Particle-in-cell

Last updated
Ebola Virus Particles Ebola Virus Particles (3).jpg
Ebola Virus Particles

In plasma physics, 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.

Contents

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]

Technical aspects

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:

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 P3M.

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.

Basics of the PIC plasma simulation technique

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.

Super-particles

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 acceleration from 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.

The particle mover

Even with super-particles, the number of simulated particles is usually very large (> 105), 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. In PIC simulation the leapfrog method is used, a second-order explicit method. [6] Also the Boris algorithm is used which cancel out the magnetic field in the Newton-Lorentz equation. [7] [8]

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 .

The equations of the Boris scheme which are substitute in the above equations 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 realized that the excellent long term accuracy of nonrelativistic 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. It has also been shown [9] that one can improve on the relativistic Boris push to make it both volume preserving and have a constant-velocity solution in crossed E and B fields.

The field solver

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.

Particle and field weighting

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: [10] 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 [10]

Collisions

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, [11] 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, [12] [13] which does not analyze all particles but uses the maximum collision probability for each charged species instead.

Accuracy and stability conditions

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. [10] [14] 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.

Applications

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. [15] [16]

Electromagnetic particle-in-cell computational applications

Computational applicationWeb siteLicenseAvailabilityCanonical Reference
SHARP [17] Proprietary doi : 10.3847/1538-4357/aa6d13
ALaDyn [18] GPLv3+Open Repo: [19] doi : 10.5281/zenodo.49553
EPOCH [20] GPLv3Open Repo: [21] doi : 10.1088/0741-3335/57/11/113001
FBPIC [22] 3-Clause-BSD-LBNLOpen Repo: [23] doi : 10.1016/j.cpc.2016.02.007
LSP [24] ProprietaryAvailable from ATK doi : 10.1016/S0168-9002(01)00024-9
MAGIC [25] ProprietaryAvailable from ATK doi : 10.1016/0010-4655(95)00010-D
OSIRIS [26] GNU AGPLOpen Repo [27] doi : 10.1007/3-540-47789-6_36
PICCANTE [28] GPLv3+Open Repo: [29] doi : 10.5281/zenodo.48703
PICLas [30] GPLv3+Open Repo: [31] doi : 10.1016/j.crme.2014.07.005

doi : 10.1063/1.5097638

PICMC [32] ProprietaryAvailable from Fraunhofer IST
PIConGPU [33] GPLv3+Open Repo: [34] doi : 10.1145/2503210.2504564
SMILEI [35] CeCILL-BOpen Repo: [36] doi : 10.1016/j.cpc.2017.09.024
iPIC3D [37] Apache License 2.0Open Repo: [38] doi : 10.1016/j.matcom.2009.08.038
The Virtual Laser Plasma Lab (VLPL) [39] ProprietaryUnknown doi : 10.1017/S0022377899007515
Tristan v2 [40] 3-Clause-BSDOpen source, [41] but also has a private version with QED/radiative [42] modules doi : 10.5281/zenodo.7566725 [43]
VizGrain [44] ProprietaryCommercially available from Esgee Technologies Inc.
VPIC [45] 3-Clause-BSDOpen Repo: [46] doi : 10.1063/1.2840133
VSim (Vorpal) [47] ProprietaryAvailable from Tech-X Corporation doi : 10.1016/j.jcp.2003.11.004
Warp [48] 3-Clause-BSD-LBNLOpen Repo: [49] doi : 10.1063/1.860024
WarpX [50] 3-Clause-BSD-LBNLOpen Repo: [51] doi : 10.1016/j.nima.2018.01.035
ZPIC [52] AGPLv3+Open Repo: [53]
ultraPICAProprietaryCommercially available from Plasma Taiwan Innovation Corporation.

See also

Related Research Articles

In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as where the vector r is the relative position between the charges. This interaction complicates the theoretical treatment of the fluid. For example, a naive quantum mechanical calculation of the ground-state energy density yields infinity, which is unreasonable. The difficulty lies in the fact that even though the Coulomb force diminishes with distance as 1/r2, the average number of particles at each distance r is proportional to r2, assuming the fluid is fairly isotropic. As a result, a charge fluctuation at any one point has non-negligible effects at large distances.

<span class="mw-page-title-main">Computational fluid dynamics</span> Analysis and solving of problems that involve fluid flows

Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is typically performed using experimental apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as flight tests.

