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]
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Technical aspects
For many types of problems, the PIC method 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 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.
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.
PIC simulations have also been applied outside of plasma physics to problems in solid and fluid mechanics. [4] [5]
See also
References
- ^ 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: 403. doi:.
- ^ Hideo Okuda (1972). "Nonphysical noises and instabilities in plasma simulation due to a spatial grid". Journal of Computational Physics 10: 475. doi:.
- ^ Liu, G.R.; M.B. Liu (2003). Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific. ISBN 981-238-456-1.
- ^ Harlow, F. H. (1964), "The particle-in-cell computing method for fluid dynamics", Methods Comput. Phys. 3: 319–343
- 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 0852743920.
External links
- Powerful, free open source 3D Particle-In-Cell code for spacecraft plasma interactions
- Simple Particle-In-Cell code in MATLAB (GWU)
- Plasma Theory and Simulation Group (Berkeley) Contains links to freely-available software.
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