VIDEOS 1 TO 50

"Kinetic Plasma Simulations with the Particle-in-Cell Method I" - Spitkovsky

Published: 2016/07/21

Channel: Institute for Advanced Study

The Affine Particle-In-Cell Method

Published: 2016/03/15

Channel: TeranGroup

800 million particles on 16 GPUs (PIConGPU 3D)

Published: 2011/11/13

Channel: psychocoder

Vlasov-Poisson Equations - PIC (Particle In Cell) Method

Published: 2010/08/17

Channel: kwangjin1014

Particle in cell plasma simulation

Published: 2013/11/27

Channel: Kimi Tsai

Kinetic Plasma Simulations with the Particle-in-Cell Methods II - Spitkovsky

Published: 2016/07/27

Channel: Institute for Advanced Study

Petascale Particle-in-Cell Simulation of Kinetic Effects in High Energy Density Plasmas

Published: 2015/06/04

Channel: NCSAatIllinois

Introduction to the Particle-in-Cell Method of Plasma Simulation by Kartik Patel

Published: 2017/04/07

Channel: International Centre for Theoretical Sciences

Frank Tsung (PI: Warren Mori) Transformative Petascale Particle-in-cell Simulations

Published: 2017/06/09

Channel: NCSAatIllinois

Particle in Cell reconnection

Published: 2014/06/11

Channel: Turbulence Research for Environmental & Astrophysical Transport

Petascale Particle-in-Cell Simulation of Kinetic Effects in High Energy Density Plasmas

Published: 2016/07/12

Channel: NCSAatIllinois

Laser Wakefield Acceleration LWFA of electrons

Published: 2011/07/29

Channel: Taylor Ratliff

particle in cell 232x114 cell wetbed simulation GMRES solver

Published: 2014/08/07

Channel: Sammypops

Vispic Particle in Cell simulation version May 11

Published: 2012/05/11

Channel: Luc Provoost

Particle-in-cell test running

Published: 2014/02/27

Channel: Alexander Volkov

gesa40kV Particle in cell simulations

Published: 2015/11/16

Channel: Владислав Алцыбеев

IMG1677_no particle cell contrast

Published: 2013/05/17

Channel: Philip Lee

Petascale Particle-in-Cell Simulations of Kinetic Effects in Plasmas

Published: 2014/07/10

Channel: NCSAatIllinois

particle in cell 76x36 GMRES solver wetbed simulation

Published: 2014/08/07

Channel: Sammypops

Particle in Cell

Published: 2017/01/25

Channel: Chris Rinox - Topic

Formation of Sheath in Magnetised Plasma

Published: 2013/08/06

Channel: Sayan Adhikari

An Angular Momentum Conserving Affine-Particle-In-Cell Method

Published: 2016/04/05

Channel: Chenfanfu Jiang

New understanding of plasma phase-space holes via hole-tracking Particle-In-Cell simulation

Published: 2017/03/31

Channel: MIT Department of Nuclear Science and Engineering

Optimizing Particle in Cell Codes for Intel Xeon Phi Processors

Published: 2016/09/26

Channel: ANL Training

particle in cell 232x114 GMRES solver wetbed simulation

Published: 2014/08/07

Channel: Sammypops

The Affine Particle In Cell Method

Published: 2017/04/06

Channel: Research in Science and Technology

gesa25kV Particle in cell simulations

Published: 2015/11/16

Channel: Владислав Алцыбеев

466 million particles on 16 GPUs (PIConGPU)

Published: 2010/09/23

Channel: psychocoder

Modeling astrophysical plasmas using particle in cell method by Ujjwal Sinha

Published: 2017/03/15

Channel: International Centre for Theoretical Sciences

Particle-in-Cell 1D Two Stream Instability

Published: 2014/02/15

Channel: Sinéad Mannion

2D particle in cell simulation for PILS

Published: 2013/03/17

Channel: hanused

Particle in cell movimiento de particulas en un espacio de fase

Published: 2015/10/22

Channel: Yen Gomez-Villegas

Underworld Simulation of Indentor / Collision

Published: 2015/12/01

Channel: Louis Moresi

Zoltán Juhász - Highly Parallel GPU-based Particle-in-Cell/MCC Plasma Simulation

Published: 2017/08/28

Channel: Wigner GPU Lab

CFD Support Preprocessing Training Tutorial Cyclone Lagrange MPPIC Multiphase Particle In Cell

