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.
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.
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.
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.)
Thus, the pusher is required to be of high accuracy and speed and much effort is spent on optimizing the different schemes.
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.
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.
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 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.
The PDEs are treated as an eigenvalue problem and initially a trial solution is calculated using basis functions that are localized in each element.
Particles can be situated anywhere on the continuous domain, but macro-quantities are calculated only on the mesh points, just as the fields are.
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.
Simulating the interaction for every pair of a big system would be computationally too expensive, so several Monte Carlo methods have been developed instead.
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
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.
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.
PIC simulations have also been applied outside of plasma physics to problems in solid and fluid mechanics.