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

Hakim, Ammar; Francisquez, Manaure; Juno, James; Hammett, Gregory W.

doi:10.1017/s0022377820000586

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

Journal Article · Fri Jul 17 00:00:00 EDT 2020 · Journal of Plasma Physics

DOI:https://doi.org/10.1017/s0022377820000586· OSTI ID:1668773

^[1]; ^[2]; ^[3]; Hammett, Gregory W. ^[1]

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)

Here, 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.

View Accepted Manuscript (DOE)

Research Organization:: Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)

Sponsoring Organization:: USDOE Office of Science (SC); US Air Force Office of Scientific Research (AFOSR); National Aeronautic and Space Administration (NASA); National Science Foundation (NSF)

Grant/Contract Number:: AC02-06CH11357; FG02-91ER54109; AC02-09CH11466; SC0010508; FC02-08ER54966

OSTI ID:: 1668773

Journal Information:: Journal of Plasma Physics, Journal Name: Journal of Plasma Physics Journal Issue: 4 Vol. 86; ISSN 0022-3778

Publisher:: Cambridge University PressCopyright Statement

Country of Publication:: United States

Language:: English

References (31)

Progress with the COGENT Edge Kinetic Code: Collision Operator Options Dorf, M. A.; Cohen, R. H.; Compton, J. C. Contributions to Plasma Physics, Vol. 52, Issue 5-6 https://doi.org/10.1002/ctpp.201210042	journal	June 2012
The Mathematical Theory of Finite Element Methods Brenner, Susanne C.; Scott, L. Ridgway Texts in Applied Mathematics https://doi.org/10.1007/978-0-387-75934-0	book	January 2008
The Serendipity Family of Finite Elements Arnold, Douglas N.; Awanou, Gerard Foundations of Computational Mathematics, Vol. 11, Issue 3 https://doi.org/10.1007/s10208-011-9087-3	journal	March 2011
FPPAC: A two-dimensional multispecies nonlinear Fokker-Planck package McCoy, M. G.; Mirin, A. A.; Killeen, J. Computer Physics Communications, Vol. 24, Issue 1 https://doi.org/10.1016/0010-4655(81)90105-3	journal	September 1981
Conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation Idomura, Yasuhiro; Ida, Masato; Kano, Takuma Computer Physics Communications, Vol. 179, Issue 6 https://doi.org/10.1016/j.cpc.2008.04.005	journal	September 2008
Improved collision operator for plasma kinetic simulations with multi-species ions and electrons Nakata, Motoki; Nunami, Masanori; Watanabe, Tomo-Hiko Computer Physics Communications, Vol. 197 https://doi.org/10.1016/j.cpc.2015.08.007	journal	December 2015
A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation Taitano, W. T.; Chacón, L.; Simakov, A. N. Journal of Computational Physics, Vol. 297 https://doi.org/10.1016/j.jcp.2015.05.025	journal	September 2015
A fully non-linear multi-species Fokker–Planck–Landau collision operator for simulation of fusion plasma Hager, Robert; Yoon, E. S.; Ku, S. Journal of Computational Physics, Vol. 315 https://doi.org/10.1016/j.jcp.2016.03.064	journal	June 2016
Discontinuous Galerkin algorithms for fully kinetic plasmas Juno, J.; Hakim, A.; TenBarge, J. Journal of Computational Physics, Vol. 353 https://doi.org/10.1016/j.jcp.2017.10.009	journal	January 2018
Collisional relaxation: Landau versus Dougherty operator Pezzi, Oreste; Valentini, F.; Veltri, P. Journal of Plasma Physics, Vol. 81, Issue 1 https://doi.org/10.1017/S0022377814000877	journal	October 2014
Gyrokinetic continuum simulation of turbulence in a straight open-field-line plasma Shi, E. L.; Hammett, G. W.; Stoltzfus-Dueck, T. Journal of Plasma Physics, Vol. 83, Issue 3 https://doi.org/10.1017/S002237781700037X	journal	May 2017
A drift-kinetic analytical model for scrape-off layer plasma dynamics at arbitrary collisionality Jorge, R.; Ricci, P.; Loureiro, N. F. Journal of Plasma Physics, Vol. 83, Issue 6 https://doi.org/10.1017/S002237781700085X	journal	November 2017