Published: 2016/10/05

Channel: CFD Support

gesa24kV 60 ns Particle in cell simulations

Published: 2015/11/18

Channel: Владислав Алцыбеев

Particle plasma simulation

Published: 2012/03/05

Channel: DrAD1907

Zoltán Juhász - Highly Parallel GPU-based Particle-in-Cell/MCC Plasma Simulation Videos

Published: 2017/09/19

Channel: Wigner GPU Lab

picongpu_4x4_8192x8192_100000_50.mpeg

Published: 2010/08/26

Channel: psychocoder

1-D Parcile-in-Cell simulation of collision-less plasma shock

Published: 2011/05/12

Channel: ymatumot

Hybrid Optimization of an IEC Beamline using 2D3V Particle-in-cell Simulation

Published: 2017/08/28

Channel: Andrew Chap

DAISI. Diode with elliptical emitter. PIC simulations.

Published: 2017/01/16

Channel: Владислав Алцыбеев

Plasma cloud expansion in the uniform magnetic field

Published: 2012/06/10

Channel: David Osipyan

iPIC Simulation 1

Published: 2016/03/15

Channel: EPiGRAM Project

Simple PIC simulation

Published: 2007/05/24

Channel: Nils Larsgård

iPIC Simulation 3

Published: 2016/03/15

Channel: EPiGRAM Project

Two Stream Instability Simulation

Published: 2014/05/07

Channel: Frans Ebersohn

iPIC Simulation 2

Published: 2016/03/15

Channel: EPiGRAM Project

FLIP

Published: 2016/04/04

Channel: Duosheng Yu

PenningCircle nb log vs xy

Published: 2016/11/01

Channel: Crow Radio and Plasma Science

The **particle-in-cell** (**PIC**) method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles (or fluid elements) in a Lagrangian frame are tracked in continuous phase space, whereas moments of the distribution such as densities and currents are computed simultaneously on Eulerian (stationary) mesh points.

PIC methods were already in use as early as 1955,^{[1]} even before the first Fortran compilers were available. The method gained popularity for plasma simulation in the late 1950s and early 1960s by Buneman, Dawson, Hockney, Birdsall, Morse and others. In plasma physics applications, the method amounts to following the trajectories of charged particles in self-consistent electromagnetic (or electrostatic) fields computed on a fixed mesh. ^{[2]}

For many types of problems, the classical PIC method invented by Buneman, Dawson, Hockney, Birdsall, Morse and others is relatively intuitive and straightforward to implement. This probably accounts for much of its success, particularly for plasma simulation, for which the method typically includes the following procedures:

- Integration of the equations of motion.
- Interpolation of charge and current source terms to the field mesh.
- Computation of the fields on mesh points.
- Interpolation of the fields from the mesh to the particle locations.

Models which include interactions of particles only through the average fields are called **PM** (particle-mesh). Those which include direct binary interactions are **PP** (particle-particle). Models with both types of interactions are called **PP-PM** or **P ^{3}M**.

Since the early days, it has been recognized that the PIC method is susceptible to error from so-called *discrete particle noise*. ^{[3]} This error is statistical in nature, and today it remains less-well understood than for traditional fixed-grid methods, such as Eulerian or semi-Lagrangian schemes.

Modern geometric PIC algorithms are based on a very different theoretical framework. These algorithms use tools of discrete manifold, interpolating differential forms, and canonical or non-canonical symplectic integrators to guarantee gauge invariant and conservation of charge, energy-momentum, and more importantly the infinitely dimensional symplectic structure of the particle-field system. ^{[4]} ^{[5]} These desired features are attributed to the fact that geometric PIC algorithms are built on the more fundamental ﬁeld-theoretical framework and are directly linked to the perfect form, i.e., the variational principle of physics.

Inside the plasma research community, systems of different species (electrons, ions, neutrals, molecules, dust particles, etc.) are investigated. The set of equations associated with PIC codes are therefore the Lorentz force as the equation of motion, solved in the so-called *pusher* or *particle mover* of the code, and Maxwell's equations determining the electric and magnetic fields, calculated in the *(field) solver*.

The real systems studied are often extremely large in terms of the number of particles they contain. In order to make simulations efficient or at all possible, so-called *super-particles* are used. A super-particle (or *macroparticle*) is a computational particle that represents many real particles; it may be millions of electrons or ions in the case of a plasma simulation, or, for instance, a vortex element in a fluid simulation. It is allowed to rescale the number of particles, because the Lorentz force depends only on the charge-to-mass ratio, so a super-particle will follow the same trajectory as a real particle would.

