The collisional particle-in-cell method for the Vlasov–Maxwell–Landau equations

Bailo, Rafael; Carrillo, José A.; Hu, Jingwei

doi:10.1017/S0022377824001077

Title: The collisional particle-in-cell method for the Vlasov–Maxwell–Landau equations

Journal Article · 18 October 2024 · Journal of Plasma Physics

DOI: https://doi.org/10.1017/S0022377824001077 · OSTI ID:2472872

;

We introduce an extension of the particle-in-cell method that captures the Landau collisional effects in the Vlasov–Maxwell–Landau equations. The method arises from a regularisation of the variational formulation of the Landau equation, leading to a discretisation of the collision operator that conserves mass, charge, momentum and energy, while increasing the (regularised) entropy. The collisional effects appear as a fully deterministic effective force, thus the method does not require any transport–collision splitting. The scheme can be used in arbitrary dimension, and for a general interaction, including the Coulomb case. We validate the scheme on scenarios such as the Landau damping, the two-stream instability and the Weibel instability, demonstrating its effectiveness in the numerical simulation of plasma.

View Journal Article

Cite

Export

Save

Sponsoring Organization:: USDOE

Grant/Contract Number:: SC0023164

OSTI ID:: 2472872

Journal Information:: Journal of Plasma Physics, Journal Name: Journal of Plasma Physics Journal Issue: 4 Vol. 90; ISSN 0022-3778

Publisher:: Cambridge University Press (CUP)Copyright Statement

Country of Publication:: United Kingdom

Language:: English

References (62)

