1947
- Crank, J.; Nicolson, P.
- Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 43, Issue 1
Two-way coupling of a global Hall magnetohydrodynamics model with a local implicit particle-in-cell model
journal, July 2014
- Daldorff, Lars K. S.; Tóth, Gábor; Gombosi, Tamas I.
- Journal of Computational Physics, Vol. 268
Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form
journal, November 2015
- Delzanno, G. L.
- Journal of Computational Physics, Vol. 301
Nonlinear Effects from Vlasov's Equation
journal, January 1963
- Engelmann, F.; Feix, M.; Minardi, E.
- Physics of Fluids, Vol. 6, Issue 2
Conservative Numerical Schemes for the Vlasov Equation
journal, September 2001
- Filbet, Francis; Sonnendrücker, Eric; Bertrand, Pierre
- Journal of Computational Physics, Vol. 172, Issue 1
Spectral velocity discretizations for the Vlasov-Maxwell equations
journal, January 1996
- Holloway, James Paul
- Transport Theory and Statistical Physics, Vol. 25, Issue 1
A numerical method based on the Fourier-Fourier transform approach for modeling 1-D electron plasma evolution
journal, May 1983
- Klimas, Alexander J.
- Journal of Computational Physics, Vol. 50, Issue 2
Jacobian-free Newton–Krylov methods: a survey of approaches and applications
journal, January 2004
- Knoll, D. A.; Keyes, D. E.
- Journal of Computational Physics, Vol. 193, Issue 2
Fast Collisionless Reconnection and Electron Heating in Strongly Magnetized Plasmas
journal, July 2013
- Loureiro, N. F.; Schekochihin, A. A.; Zocco, A.
- Physical Review Letters, Vol. 111, Issue 2
The Fluid-Kinetic Particle-in-Cell method for plasma simulations
journal, August 2014
- Markidis, Stefano; Henri, Pierre; Lapenta, Giovanni
- Journal of Computational Physics, Vol. 271
Summation by parts operators for finite difference approximations of second derivatives
journal, September 2004
- Mattsson, Ken; Nordström, Jan
- Journal of Computational Physics, Vol. 199, Issue 2
Finite volume methods, unstructured meshes and strict stability for hyperbolic problems
journal, June 2003
- Nordström, Jan; Forsberg, Karl; Adamsson, Carl
- Applied Numerical Mathematics, Vol. 45, Issue 4
Fourier–Hermite spectral representation for the Vlasov–Poisson system in the weakly collisional limit
journal, February 2015
- Parker, Joseph T.; Dellar, Paul J.
- Journal of Plasma Physics, Vol. 81, Issue 2
A Review of Neutron Transport Approximations
journal, April 1982
- Sanchez, R.; McCormick, N. J.
- Nuclear Science and Engineering, Vol. 80, Issue 4
Vlasov Simulations Using Velocity-Scaled Hermite Representations
journal, August 1998
- Schumer, Joseph W.; Holloway, James Paul
- Journal of Computational Physics, Vol. 144, Issue 2
The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation
journal, March 1999
- Sonnendrücker, Eric; Roche, Jean; Bertrand, Pierre
- Journal of Computational Physics, Vol. 149, Issue 2
Summation by Parts for Finite Difference Approximations for d/dx
journal, January 1994
- Strand, Bo
- Journal of Computational Physics, Vol. 110, Issue 1
Review of summation-by-parts schemes for initial–boundary-value problems
journal, July 2014
- Svärd, Magnus; Nordström, Jan
- Journal of Computational Physics, Vol. 268
Spectral Solver for Multi-scale Plasma Physics Simulations with Dynamically Adaptive Number of Moments
journal, January 2015
- Vencels, Juris; Delzanno, Gian Luca; Johnson, Alec
- Procedia Computer Science, Vol. 51
SpectralPlasmaSolver: a Spectral Code for Multiscale Simulations of Collisionless, Magnetized Plasmas
journal, May 2016
- Vencels, Juris; Delzanno, Gian Luca; Manzini, Gianmarco
- Journal of Physics: Conference Series, Vol. 719
Existence and uniqueness theory of the Vlasov-Poisson system with application to the problem with cylindrical symmetry
journal, November 1982
- Wollman, Stephen
- Journal of Mathematical Analysis and Applications, Vol. 90, Issue 1
Works referencing / citing this record:
Convergence of spectral discretizations of the Vlasov-Poisson system
text, January 2016
- Manzini, Gianmarco; Funaro, Daniele; Delzanno, Gian Luca
- arXiv
Annotations on the virtual element method for second-order elliptic problems
preprint, January 2016
- Manzini, Gianmarco
- arXiv
Quadratic conservative scheme for relativistic Vlasov--Maxwell system
text, January 2018
- Shiroto, Takashi; Ohnishi, Naofumi; Sentoku, Yasuhiko
- arXiv
A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials
journal, May 2019
- Fatone, Lorella; Funaro, Daniele; Manzini, Gianmarco
- Communications on Applied Mathematics and Computation, Vol. 1, Issue 3
On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system
text, January 2021
- Bessemoulin-Chatard, Marianne; Filbet, Francis
- arXiv
Conservative discontinuous Galerkin/Hermite Spectral Method for the Vlasov-Poisson System
preprint, January 2020
- Filbet, Francis; Xiong, Tao
- arXiv