The number of real particles corresponding to a super-particle must be chosen such that sufficient statistics can be collected on the particle motion. If there is a significant difference between the density of different species in the system (between ions and neutrals, for instance), separate real to super-particle ratios can be used for them.

Even with super-particles, the number of simulated particles is usually very large (> 10^{5}), and often the particle mover is the most time consuming part of PIC, since it has to be done for each particle separately. Thus, the pusher is required to be of high accuracy and speed and much effort is spent on optimizing the different schemes.

The schemes used for the particle mover can be split into two categories, implicit and explicit solvers. While implicit solvers(e.g. implicit Euler scheme) calculate the particle velocity from the already updated fields, explicit solvers use only the old force from the previous time step, and are therefore simpler and faster, but require a smaller time step. Two frequently used schemes are the leapfrog method,^{[6]} and the *Boris scheme*,^{[7]} ^{[8]} which are explicit solvers.

For plasma applications, the leapfrog method takes the following form:

where the subscript refers to "old" quantities from the previous time step, to updated quantities from the next time step (i.e. ), and velocities are calculated in-between the usual time steps .

In comparison, the equations of the Boris scheme are:

with

and .

Because of its excellent long term accuracy, the Boris algorithm is the de facto standard for advancing a charged particle. It was recently realized that the excellent long term accuracy of Boris algorithm is due to the fact it conserves phase space volume, even though it is not symplectic. The global bound on energy error typically associated with symplectic algorithms still holds for the Boris algorithm, making it an effective algorithm for the multi-scale dynamics of plasmas.

The most commonly used methods for solving Maxwell's equations (or more generally, partial differential equations (PDE)) belong to one of the following three categories:

With the FDM, the continuous domain is replaced with a discrete grid of points, on which the electric and magnetic fields are calculated. Derivatives are then approximated with differences between neighboring grid-point values and thus PDEs are turned into algebraic equations.

Using FEM, the continuous domain is divided into a discrete mesh of elements. The PDEs are treated as an eigenvalue problem and initially a trial solution is calculated using basis functions that are localized in each element. The final solution is then obtained by optimization until the required accuracy is reached.

Also spectral methods, such as the fast Fourier transform (FFT), transform the PDEs into an eigenvalue problem, but this time the basis functions are high order and defined globally over the whole domain. The domain itself is not discretized in this case, it remains continuous. Again, a trial solution is found by inserting the basis functions into the eigenvalue equation and then optimized to determine the best values of the initial trial parameters.

The name "particle-in-cell" originates in the way that plasma macro-quantities (number density, current density, etc.) are assigned to simulation particles (i.e., the *particle weighting*). Particles can be situated anywhere on the continuous domain, but macro-quantities are calculated only on the mesh points, just as the fields are. To obtain the macro-quantities, one assumes that the particles have a given "shape" determined by the shape function

where is the coordinate of the particle and the observation point. Perhaps the easiest and most used choice for the shape function is the so-called *cloud-in-cell* (CIC) scheme, which is a first order (linear) weighting scheme. Whatever the scheme is, the shape function has to satisfy the following conditions: ^{[9]} space isotropy, charge conservation, and increasing accuracy (convergence) for higher-order terms.

The fields obtained from the field solver are determined only on the grid points and can't be used directly in the particle mover to calculate the force acting on particles, but have to be interpolated via the *field weighting*:

where the subscript labels the grid point. To ensure that the forces acting on particles are self-consistently obtained, the way of calculating macro-quantities from particle positions on the grid points and interpolating fields from grid points to particle positions has to be consistent, too, since they both appear in Maxwell's equations. Above all, the field interpolation scheme should conserve momentum. This can be achieved by choosing the same weighting scheme for particles and fields and by ensuring the appropriate space symmetry (i.e. no self-force and fulfilling the action-reaction law) of the field solver at the same time ^{[9]}

As the field solver is required to be free of self-forces, inside a cell the field generated by a particle must decrease with decreasing distance from the particle, and hence inter-particle forces inside the cells are underestimated. This can be balanced with the aid of Coulomb collisions between charged particles. Simulating the interaction for every pair of a big system would be computationally too expensive, so several Monte Carlo methods have been developed instead. A widely used method is the *binary collision model*,^{[10]} in which particles are grouped according to their cell, then these particles are paired randomly, and finally the pairs are collided.

In a real plasma, many other reactions may play a role, ranging from elastic collisions, such as collisions between charged and neutral particles, over inelastic collisions, such as electron-neutral ionization collision, to chemical reactions; each of them requiring separate treatment. Most of the collision models handling charged-neutral collisions use either the *direct Monte-Carlo* scheme, in which all particles carry information about their collision probability, or the *null-collision* scheme,^{[11]}^{[12]} which does not analyze all particles but uses the maximum collision probability for each charged species instead.