Fast Algorithms for Numerical, Conservative, and Entropy Approximations of the Fokker–Planck–Landau Equation Buet, C.; Cordier, S.; Degond, P. Journal of Computational Physics, Vol. 133, Issue 2 https://doi.org/10.1006/jcph.1997.5669	journal	May 1997
Langevin Representation of Coulomb Collisions in PIC Simulations Manheimer, Wallace M.; Lampe, Martin; Joyce, Glenn Journal of Computational Physics, Vol. 138, Issue 2 https://doi.org/10.1006/jcph.1997.5834	journal	December 1997
Conservative and Entropy Decaying Numerical Scheme for the Isotropic Fokker–Planck–Landau Equation Buet, C.; Cordier, S. Journal of Computational Physics, Vol. 145, Issue 1 https://doi.org/10.1006/jcph.1998.6015	journal	September 1998
Mimetic Discretizations for Maxwell's Equations Hyman, James M.; Shashkov, Mikhail Journal of Computational Physics, Vol. 151, Issue 2 https://doi.org/10.1006/jcph.1999.6225	journal	May 1999
Fast Spectral Methods for the Fokker–Planck–Landau Collision Operator Pareschi, L.; Russo, G.; Toscani, G. Journal of Computational Physics, Vol. 165, Issue 1 https://doi.org/10.1006/jcph.2000.6612	journal	November 2000
A Numerical Method for the Accurate Solution of the Fokker–Planck–Landau Equation in the Nonhomogeneous Case Filbet, Francis; Pareschi, Lorenzo Journal of Computational Physics, Vol. 179, Issue 1 https://doi.org/10.1006/jcph.2002.7010	journal	June 2002
Introduction to Plasma Physics and Controlled Fusion Chen, Francis F. https://doi.org/10.1007/978-3-319-22309-4	book	January 2016
On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations Villani, Cédric Archive for Rational Mechanics and Analysis, Vol. 143, Issue 3 https://doi.org/10.1007/s002050050106	journal	September 1998
An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory Degond, Pierre; Lucquin-Desreux, Brigitte Numerische Mathematik, Vol. 68, Issue 2 https://doi.org/10.1007/s002110050059	journal	July 1994
Multipole expansions for the Fokker-Planck-Landau operator Lemou, M. Numerische Mathematik, Vol. 78, Issue 4 https://doi.org/10.1007/s002110050327	journal	February 1998
Study of conservation and recurrence of Runge–Kutta discontinuous Galerkin schemes for Vlasov–Poisson systems Cheng, Yingda; Gamba, Irene M.; Morrison, Philip J. Journal of Scientific Computing, Vol. 56, Issue 2 https://doi.org/10.1007/s10915-012-9680-x	journal	January 2013
Mathematics of Granular Materials Villani, Cédric Journal of Statistical Physics, Vol. 124, Issue 2-4 https://doi.org/10.1007/s10955-006-9038-6	journal	April 2006
Rigorous charge conservation for local electromagnetic field solvers Villasenor, John; Buneman, Oscar Computer Physics Communications, Vol. 69, Issue 2-3 https://doi.org/10.1016/0010-4655(92)90169-Y	journal	March 1992
A binary collision model for plasma simulation with a particle code Takizuka, Tomonor; Abe, Hirotada Journal of Computational Physics, Vol. 25, Issue 3 https://doi.org/10.1016/0021-9991(77)90099-7	journal	November 1977
Exact charge conservation scheme for Particle-in-Cell simulation with an arbitrary form-factor Esirkepov, T. Zh. Computer Physics Communications, Vol. 135, Issue 2 https://doi.org/10.1016/S0010-4655(00)00228-9	journal	April 2001
Spectral gap and exponential convergence to equilibrium for a multi-species Landau system Gualdani, Maria Pia; Zamponi, Nicola Bulletin des Sciences Mathématiques, Vol. 141, Issue 6 https://doi.org/10.1016/j.bulsci.2017.07.002	journal	August 2017
A conservative phase-space moving-grid strategy for a 1D-2V Vlasov–Fokker–Planck Solver Taitano, W. T.; Chacón, L.; Simakov, A. N. Computer Physics Communications, Vol. 258 https://doi.org/10.1016/j.cpc.2020.107547	journal	January 2021
Numerical approximation of collisional plasmas by high order methods Crouseilles, Nicolas; Filbet, Francis Journal of Computational Physics, Vol. 201, Issue 2 https://doi.org/10.1016/j.jcp.2004.06.007	journal	December 2004
A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation Jin, Shi; Yan, Bokai Journal of Computational Physics, Vol. 230, Issue 17 https://doi.org/10.1016/j.jcp.2011.04.002	journal	July 2011
An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm Chen, G.; Chacón, L.; Barnes, D. C. Journal of Computational Physics, Vol. 230, Issue 18 https://doi.org/10.1016/j.jcp.2011.05.031	journal	August 2011
The energy conserving particle-in-cell method Markidis, Stefano; Lapenta, Giovanni Journal of Computational Physics, Vol. 230, Issue 18 https://doi.org/10.1016/j.jcp.2011.05.033	journal	August 2011
Higher-order time integration of Coulomb collisions in a plasma using Langevin equations Dimits, A. M.; Cohen, B. I.; Caflisch, R. E. Journal of Computational Physics, Vol. 242 https://doi.org/10.1016/j.jcp.2013.01.038	journal	June 2013