Verlet integration is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Jean Baptiste Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics. It was also used by P. H. Cowell and A. C. C. Crommelin in 1909 to compute the orbit of Halley's Comet, and by Carl Størmer in 1907 to study the trajectories of electrical particles in a magnetic field . The Verlet integrator provides good numerical stability, as well as other properties that are important in physical systems such as time reversibility and preservation of the symplectic form on phase space, at no significant additional computational cost over the simple Euler method.

<span class="mw-page-title-main">Boltzmann equation</span> Equation of statistical mechanics

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872. The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.

<span class="mw-page-title-main">Smoothed-particle hydrodynamics</span> Method of hydrodynamics simulation

Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method, and the resolution of the method can easily be adjusted with respect to variables such as density.

In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics.

Direct simulation Monte Carlo (DSMC) method uses probabilistic Monte Carlo simulation to solve the Boltzmann equation for finite Knudsen number fluid flows.

<span class="mw-page-title-main">Mesh generation</span> Subdivision of space into cells

Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI, depending on the complexity of the domain and the type of mesh desired. A typical goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine in areas that are important for the subsequent calculations.

Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions in periodic systems. It was first developed as the method for calculating the electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system to accurately calculate the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.

<i>N</i>-body simulation Simulation of a dynamical system of particles

In physics and astronomy, an N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity. N-body simulations are widely used tools in astrophysics, from investigating the dynamics of few-body systems like the Earth-Moon-Sun system to understanding the evolution of the large-scale structure of the universe. In physical cosmology, N-body simulations are used to study processes of non-linear structure formation such as galaxy filaments and galaxy halos from the influence of dark matter. Direct N-body simulations are used to study the dynamical evolution of star clusters.

<span class="mw-page-title-main">Lattice Boltzmann methods</span> Class of computational fluid dynamics methods

The lattice Boltzmann methods (LBM), originated from the lattice gas automata (LGA) method (Hardy-Pomeau-Pazzis and Frisch-Hasslacher-Pomeau models), is a class of computational fluid dynamics (CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations directly, a fluid density on a lattice is simulated with streaming and collision (relaxation) processes. The method is versatile as the model fluid can straightforwardly be made to mimic common fluid behaviour like vapour/liquid coexistence, and so fluid systems such as liquid droplets can be simulated. Also, fluids in complex environments such as porous media can be straightforwardly simulated, whereas with complex boundaries other CFD methods can be hard to work with.

In the study of partial differential equations, the MUSCL scheme is a finite volume method that can provide highly accurate numerical solutions for a given system, even in cases where the solutions exhibit shocks, discontinuities, or large gradients. MUSCL stands for Monotonic Upstream-centered Scheme for Conservation Laws, and the term was introduced in a seminal paper by Bram van Leer. In this paper he constructed the first high-order, total variation diminishing (TVD) scheme where he obtained second order spatial accuracy.

In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are: (i) choose novel unconstrained coordinates, (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods.

Gyrokinetic ElectroMagnetic (GEM) is a gyrokinetic plasma turbulence simulation that uses the particle-in-cell method. It is used to study waves, instabilities and nonlinear behavior of tokamak fusion plasmas. Information about GEM can be found at the GEM web page. There are two versions of GEM, one is a flux-tube version and the other one is a global general geometry version. Both versions of GEM use a field-aligned coordinate system. Ions are treated kinetically, but averaged over their gyro-obits and electrons are treated as drift-kinetic.

In computational fluid dynamics, the immersed boundary method originally referred to an approach developed by Charles Peskin in 1972 to simulate fluid-structure (fiber) interactions. Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations. In the immersed boundary method the fluid is represented in an Eulerian coordinate system and the structure is represented in Lagrangian coordinates. For Newtonian fluids governed by the Navier–Stokes equations, the fluid equations are

Gyrokinetics is a theoretical framework to study plasma behavior on perpendicular spatial scales comparable to the gyroradius and frequencies much lower than the particle cyclotron frequencies. These particular scales have been experimentally shown to be appropriate for modeling plasma turbulence. The trajectory of charged particles in a magnetic field is a helix that winds around the field line. This trajectory can be decomposed into a relatively slow motion of the guiding center along the field line and a fast circular motion, called gyromotion. For most plasma behavior, this gyromotion is irrelevant. Averaging over this gyromotion reduces the equations to six dimensions rather than the seven. Because of this simplification, gyrokinetics governs the evolution of charged rings with a guiding center position, instead of gyrating charged particles.