Effect of a weak ion collisionality on the dynamics of kinetic electrostatic shocks Sundström, Andréas; Juno, James; TenBarge, Jason M. Journal of Plasma Physics, Vol. 85, Issue 1 https://doi.org/10.1017/S0022377819000023	journal	February 2019
Improved Bhatnagar-Gross-Krook model of electron-ion collisions Greene, John M. Physics of Fluids, Vol. 16, Issue 11 https://doi.org/10.1063/1.1694254	journal	January 1973
Heating of a Confined Plasma by Oscillating Electromagnetic Fields Berger, J. M.; Newcomb, W. A.; Dawson, J. M. Physics of Fluids, Vol. 1, Issue 4 https://doi.org/10.1063/1.1705888	journal	January 1958
Transition between Landau and Van Kampen Treatments of the Vlasov Equation Grant, Frederick C. Physics of Fluids, Vol. 10, Issue 6 https://doi.org/10.1063/1.1762288	journal	January 1967
Eigenfunctions and eigenvalues of the Dougherty collision operator Anderson, M. W.; O’Neil, T. M. Physics of Plasmas, Vol. 14, Issue 5 https://doi.org/10.1063/1.2727463	journal	May 2007
Model Fokker-Planck Equation for a Plasma and Its Solution Dougherty, J. P. Physics of Fluids, Vol. 7, Issue 11 https://doi.org/10.1063/1.2746779	journal	January 1964
Linearized model Fokker–Planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests Barnes, M.; Abel, I. G.; Dorland, W. Physics of Plasmas, Vol. 16, Issue 7 https://doi.org/10.1063/1.3155085	journal	July 2009
Conservative discretization of the Landau collision integral Hirvijoki, E.; Adams, M. F. Physics of Plasmas, Vol. 24, Issue 3 https://doi.org/10.1063/1.4979122	journal	March 2017
Full- f gyrokinetic simulation of turbulence in a helical open-field-line plasma Shi, E. L.; Hammett, G. W.; Stoltzfus-Dueck, T. Physics of Plasmas, Vol. 26, Issue 1 https://doi.org/10.1063/1.5074179	journal	January 2019
Gyrokinetic continuum simulations of plasma turbulence in the Texas Helimak Bernard, T. N.; Shi, E. L.; Gentle, K. W. Physics of Plasmas, Vol. 26, Issue 4 https://doi.org/10.1063/1.5085457	journal	April 2019
Theory of first‐order plasma heating by collisional magnetic pumping Laroussi, M.; Roth, J. Reece Physics of Fluids B: Plasma Physics, Vol. 1, Issue 5 https://doi.org/10.1063/1.859025	journal	May 1989
Plasma Instabilities and Magnetic Field Growth in Clusters of Galaxies Schekochihin, A. A.; Cowley, S. C.; Kulsrud, R. M. The Astrophysical Journal, Vol. 629, Issue 1 https://doi.org/10.1086/431202	journal	August 2005
Developments in the gyrofluid approach to Tokamak turbulence simulations Hammett, G. W.; Beer, M. A.; Dorland, W. Plasma Physics and Controlled Fusion, Vol. 35, Issue 8 https://doi.org/10.1088/0741-3335/35/8/006	journal	August 1993
Implications of advanced collision operators for gyrokinetic simulation Belli, E. A.; Candy, J. Plasma Physics and Controlled Fusion, Vol. 59, Issue 4 https://doi.org/10.1088/1361-6587/aa5c94	journal	February 2017
Fokker-Planck Equation for an Inverse-Square Force Rosenbluth, Marshall N.; MacDonald, William M.; Judd, David L. Physical Review, Vol. 107, Issue 1 https://doi.org/10.1103/PhysRev.107.1	journal	July 1957
Plasma Oscillations with Diffusion in Velocity Space Lenard, A.; Bernstein, Ira B. Physical Review, Vol. 112, Issue 5 https://doi.org/10.1103/PhysRev.112.1456	journal	December 1958
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems Cockburn, Bernardo; Shu, Chi-Wang SIAM Journal on Numerical Analysis, Vol. 35, Issue 6 https://doi.org/10.1137/S0036142997316712	journal	December 1998
Plasma Physics for Astrophysics Kulsrud, Russell M. Princeton Series in Astrophysics, Vol. 66 https://doi.org/10.1515/9780691213354	journal	January 2005
Magnetic Pumping as a Source of Particle Heating and Power-law Distributions in the Solar Wind Lichko, E.; Egedal, J.; Daughton, W. The Astrophysical Journal, Vol. 850, Issue 2 https://doi.org/10.3847/2041-8213/aa9a33	journal	November 2017

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