Multilevel Monte Carlo simulation of Coulomb collisions Rosin, M. S.; Ricketson, L. F.; Dimits, A. M. Journal of Computational Physics, Vol. 274 https://doi.org/10.1016/j.jcp.2014.05.030	journal	October 2014
Energy-conserving discontinuous Galerkin methods for the Vlasov–Maxwell system Cheng, Yingda; Christlieb, Andrew J.; Zhong, Xinghui Journal of Computational Physics, Vol. 279 https://doi.org/10.1016/j.jcp.2014.08.041	journal	December 2014
Charge-and-energy conserving moment-based accelerator for a multi-species Vlasov–Fokker–Planck–Ampère system, part II: Collisional aspects Taitano, William T.; Knoll, Dana A.; Chacón, Luis Journal of Computational Physics, Vol. 284 https://doi.org/10.1016/j.jcp.2014.09.004	journal	March 2015
Hamiltonian splitting for the Vlasov–Maxwell equations Crouseilles, Nicolas; Einkemmer, Lukas; Faou, Erwan Journal of Computational Physics, Vol. 283 https://doi.org/10.1016/j.jcp.2014.11.029	journal	February 2015
Charge-and-energy conserving moment-based accelerator for a multi-species Vlasov–Fokker–Planck–Ampère system, part I: Collisionless aspects Taitano, William T.; Chacón, Luis Journal of Computational Physics, Vol. 284 https://doi.org/10.1016/j.jcp.2014.12.038	journal	March 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 new hybrid-Lagrangian numerical scheme for gyrokinetic simulation of tokamak edge plasma Ku, S.; Hager, R.; Chang, C. S. Journal of Computational Physics, Vol. 315 https://doi.org/10.1016/j.jcp.2016.03.062	journal	June 2016
An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes Taitano, W. T.; Chacón, L.; Simakov, A. N. Journal of Computational Physics, Vol. 318 https://doi.org/10.1016/j.jcp.2016.03.071	journal	August 2016
Exactly energy conserving semi-implicit particle in cell formulation Lapenta, Giovanni Journal of Computational Physics, Vol. 334 https://doi.org/10.1016/j.jcp.2017.01.002	journal	April 2017
A conservative scheme for Vlasov Poisson Landau modeling collisional plasmas Zhang, Chenglong; Gamba, Irene M. Journal of Computational Physics, Vol. 340 https://doi.org/10.1016/j.jcp.2017.03.046	journal	July 2017
Random Batch Methods (RBM) for interacting particle systems Jin, Shi; Li, Lei; Liu, Jian-Guo Journal of Computational Physics, Vol. 400 https://doi.org/10.1016/j.jcp.2019.108877	journal	January 2020
A semi-implicit, energy- and charge-conserving particle-in-cell algorithm for the relativistic Vlasov-Maxwell equations Chen, G.; Chacón, L.; Yin, L. Journal of Computational Physics, Vol. 407 https://doi.org/10.1016/j.jcp.2020.109228	journal	April 2020
An energy-conserving and asymptotic-preserving charged-particle orbit implicit time integrator for arbitrary electromagnetic fields Ricketson, L. F.; Chacón, L. Journal of Computational Physics, Vol. 418 https://doi.org/10.1016/j.jcp.2020.109639	journal	October 2020
Energy-conserving time propagation for a structure-preserving particle-in-cell Vlasov–Maxwell solver Kormann, Katharina; Sonnendrücker, Eric Journal of Computational Physics, Vol. 425 https://doi.org/10.1016/j.jcp.2020.109890	journal	January 2021
A stochastic kinetic scheme for multi-scale plasma transport with uncertainty quantification Xiao, Tianbai; Frank, Martin Journal of Computational Physics, Vol. 432 https://doi.org/10.1016/j.jcp.2021.110139	journal	May 2021
Hermite spectral method for Fokker-Planck-Landau equation modeling collisional plasma Li, Ruo; Ren, Yinuo; Wang, Yanli Journal of Computational Physics, Vol. 434 https://doi.org/10.1016/j.jcp.2021.110235	journal	June 2021
Projective and telescopic projective integration for non-linear kinetic mixtures Bailo, Rafael; Rey, Thomas Journal of Computational Physics, Vol. 458 https://doi.org/10.1016/j.jcp.2022.111082	journal	June 2022
Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties Medaglia, Andrea; Pareschi, Lorenzo; Zanella, Mattia Journal of Computational Physics, Vol. 479 https://doi.org/10.1016/j.jcp.2023.112011	journal	April 2023
Boltzmann to Landau from the gradient flow perspective Carrillo, José A.; Delgadino, Matias G.; Wu, Jeremy Nonlinear Analysis, Vol. 219 https://doi.org/10.1016/j.na.2022.112824	journal	June 2022
Numerical methods for plasma physics in collisional regimes Dimarco, G.; Li, Q.; Pareschi, L. Journal of Plasma Physics, Vol. 81, Issue 1 https://doi.org/10.1017/S0022377814000762	journal	October 2014
GEMPIC: geometric electromagnetic particle-in-cell methods Kraus, Michael; Kormann, Katharina; Morrison, Philip J. Journal of Plasma Physics, Vol. 83, Issue 4 https://doi.org/10.1017/S002237781700040X	journal	July 2017
