skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Conservative discontinuous Galerkin schemes for nonlinear Dougherty–Fokker–Planck collision operators

Abstract

In this paper, we present a novel discontinuous Galerkin algorithm for the solution of a class of Fokker–Planck collision operators. These operators arise in many fields of physics, and our particular application is for kinetic plasma simulations. In particular, we focus on an operator often known as the ‘Lenard–Bernstein’ or ‘Dougherty’ operator. Several novel algorithmic innovations, based on the concept of weak equality, are reported. These weak equalities are used to define weak operators that compute primitive moments, and are also used to determine a reconstruction procedure that allows an efficient and accurate discretization of the diffusion term. We show that when two integrations by parts are used to construct the discrete weak form, and finite velocity-space extents are accounted for, a scheme that conserves density, momentum and energy exactly is obtained. One novel feature is that the requirements of momentum and energy conservation lead to unique formulas to compute primitive moments. Careful definition of discretized moments also ensure that energy is conserved in the piecewise linear case, even though the kinetic-energy term, $$v^{2}$$ is not included in the basis set used in the discretization. A series of benchmark problems is presented and shows that the scheme conserves momentum and energy to machine precision. Empirical evidence also indicates that entropy is a non-decreasing function. The collision terms are combined with the Vlasov equation to study collisional Landau damping and plasma heating via magnetic pumping.

Authors:
ORCiD logo [1]; ORCiD logo [2]; ORCiD logo [3];  [1]
  1. Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
  2. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States). Plasma Science and Fusion Center; Dartmouth College, Hanover, NH (United States)
  3. Univ. of Maryland, College Park, MD (United States)
Publication Date:
Research Org.:
Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
Sponsoring Org.:
USDOE Office of Science (SC); US Air Force Office of Scientific Research (AFOSR); National Aeronautic and Space Administration (NASA); National Science Foundation (NSF)
OSTI Identifier:
1668773
Grant/Contract Number:  
AC02-06CH11357; FG02-91-ER54109; AC02-09CH11466; SC-0010508; FC02-08ER54966; 80NSSC17K0428; ACI-1548562; FA9550-15-1-0193
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of Plasma Physics
Additional Journal Information:
Journal Volume: 86; Journal Issue: 4; Journal ID: ISSN 0022-3778
Publisher:
Cambridge University Press
Country of Publication:
United States
Language:
English
Subject:
plasma dynamics; plasma nonlinear phenomena; plasma simulation

Citation Formats

Hakim, Ammar, Francisquez, Manaure, Juno, James, and Hammett, Gregory W. Conservative discontinuous Galerkin schemes for nonlinear Dougherty–Fokker–Planck collision operators. United States: N. p., 2020. Web. doi:10.1017/s0022377820000586.
Hakim, Ammar, Francisquez, Manaure, Juno, James, & Hammett, Gregory W. Conservative discontinuous Galerkin schemes for nonlinear Dougherty–Fokker–Planck collision operators. United States. doi:10.1017/s0022377820000586.
Hakim, Ammar, Francisquez, Manaure, Juno, James, and Hammett, Gregory W. Fri . "Conservative discontinuous Galerkin schemes for nonlinear Dougherty–Fokker–Planck collision operators". United States. doi:10.1017/s0022377820000586.
@article{osti_1668773,
title = {Conservative discontinuous Galerkin schemes for nonlinear Dougherty–Fokker–Planck collision operators},
author = {Hakim, Ammar and Francisquez, Manaure and Juno, James and Hammett, Gregory W.},
abstractNote = {In this paper, we present a novel discontinuous Galerkin algorithm for the solution of a class of Fokker–Planck collision operators. These operators arise in many fields of physics, and our particular application is for kinetic plasma simulations. In particular, we focus on an operator often known as the ‘Lenard–Bernstein’ or ‘Dougherty’ operator. Several novel algorithmic innovations, based on the concept of weak equality, are reported. These weak equalities are used to define weak operators that compute primitive moments, and are also used to determine a reconstruction procedure that allows an efficient and accurate discretization of the diffusion term. We show that when two integrations by parts are used to construct the discrete weak form, and finite velocity-space extents are accounted for, a scheme that conserves density, momentum and energy exactly is obtained. One novel feature is that the requirements of momentum and energy conservation lead to unique formulas to compute primitive moments. Careful definition of discretized moments also ensure that energy is conserved in the piecewise linear case, even though the kinetic-energy term, $v^{2}$ is not included in the basis set used in the discretization. A series of benchmark problems is presented and shows that the scheme conserves momentum and energy to machine precision. Empirical evidence also indicates that entropy is a non-decreasing function. The collision terms are combined with the Vlasov equation to study collisional Landau damping and plasma heating via magnetic pumping.},
doi = {10.1017/s0022377820000586},
journal = {Journal of Plasma Physics},
issn = {0022-3778},
number = 4,
volume = 86,
place = {United States},
year = {2020},
month = {7}
}

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on July 17, 2021
Publisher's Version of Record

Save / Share:

Works referenced in this record:

