Hostname: page-component-76d6cb85b7-jhrpq Total loading time: 0 Render date: 2026-07-13T05:13:33.843Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Anderson, M. W. & O’Neil, T. M. 2007 Eigenfunctions and eigenvalues of the Dougherty collision operator. Phys. Plasmas 14 (5), 052103.CrossRefGoogle Scholar
Arnold, D. N. & Awanou, G. 2011 The serendipity family of finite elements. Found. Comput. Math. 11 (3), 337–344.CrossRefGoogle Scholar
Barnes, M., Abel, I. G., Dorland, W., Ernst, D. R., Hammett, G. W., Ricci, P., Rogers, B. N., Schekochihin, A. A. & Tatsuno, T. 2009 Linearized model Fokker–Planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests. Phys. Plasmas 16 (7), 072107.CrossRefGoogle Scholar
Belli, E. A. & Candy, J. 2017 Implications of advanced collision operators for gyrokinetic simulation. Plasma Phys. Control. Fusion 59 (4), 045005.CrossRefGoogle Scholar
Berger, J. M., Newcomb, W. A., Dawson, J. M., Frieman, E. A., Kulsrud, R. M. & Lenard, A. 1958 Heating of a confined plasma by oscillating electromagnetic fields. Phys. Fluids 1 (4), 301–307.CrossRefGoogle Scholar
Bernard, T. N., Shi, E. L., Gentle, K. W., Hakim, A., Hammett, G. W., Stoltzfus-Dueck, T. & Taylor, E. I. 2019 Gyrokinetic continuum simulations of plasma turbulence in the texas Helimak. Phys. Plasmas 26 (4), 042301.CrossRefGoogle Scholar
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205.Google Scholar
Brenner, S. & Scott, R. 2008 The Mathematical Theory of Finite Element Methods. Springer.CrossRefGoogle Scholar
Cockburn, B. & Shu, C.-W. 1998 The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (6), 2440–2463.CrossRefGoogle Scholar
Dorf, M. A., Cohen, R. H., Compton, J. C., Dorr, M., Rognlien, T. D., Angus, J., Krasheninnikov, S., Colella, P., Martin, D. & McCorquodale, P. 2012 Progress with the COGENT edge kinetic code: collision operator options. Contrib. Plasma Phys. 52 (56), 518–522.CrossRefGoogle Scholar
Dougherty, J. P. 1964 Model Fokker–Planck equation for a plasma and its solution. Phys. Fluids 7 (11), 1788–1799.CrossRefGoogle Scholar
Francisquez, M., Bernard, T. N., Mandell, N. R., Hammett, G. W. & Hakim, A. 2020 Nucl. Fusion doi:10.1088/1741-4326/aba0c9.Google Scholar
Grant, F. C. & Feix, M. R. 1967 Transition between Landau and Van Kampen treatments of the Vlasov equation. Phys. Fluids 10, 1356–1357.CrossRefGoogle Scholar
Greene, J. M. 1973 Improved Bhatnaga–Gross–Krook model of electron-ion collisions. Phys. Fluids 16 (11), 2022–2023.CrossRefGoogle Scholar
Hager, R., Yoon, E. S., Ku, S., D’Azevedo, E. F., Worley, P. H. & Chang, C. S. 2016 A fully non-linear multi-species Fokker–Planck–Landau collision operator for simulation of fusion plasma. J. Comput. Phys. 315 (C), 644–660.CrossRefGoogle Scholar
Hakim, A. & Juno, J.2020 Alias-free, matrix-free, and quadrature-free discontinuous Galerkin algorithms for (plasma) kinetic equations. Preprint, arXiv:2004.09019.Google Scholar
Hakim, A. H., Hammett, G. W. & Shi, E. L.2014 On discontinuous Galerkin discretizations of second-order derivatives. Preprint, arXiv:1405.5907.Google Scholar
Hammett, G. W.1986 Fast ion studies of ion cyclotron heating in the PLT Tokamak. PhD thesis, Princeton University.Google Scholar
Hammett, G. W., Beer, M. A., Dorland, W., Cowley, S. C. & Smith, S. A. 1993 Developments in the gyrofluid approach to Tokamak turbulence simulations. Plasma Phys. Control. Fusion 35, 973–985.CrossRefGoogle Scholar
Harris, S. 2004 An Introduction to the Theory of the Boltzmann Equation. Dover Publications.Google Scholar
Helander, P. & Sigmar, D. J. 2005 Collisional Transport in Magnetized Plasmas, Cambridge Monographs on Plasma Physics, 2002.Google Scholar
Hirvijoki, E. & Adams, M. F. 2017 Conservative discretization of the Landau collision integral. Phys. Plasmas 24 (3), 032121-8.CrossRefGoogle Scholar
Hirvijoki, E., Burby, J. W. & Kraus, M.2018 Energy-, momentum-, density-, and positivity-preserving spatio-temporal discretizations for the nonlinear Landau collision operator with exact H-theorems. Preprint, arXiv:1804.08546.Google Scholar
Idomura, Y., Ida, M., Kano, T., Aiba, N. & Tokuda, S. 2008 Conservative global gyrokinetic toroidal full-$f$ five-dimensional Vlasov simulation. Comput. Phys. Commun. 179 (6), 391–403.CrossRefGoogle Scholar
