There are a class of numerical methods that make use of the characteristic lines of hyperbolic Partial Differential Equations (PDEs). These methods reduce a hyperbolic PDE to a family of ordinary differential equations that can be integrated from initial values to get the the solution of the original partial differential equation. By using the integration along the characteristic lines, all partial derivatives of the unknown distribution function with respect to phase-space coordinates are avoided, and thus this method does not need a regular phase-space mesh to construct numerical approximation to the phase-space partial differential operators. This enables us to adopt random sampling of the phase-space, which has the advantage of reducing the error in evaluating high-dimensional phase-space integration. In this sense, this method can be considered as a kind of “mesh-free” method.
Since the characteristic lines are usually identical to the orbits of particles/elements in the phase-space, these methods can be generally called “particle methods”. In the core of particle methods, are the method of characteristics and Monte-Carlo sampling and integration.
The computational nodes (often called markers/particles/super-particles) in particle methods follow the particle orbits and thus their positions in phase-space evolve with time, which is one of the differences between the particle methods and the usual Euler-grid-based methods that usually uses fixed grid-points. The evolving computational nodes in particle method are often called Lagrangian markers while the fixed grids are often called Euler grid-points. Therefore, particle method is a kind of mesh-free Lagrangian method.
The so-called Particle-In-Cell (PIC) method, however, is a kind of incomplete particle method, in which both evolving nodes (mesh-free) and fixed grid-points (mesh) are used. Specificly, the kinetic equation for the particle distribution function in the phase space is solved by using mesh-free particle methods, whereas Maxwell’s eqaution for the electromagnetic field are solved by using spatial mesh. To obtain the source terms in Maxwell’s equation at grids, we need to calculate moments of the distribution function represented by the mesh-free computational nodes. In PIC simulations, the values of a moment at a spatial grid is approximated by the averaged value of the moment in the corresponding spatial cell. This averaging procedure, along with the use of finite spatial shape function of markers (discussed later), has the effect of reducing (ideally removing) collisions between particles. Collisions, if to be modeled, should be modeled by other means.
For instance, in solving the Vlasov-Poisson equations by the PIC method, the Vlasov equation for the phase-space distribution function is solved by using the evolving nodes while the Poisson equation for the spatial electric potential is solved by using the fixed grid-points. The special nomenclature “particle-in-cell” refers to the particular way of inferring the value of charge density at the fixed spatial grid-points from the evolving coordinates of computational markers. Therefore the PIC method is a kind of hybrid particle-mesh methods.
This note discusses the basic theory of the Particle-In-Cell (PIC) simulation, along with practical implementation in curvilinear coordinate system. In the process of learning the PIC simulation method, I have developed several toy Fortran codes to test what I have learned. The numerical results given in these notes are obtained by using these codes. A copy of these codes can be found at http://theory.ipp.ac.cn/˜yj/codes/pic_1D_src.tar