Discontinuous Galerkin algorithms for fully kinetic plasmas
journal, January 2018


Implications of advanced collision operators for gyrokinetic simulation
journal, February 2017


Progress with the COGENT Edge Kinetic Code: Collision Operator Options
journal, June 2012

  • Dorf, M. A.; Cohen, R. H.; Compton, J. C.
  • Contributions to Plasma Physics, Vol. 52, Issue 5-6
  • DOI: 10.1002/ctpp.201210042

Model Fokker-Planck Equation for a Plasma and Its Solution
journal, January 1964


Effect of a weak ion collisionality on the dynamics of kinetic electrostatic shocks
journal, February 2019

  • Sundström, Andréas; Juno, James; TenBarge, Jason M.
  • Journal of Plasma Physics, Vol. 85, Issue 1
  • DOI: 10.1017/S0022377819000023

FPPAC: A two-dimensional multispecies nonlinear Fokker-Planck package
journal, September 1981


Improved Bhatnagar-Gross-Krook model of electron-ion collisions
journal, January 1973


Gyrokinetic continuum simulations of plasma turbulence in the Texas Helimak
journal, April 2019

  • Bernard, T. N.; Shi, E. L.; Gentle, K. W.
  • Physics of Plasmas, Vol. 26, Issue 4
  • DOI: 10.1063/1.5085457

A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation
journal, September 2015


Improved collision operator for plasma kinetic simulations with multi-species ions and electrons
journal, December 2015

  • Nakata, Motoki; Nunami, Masanori; Watanabe, Tomo-Hiko
  • Computer Physics Communications, Vol. 197
  • DOI: 10.1016/j.cpc.2015.08.007

Gyrokinetic continuum simulation of turbulence in a straight open-field-line plasma
journal, May 2017


Eigenfunctions and eigenvalues of the Dougherty collision operator
journal, May 2007

  • Anderson, M. W.; O’Neil, T. M.
  • Physics of Plasmas, Vol. 14, Issue 5
  • DOI: 10.1063/1.2727463

Heating of a Confined Plasma by Oscillating Electromagnetic Fields
journal, January 1958

  • Berger, J. M.; Newcomb, W. A.; Dawson, J. M.
  • Physics of Fluids, Vol. 1, Issue 4
  • DOI: 10.1063/1.1705888

Conservative discretization of the Landau collision integral
journal, March 2017

  • Hirvijoki, E.; Adams, M. F.
  • Physics of Plasmas, Vol. 24, Issue 3
  • DOI: 10.1063/1.4979122

Magnetic Pumping as a Source of Particle Heating and Power-law Distributions in the Solar Wind
journal, November 2017


Fokker-Planck Equation for an Inverse-Square Force
journal, July 1957

  • Rosenbluth, Marshall N.; MacDonald, William M.; Judd, David L.
  • Physical Review, Vol. 107, Issue 1
  • DOI: 10.1103/PhysRev.107.1

Transition between Landau and Van Kampen Treatments of the Vlasov Equation
journal, January 1967


The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
journal, December 1998


Conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation
journal, September 2008

  • Idomura, Yasuhiro; Ida, Masato; Kano, Takuma
  • Computer Physics Communications, Vol. 179, Issue 6
  • DOI: 10.1016/j.cpc.2008.04.005

Full- f gyrokinetic simulation of turbulence in a helical open-field-line plasma
journal, January 2019

  • Shi, E. L.; Hammett, G. W.; Stoltzfus-Dueck, T.
  • Physics of Plasmas, Vol. 26, Issue 1
  • DOI: 10.1063/1.5074179

The Serendipity Family of Finite Elements
journal, March 2011

  • Arnold, Douglas N.; Awanou, Gerard
  • Foundations of Computational Mathematics, Vol. 11, Issue 3
  • DOI: 10.1007/s10208-011-9087-3

A fully non-linear multi-species Fokker–Planck–Landau collision operator for simulation of fusion plasma
journal, June 2016


A drift-kinetic analytical model for scrape-off layer plasma dynamics at arbitrary collisionality
journal, November 2017


Theory of first‐order plasma heating by collisional magnetic pumping
journal, May 1989

  • Laroussi, M.; Roth, J. Reece
  • Physics of Fluids B: Plasma Physics, Vol. 1, Issue 5
  • DOI: 10.1063/1.859025

Collisional relaxation: Landau versus Dougherty operator
journal, October 2014


Plasma Instabilities and Magnetic Field Growth in Clusters of Galaxies
journal, August 2005

  • Schekochihin, A. A.; Cowley, S. C.; Kulsrud, R. M.
  • The Astrophysical Journal, Vol. 629, Issue 1
  • DOI: 10.1086/431202

Plasma Oscillations with Diffusion in Velocity Space
journal, December 1958


Developments in the gyrofluid approach to Tokamak turbulence simulations
journal, August 1993


Linearized model Fokker–Planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests
journal, July 2009

  • Barnes, M.; Abel, I. G.; Dorland, W.
  • Physics of Plasmas, Vol. 16, Issue 7
  • DOI: 10.1063/1.3155085