Encoder–decoder neural network for solving the nonlinear Fokker–Planck–Landau collision operator in XGC Miller, M. A.; Churchill, R. M.; Dener, A. Journal of Plasma Physics, Vol. 87, Issue 2 https://doi.org/10.1017/S0022377821000155	journal	March 2021
Numerical methods for kinetic equations Dimarco, G.; Pareschi, L. Acta Numerica, Vol. 23 https://doi.org/10.1017/S0962492914000063	journal	May 2014
Spontaneous rotation sources in a quiescent tokamak edge plasma Chang, C. S.; Ku, S. Physics of Plasmas, Vol. 15, Issue 6 https://doi.org/10.1063/1.2937116	journal	June 2008
Effect of collisions on the two-stream instability in a finite length plasma Sydorenko, D.; Kaganovich, I. D.; Ventzek, P. L. G. Physics of Plasmas, Vol. 23, Issue 12 https://doi.org/10.1063/1.4972543	journal	December 2016
Multispecies structure-preserving particle discretization of the Landau collision operator Zonta, Filippo; Pusztay, Joseph V.; Hirvijoki, Eero Physics of Plasmas, Vol. 29, Issue 12 https://doi.org/10.1063/5.0105182	journal	December 2022
Two-stream instability with a growth rate insensitive to collisions in a dissipative plasma jet Zhou, Yi; Bellan, Paul M. Physics of Plasmas, Vol. 30, Issue 5 https://doi.org/10.1063/5.0146806	journal	May 2023
Full-f gyrokinetic particle simulation of centrally heated global ITG turbulence from magnetic axis to edge pedestal top in a realistic tokamak geometry Ku, S.; Chang, C. S.; Diamond, P. H. Nuclear Fusion, Vol. 49, Issue 11 https://doi.org/10.1088/0029-5515/49/11/115021	journal	September 2009
Kinetic theory of particle-in-cell simulation plasma and the ensemble averaging technique Touati, Michaël; Codur, Romain; Tsung, Frank Plasma Physics and Controlled Fusion, Vol. 64, Issue 11 https://doi.org/10.1088/1361-6587/ac9016	journal	October 2022
Molecular Gas Dynamics And The Direct Simulation Of Gas Flows Bird, G. A. Oxford Engineering Science Series, Vol. 2 https://doi.org/10.1093/oso/9780198561958.001.0001	book	May 1994
Theory of cumulative small-angle collisions in plasmas Nanbu, K. Physical Review E, Vol. 55, Issue 4 https://doi.org/10.1103/PhysRevE.55.4642	journal	April 1997
Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation Bobylev, A. V.; Nanbu, K. Physical Review E, Vol. 61, Issue 4 https://doi.org/10.1103/PhysRevE.61.4576	journal	April 2000
Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media IEEE Transactions on Antennas and Propagation, Vol. 14, Issue 3 https://doi.org/10.1109/TAP.1966.1138693	journal	May 1966
Implicit Schemes for the Fokker--Planck--Landau Equation Lemou, Mohammed; Mieussens, Luc SIAM Journal on Scientific Computing, Vol. 27, Issue 3 https://doi.org/10.1137/040609422	journal	January 2005
A Hybrid Method for Accelerated Simulation of Coulomb Collisions in a Plasma Caflisch, Russel; Wang, Chiaming; Dimarco, Giacomo Multiscale Modeling & Simulation, Vol. 7, Issue 2 https://doi.org/10.1137/070704939	journal	January 2008
Convergence of a particle method for a regularized spatially homogeneous Landau equation Carrillo, José A.; Delgadino, Matias G.; Wu, Jeremy S. H. Mathematical Models and Methods in Applied Sciences, Vol. 33, Issue 05 https://doi.org/10.1142/S0218202523500215	journal	March 2023
Plasma Physics via Computer Simulation Birdsall, C. K.; Langdon, A. B. https://doi.org/10.1201/9781315275048	book	January 1991
The Landau equation as a gradient Flow Carrillo, José A.; Delgadino, Matias G.; Desvillettes, Laurent Analysis & PDE, Vol. 17, Issue 4 https://doi.org/10.2140/apde.2024.17.1331	journal	May 2024
A Unified Gas Kinetic Scheme for Continuum and Rarefied Flows V: Multiscale and Multi-Component Plasma Transport Liu, Chang; Xu, Kun Communications in Computational Physics, Vol. 22, Issue 5 https://doi.org/10.4208/cicp.OA-2017-0102	journal	October 2017
Random Batch Particle Methods for the Homogeneous Landau Equation José Antonio Carrillo, José Antonio Carrillo; Shi Jin, Shi Jin; Yijia Tang, Yijia Tang Communications in Computational Physics, Vol. 31, Issue 4 https://doi.org/10.4208/cicp.OA-2021-0200	journal	January 2022

Similar Records

Collisional damping in heated nuclei within the Landau-Vlasov kinetic theory

Journal Article · 1995 · Physical Review, C · OSTI ID:122381

Kolomietz, V M; Plujko, V A; Shlomo, S

Electron-ion collisional effect on Weibel instability in a Kappa distributed unmagnetized plasma

Journal Article · 2014 · Physics of Plasmas · OSTI ID:22253296

Das, Nilakshi

Hamiltonian time integrators for Vlasov-Maxwell equations

Journal Article · 2015 · Physics of Plasmas · OSTI ID:22489987

He, Yang; Xiao, Jianyuan; Zhang, Ruili; +3 more

Title: The collisional particle-in-cell method for the Vlasov–Maxwell–Landau equations

Citation Formats

References (62)

Similar Records

Related Subjects