The multiphase particle-in-cell method (MP-PIC) is a numerical method for modeling particle-fluid and particle-particle interactions in a computational fluid dynamics (CFD) calculation. The MP-PIC method achieves greater stability than its particle-in-cell predecessor by simultaneously treating the solid particles as computational particles and as a continuum. In the MP-PIC approach, the particle properties are mapped from the Lagrangian coordinates to an Eulerian grid through the use of interpolation functions. After evaluation of the continuum derivative terms, the particle properties are mapped back to the individual particles. This method has proven to be stable in dense particle flows, computationally efficient, and physically accurate. This has allowed the MP-PIC method to be used as particle-flow solver for the simulation of industrial-scale chemical processes involving particle-fluid flows.

<span class="mw-page-title-main">Hamiltonian Monte Carlo</span> Sampling algorithm

The Hamiltonian Monte Carlo algorithm is a Markov chain Monte Carlo method for obtaining a sequence of random samples which converge to being distributed according to a target probability distribution for which direct sampling is difficult. This sequence can be used to estimate integrals with respect to the target distribution.

Hybrid stochastic simulations are a sub-class of stochastic simulations. These simulations combine existing stochastic simulations with other stochastic simulations or algorithms. Generally they are used for physics and physics-related research. The goal of a hybrid stochastic simulation varies based on context, however they typically aim to either improve accuracy or reduce computational complexity. The first hybrid stochastic simulation was developed in 1985.

<span class="mw-page-title-main">Momentum mapping format</span>

Momentum mapping format is a key technique in the Material Point Method (MPM) for transferring physical quantities such as momentum, mass, and stress between a material point and a background grid.

