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 velocityspace 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 kineticenergy 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 nondecreasing function. The collision terms are combined with the Vlasov equation to study collisional Landau damping and plasma heating via magnetic pumping.
 Authors:

 Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
 Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States). Plasma Science and Fusion Center; Dartmouth College, Hanover, NH (United States)
 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:
 AC0206CH11357; FG0291ER54109; AC0209CH11466; SC0010508; FC0208ER54966; 80NSSC17K0428; ACI1548562; FA95501510193
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Journal of Plasma Physics
 Additional Journal Information:
 Journal Volume: 86; Journal Issue: 4; Journal ID: ISSN 00223778
 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 velocityspace 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 kineticenergy 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 nondecreasing 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 = {00223778},
number = 4,
volume = 86,
place = {United States},
year = {2020},
month = {7}
}
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