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

Published online by Cambridge University Press:  17 July 2020

Ammar Hakim Open the ORCID record for Ammar Hakim [Opens in a new window] ,
Manaure Francisquez Open the ORCID record for Manaure Francisquez [Opens in a new window] ,
James Juno Open the ORCID record for James Juno [Opens in a new window]  and
Gregory W. Hammett
Ammar Hakim*
Affiliation:
Princeton Plasma Physics Laboratory, Princeton, NJ 08543-0451, USA
Manaure Francisquez
Affiliation:
Princeton Plasma Physics Laboratory, Princeton, NJ 08543-0451, USA MIT Plasma Science and Fusion Center, Cambridge, MA 02139, USA
James Juno
Affiliation:
IREAP, University of Maryland, College Park, MD 20742, USA
Gregory W. Hammett
Affiliation:
Princeton Plasma Physics Laboratory, Princeton, NJ 08543-0451, USA
*
†Email address for correspondence: ahakim@pppl.gov
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Abstract

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.

Keywords

plasma dynamicsplasma nonlinear phenomenaplasma simulation
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 4 , August 2020 , 905860403
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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