As in every simulation method, also in PIC, the time step and the grid size must be well chosen, so that the time and length scale phenomena of interest are properly resolved in the problem. In addition, time step and grid size affect the speed and accuracy of the code.

For an electrostatic plasma simulation using an explicit time integration scheme (e.g. leapfrog, which is most commonly used), two important conditions regarding the grid size and the time step should be fulfilled in order to ensure the stability of the solution:

which can be derived considering the harmonic oscillations of a one-dimensional unmagnetized plasma. The latter conditions is strictly required but practical considerations related to energy conservation suggest to use a much stricter constraint where the factor 2 is replaced by a number one order of magnitude smaller. The use of is typical.^{[9]}^{[13]} Not surprisingly, the natural time scale in the plasma is given by the inverse plasma frequency and length scale by the Debye length .

For an explicit electromagnetic plasma simulation, the time step must also satisfy the CFL condition:

where , and is the speed of light.

Within plasma physics, PIC simulation has been used successfully to study laser-plasma interactions, electron acceleration and ion heating in the auroral ionosphere, magnetohydrodynamics, magnetic reconnection, as well as ion-temperature-gradient and other microinstabilities in tokamaks, furthermore vacuum discharges, and dusty plasmas.

Hybrid models may use the PIC method for the kinetic treatment of some species, while other species (that are Maxwellian) are simulated with a fluid model.

PIC simulations have also been applied outside of plasma physics to problems in solid and fluid mechanics. ^{[14]} ^{[15]}

Computational application | Web site | License | Availability | Canonical Reference |
---|---|---|---|---|

SHARP | https://arxiv.org/abs/1702.04732 | DOI: 10.3847/1538-4357/aa6d13 | ||

ALaDyn | http://aladyn.github.io/ALaDyn/ | GPLv3+ | Open Repo: https://github.com/ALaDyn/ALaDyn | DOI: 10.5281/zenodo.49553 |

EPOCH | http://www.ccpp.ac.uk/codes.html | GPL | Closed (Collaborators): https://cfsa-pmw.warwick.ac.uk | DOI: 10.1088/0741-3335/57/11/113001 |

FBPIC | https://fbpic.github.io/ | 3-Clause-BSD-LBNL | Open Repo: https://github.com/fbpic/fbpic | DOI: 10.1016/j.cpc.2016.02.007 |

LSP | http://www.mrcwdc.com/lsp | Proprietary | Available from ATK | DOI: 10.1016/S0168-9002(01)00024-9 |

MAGIC | http://www.mrcwdc.com/magic | Proprietary | Available from ATK | DOI: 10.1016/0010-4655(95)00010-D |

OSIRIS | http://picksc.idre.ucla.edu/software/software-production-codes/osiris | Proprietary | Closed (Collaborators) | DOI: 10.1007/3-540-47789-6_36 |

PICCANTE | http://aladyn.github.io/piccante/ | GPLv3+ | Open Repo: https://github.com/ALaDyn/piccante | DOI: 10.5281/zenodo.48703 |

PIConGPU | http://picongpu.hzdr.de/ | GPLv3+ | Open Repo: https://github.com/ComputationalRadiationPhysics/picongpu/ | DOI: 10.1145/2503210.2504564 |

SMILEI | http://www.maisondelasimulation.fr/smilei/ | CeCILL (equivalent GPL) | Open Repo: https://github.com/SmileiPIC/Smilei | |

The Virtual Laser Plasma Library | http://www2.mpq.mpg.de/lpg/research/RelLasPlas/Rel-Las-Plas.html | Proprietary | Unknown | DOI: 10.1017/S0022377899007515 |

VSim (Vorpal) | https://txcorp.com/vsim | Proprietary | Available from Tech-X Corporation | DOI: 10.1016/j.jcp.2003.11.004 |

WARP | http://warp.lbl.gov/ | 3-Clause-BSD-LBNL | Open Repo: https://bitbucket.org/berkeleylab/warp | DOI: 10.1063/1.860024 |