Jorge, R., Ricci, P. & Loureiro, N. F. 2017 A drift-kinetic analytical model for scrape-off layer plasma dynamics at arbitrary collisionality. J. Plasma Phys. 83 (6), 905830606.CrossRefGoogle Scholar
Juno, J., Hakim, A., TenBarge, J., Shi, E. & Dorland, W. 2018 Discontinuous Galerkin algorithms for fully kinetic plasmas. J. Comput. Phys. 353, 110–147.CrossRefGoogle Scholar
Killeen, J., Kerbel, G. D., McCoy, M. G. & Mirin, A. A. 1986 Computational Methods for Kinetic Models of Magnetically Confined Plasmas. Springer.CrossRefGoogle Scholar
Kulsrud, R. M. 2005 Plasma Physics for Astrophysics. Princeton University Press.CrossRefGoogle Scholar
Landau, L. 1936 Kinetic equation for the Coulomb effect. Phys. Z. Sowjetunion 10, 154.Google Scholar
Laroussi, M. & Roth, J. R. 1989 Theory of first-order plasma heating by collisional magnetic pumping. Phys. Fluids B 1 (5), 1034–1041.CrossRefGoogle Scholar
van Leer, B. & Lo, M. 2007 A discontinuous Galerkin method for diffusion based on recovery. In 18th AIAA Comput. Fluid Dyn. Conf., American Institute of Aeronautics.Google Scholar
van Leer, B. & Nomura, S. 2005 Discontinuous Galerkin for diffusion. In 17th AIAA Comput. Fluid Dyn. Conf., American Institute of Aeronautics.Google Scholar
Lenard, A. & Bernstein, I. B. 1958 Plasma oscillations with diffusion in velocity space. Phys. Rev. 112 (5), 1456–1459.CrossRefGoogle Scholar
Lichko, E., Egedal, J., Daughton, W. & Kasper, J. 2017 Magnetic pumping as a source of particle heating and power-law distributions in the solar wind. Astrophys. J. Lett. 850 (2), L28.CrossRefGoogle Scholar
McCoy, M. G., Mirin, A. A. & Killeen, J. 1981 FPPAC: A two-dimensional multispecies nonlinear Fokker–Planck package. Comput. Phys. Commun. 24 (1), 37–61.CrossRefGoogle Scholar
Nakata, M., Nunami, M., Watanabe, T.-H. & Sugama, H. 2015 Improved collision operator for plasma kinetic simulations with multi-species ions and electrons. Comput. Phys. Commun. 197, 61–72.CrossRefGoogle Scholar
Pan, Q. & Ernst, D. R. 2019 Gyrokinetic Landau collision operator in conservative form. Phys. Rev. E 99, 023201.Google ScholarPubMed
Patarroyo, K. Y.2019 A digression on Hermite polynomials. Preprint, arXiv:1901.01648.Google Scholar
Pezzi, O., Valentini, F. & Veltri, P. 2015 Collisional relaxation: Landau versus Dougherty. J. Plasma Phys. 81 (1), 305810107.CrossRefGoogle Scholar
Pusztai, I., Juno, J., Brandenburg, A., TenBarge, J. M., Hakim, A., Francisquez, M. & Sundström, A.2020 Dynamo in weakly collisional non-magnetized plasmas impeded by Landau damping of magnetic fields. Preprint, arXiv:2001.11929.Google Scholar
Rosenbluth, M. N., MacDonald, W. M. & Judd, D. L. 1957 Fokker–Planck equation for an inverse-square force. Phys. Rev. 107 (1), 1–6.CrossRefGoogle Scholar
Schekochihin, A. A., Cowley, S. C., Kulsrud, R. M., Hammett, G. W. & Sharma, P. 2005 Plasma instabilities and magnetic field growth in clusters of galaxies. Astrophys. J. 629, 139–142.CrossRefGoogle Scholar
Shi, E. L.2017 Gyrokinetic continuum simulation of turbulence in open-field-line plasmas. PhD thesis, Princeton University.CrossRefGoogle Scholar
Shi, E. L., Hammett, G. W., Stoltzfus-Dueck, T. & Hakim, A. 2017 Gyrokinetic continuum simulation of turbulence in a straight open-field-line plasma. J. Plasma Phys. 83, 905830304.CrossRefGoogle Scholar
Shi, E. L., Hammett, G. W., Stoltzfus-Dueck, T. & Hakim, A. 2019 Full-f gyrokinetic simulation of turbulence in a helical open-field-line plasma. Phys. Plasmas 26 (1), 012307.CrossRefGoogle Scholar
Shu, C. W. 2002 A survey of strong stability-preserving high-order time discretization methods. In Collected Lectures on the Preservation of Stability under Discretization. Society of Industrial and Applied Mathematics (SIAM).Google Scholar
Sundström, A., Juno, J., TenBarge, J. M. & Pusztai, I. 2019 Effect of a weak ion collisionality on the dynamics of kinetic electrostatic shocks. J. Plasma Phys. 85 (1), 905850108.CrossRefGoogle Scholar
Taitano, W. T., Chacón, L., Simakov, A. N. & Molvig, K. 2015 A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation. J. Comput. Phys. 297 (C), 357–380.CrossRefGoogle Scholar
The Gkeyll team2020 The Gkeyll code. http://gkeyll.readthedocs.io.Google Scholar
Wang, L., Hakim, A. H., Bhattacharjee, A. & Germaschewski, K. 2015 Comparison of multi-fluid moment models with particle-in-cell simulations of collisionless magnetic reconnection. Phys. Plasmas 22 (1), 012108-13.Google Scholar