References

  1. F.H. Harlow (1955). "A Machine Calculation Method for Hydrodynamic Problems". Los Alamos Scientific Laboratory report LAMS-1956.{{cite journal}}: Cite journal requires |journal= (help)
  2. Dawson, J.M. (1983). "Particle simulation of plasmas". Reviews of Modern Physics. 55 (2): 403–447. Bibcode:1983RvMP...55..403D. doi:10.1103/RevModPhys.55.403.
  3. Hideo Okuda (1972). "Nonphysical noises and instabilities in plasma simulation due to a spatial grid". Journal of Computational Physics. 10 (3): 475–486. Bibcode:1972JCoPh..10..475O. doi:10.1016/0021-9991(72)90048-4.
  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. S2CID   29190330.
  5. 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. S2CID   12893515.
  6. Birdsall, Charles K.; A. Bruce Langdon (1985). Plasma Physics via Computer Simulation. McGraw-Hill. ISBN   0-07-005371-5.
  7. 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.
  8. Qin, H.; et al. (2013). "Why is Boris algorithm so good?" (PDF). Physics of Plasmas. 20 (5): 084503. Bibcode:2013PhPl...20h4503Q. doi:10.1063/1.4818428.
  9. Higuera, Adam V.; John R. Cary (2017). "Structure-preserving second-order integration of relativistic charged particle trajectories in electromagnetic fields". Physics of Plasmas. 24 (5): 052104. Bibcode:2004JCoPh.196..448N. doi:10.1016/j.jcp.2003.11.004.
  10. 1 2 3 Tskhakaya, David (2008). "Chapter 6: The Particle-in-Cell Method". In Fehske, Holger; Schneider, Ralf; Weiße, Alexander (eds.). Computational Many-Particle Physics. Lecture Notes in Physics 739. Vol. 739. Springer, Berlin Heidelberg. doi:10.1007/978-3-540-74686-7 (inactive 2024-05-03). ISBN   978-3-540-74685-0.{{cite book}}: CS1 maint: DOI inactive as of May 2024 (link)
  11. 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.
  12. 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.
  13. 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.
  14. 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. S2CID   221030792.
  15. Liu, G.R.; M.B. Liu (2003). Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific. ISBN   981-238-456-1.
  16. 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 (inactive 2024-05-03). S2CID   121512234.{{cite journal}}: CS1 maint: DOI inactive as of May 2024 (link)
  17. Shalaby, Mohamad; Broderick, Avery E.; Chang, Philip; Pfrommer, Christoph; Lamberts, Astrid; Puchwein, Ewald (23 May 2017). "SHARP: A Spatially Higher-order, Relativistic Particle-in-Cell Code". The Astrophysical Journal. 841 (1): 52. arXiv: 1702.04732 . Bibcode:2017ApJ...841...52S. doi: 10.3847/1538-4357/aa6d13 . S2CID   119073489.
  18. "ALaDyn". ALaDyn. Retrieved 1 December 2017.
  19. "ALaDyn: A High-Accuracy PIC Code for the Maxwell-Vlasov Equations". GitHub.com. 18 November 2017. Retrieved 1 December 2017.
  20. "EPOCH". epochpic. Retrieved 14 March 2024.
  21. "EPOCH". GitHub.com. Retrieved 14 March 2024.
  22. "FBPIC documentation — FBPIC 0.6.0 documentation". fbpic.github.io. Retrieved 1 December 2017.
  23. "fbpic: Spectral, quasi-3D Particle-In-Cell code, for CPU and GPU". GitHub.com. 8 November 2017. Retrieved 1 December 2017.
  24. "Orbital ATK". Mrcwdc.com. Retrieved 1 December 2017.
  25. "Orbital ATK". Mrcwdc.com. Retrieved 1 December 2017.
  26. "OSIRIS open-source - OSIRIS". osiris-code.github.io. Retrieved 13 December 2023.
  27. "osiris-code/osiris: OSIRIS Particle-In-Cell code". GitHub.com. Retrieved 13 December 2023.
  28. "Piccante". Aladyn.github.io. Retrieved 1 December 2017.
  29. "piccante: a spicy massively parallel fully-relativistic electromagnetic 3D particle-in-cell code". GitHub.com. 14 November 2017. Retrieved 1 December 2017.
  30. "PICLas".
  31. "piclas-framework/piclas". GitHub .
  32. "Fraunhofer IST Team Simulation". ist.fraunhofer.de. Retrieved 7 August 2024.
  33. "PIConGPU - Particle-in-Cell Simulations for the Exascale Era - Helmholtz-Zentrum Dresden-Rossendorf, HZDR". picongpu.hzdr.de. Retrieved 1 December 2017.
  34. "ComputationalRadiationPhysics / PIConGPU — GitHub". GitHub.com. 28 November 2017. Retrieved 1 December 2017.
  35. "Smilei — A Particle-In-Cell code for plasma simulation". Maisondelasimulation.fr. Retrieved 1 December 2017.
  36. "SmileiPIC / Smilei — GitHub". GitHub.com. 29 October 2019. Retrieved 29 October 2019.
  37. Markidis, Stefano; Lapenta, Giovanni; Rizwan-uddin (17 Oct 2009). "Multi-scale simulations of plasma with iPIC3D". Mathematics and Computers in Simulation. 80 (7): 1509. doi:10.1016/j.matcom.2009.08.038.
  38. "iPic3D — GitHub". GitHub.com. 31 January 2020. Retrieved 31 January 2020.
  39. Dreher, Matthias. "Relativistic Laser Plasma". 2.mpq.mpg.de. Retrieved 1 December 2017.
  40. "Tristan v2 wiki | Tristan v2". princetonuniversity.github.io. Retrieved 2022-12-15.
  41. "Tristan v2 public github page". GitHub .
  42. "QED Module | Tristan v2". princetonuniversity.github.io. Retrieved 2022-12-15.
  43. "Tristan v2: Citation.md". GitHub .
  44. "VizGrain". esgeetech.com. Retrieved 1 December 2017.
  45. "VPIC". github.com. Retrieved 1 July 2019.
  46. "LANL / VPIC — GitHub". github.com. Retrieved 29 October 2019.
  47. "Tech-X - VSim". Txcorp.com. Retrieved 1 December 2017.
  48. "Warp". warp.lbl.gov. Retrieved 1 December 2017.
  49. "berkeleylab / Warp — Bitbucket". bitbucket.org. Retrieved 1 December 2017.
  50. "WarpX Documentation". ecp-warpx.github.io. Retrieved 29 October 2019.
  51. "ECP-WarpX / WarpX — GitHub". GitHub.org. Retrieved 29 October 2019.
  52. "Educational Particle-In-Cell code suite". picksc.idre.ucla.edu. Retrieved 29 October 2019.
  53. "ricardo-fonseca / ZPIC — GitHub". GitHub.org. Retrieved 29 October 2019.

Bibliography