**^**F.H. Harlow (1955). "A Machine Calculation Method for Hydrodynamic Problems". Los Alamos Scientific Laboratory report LAMS-1956.**^**Dawson, J.M. (1983). "Particle simulation of plasmas".*Reviews of Modern Physics*.**55**(2): 403. Bibcode:1983RvMP...55..403D. doi:10.1103/RevModPhys.55.403.**^**Hideo Okuda (1972). "Nonphysical noises and instabilities in plasma simulation due to a spatial grid".*Journal of Computational Physics*.**10**(3): 475. Bibcode:1972JCoPh..10..475O. doi:10.1016/0021-9991(72)90048-4.**^**Qin, H.; Liu, J.; Xiao, J.; et al. (2016). "Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov-Maxwell system".*Nuclear Fusion*.**56**(1): 014001. arXiv:1503.08334 . Bibcode:2016NucFu..56a4001Q. doi:10.1088/0029-5515/56/1/014001.**^**Xiao, J.; Qin, H.; Liu, J.; et al. (2015). "Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems".*Physics of Plasmas*.**22**(11): 12504. arXiv:1510.06972 . Bibcode:2015PhPl...22k2504X. doi:10.1063/1.4935904.**^**Birdsall, Charles K.; A. Bruce Langdon (1985).*Plasma Physics via Computer Simulation*. McGraw-Hill. ISBN 0-07-005371-5.**^**Boris, J.P. (November 1970). "Relativistic plasma simulation-optimization of a hybrid code".*Proceedings of the*4th Conference on Numerical Simulation of Plasmas. Naval Res. Lab., Washington, D.C. pp. 3–67.**^**Qin, H.; et al. (2013). "Why is Boris algorithm so good?".*Physics of Plasmas*.**20**(5): 084503. Bibcode:2013PhPl...20h4503Q. doi:10.1063/1.4818428.- ^
^{a}^{b}^{c}Tskhakaya, David (2008). "Chapter 6: The Particle-in-Cell Method". In Fehske, Holger; Schneider, Ralf; Weiße, Alexander.*Computational Many-Particle Physics*. Lecture Notes in Physics 739. Springer, Berlin Heidelberg. doi:10.1007/978-3-540-74686-7. ISBN 978-3-540-74685-0. **^**Takizuka, Tomonor; Abe, Hirotada (1977). "A binary collision model for plasma simulation with a particle code".*Journal of Computational Physics*.**25**(3): 205–219. Bibcode:1977JCoPh..25..205T. doi:10.1016/0021-9991(77)90099-7.**^**Birdsall, C.K. (1991). "Particle-in-cell charged-particle simulations, plus Monte Carlo collisions with neutral atoms, PIC-MCC".*IEEE Transactions on Plasma Science*.**19**(2): 65–85. Bibcode:1991ITPS...19...65B. doi:10.1109/27.106800. ISSN 0093-3813.**^**Vahedi, V.; Surendra, M. (1995). "A Monte Carlo collision model for the particle-in-cell method: applications to argon and oxygen discharges".*Computer Physics Communications*.**87**(1–2): 179–198. Bibcode:1995CoPhC..87..179V. doi:10.1016/0010-4655(94)00171-W. ISSN 0010-4655.**^**Tskhakaya, D.; Matyash, K.; Schneider, R.; Taccogna, F. (2007). "The Particle-In-Cell Method".*Contributions to Plasma Physics*.**47**(8-9): 563–594. Bibcode:2007CoPP...47..563T. doi:10.1002/ctpp.200710072.**^**Liu, G.R.; M.B. Liu (2003).*Smoothed Particle Hydrodynamics: A Meshfree Particle Method*. World Scientific. ISBN 981-238-456-1.**^**Byrne, F. N.; Ellison, M. A.; Reid, J. H. (1964). "The particle-in-cell computing method for fluid dynamics".*Methods Comput. Phys*.**3**(3): 319–343. Bibcode:1964SSRv....3..319B. doi:10.1007/BF00230516.

- Birdsall, Charles K.; A. Bruce Langdon (1985).
*Plasma Physics via Computer Simulation*. McGraw-Hill. ISBN 0-07-005371-5.

- Hockney, Roger W.; James W. Eastwood (1988).
*Computer Simulation Using Particles*. CRC Press. ISBN 0-85274-392-0.

- Particle-In-Cell and Kinetic Simulation Software Center (PICKSC), UCLA.
- Open source 3D Particle-In-Cell code for spacecraft plasma interactions (mandatory user registration required).
- Simple Particle-In-Cell code in MATLAB
- Plasma Theory and Simulation Group (Berkeley) Contains links to freely-available software.
- Introduction to PIC codes (Univ. of Texas)
- OpenPIC3D - 3D Hybrid Particle-In-Cell simulation of plasma dynamics

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