Simulations of plasmas with electrostatic PIC models using the finite element method

Abstract

Particle-in-cell (PIC) methods allow the study of plasma behavior by computing the trajectories of finite-size particles under the action of external and self-consistent electric and magnetic fields defined in a grid of points. In this work, the Finite Element Method (FEM) is used in order to obtain the self-consistent fields. An electrostatic PIC-FEM computational code for simulation of one-dimensional (1D) and two-dimensional (2D) plasmas was developed based on two available and independent codes: the first one a 1D PIC code that uses the Finite Difference Method and the other a FEM code developed at the Instituto de Estudos Avancados (IEAv). The Poisson equation is solved and periodic boundary conditions are used. The ion background that neutralizes the total plasma charge is kept fixed and uniformly distributed in the domain of study. The code is tested by studying the fluctuations of the plasma in thermal equilibrium. In thermal equilibrium a plasma sustain fluctuations of various collective modes of electrostatic oscillations, whose spectral distribution can be analytically obtained by using the fluctuation-dissipation theorem and the Kramers-Kronig relation. In both 1D and 2D cases, there are excellent agreement between the spectral distribution curves predicted theoretically and those obtained by simulation for finite size particles and long wavelengths. I IntroductionIn this work, we use the particle-in-cell (PIC) model, originated from the work of J.M. Dawson [1,2,3] in the late fifties, for the simulation of plasmas. In this model, the plasma is represented by thousands of particles (actually macroparticles, since each particle used in the simulation represents thousands of particles of a real experiment). From the positions and velocities of these particles at a certain instant, we calculate the current and charge densities using one of various schemes of particle distribution to a grid of points. The grid spacing ∆x must be sufficiently small to resolve the collective behavior of plasmas (typically ∆x of the order of the Debye length λ De ).Using these charge and current densities, we calculate the self-consistent electric and magnetic fields via Maxwell's equations. Then, with these fields plus the externally applied ones, we obtain the new particle positions and velocities. Finally, we proceed along this basic cycle with a ∆t sufficiently small to resolve the highest frequency of the problem (typically the plasma frequency ω pe ).In the PIC code, Maxwell's equations are usually solved by the Finite Difference Method (FDM) or by Fourier techniques (specially when periodic boundary conditions are involved). In the FDM, the fields are defined in a grid of points, the derivatives are replaced by differences in the values of the field in adjacent grid points, and difference equations on the grid substitute the differential field equations.Another approach is to apply the Finite Element Method (FEM) instead of FDM to solve the Maxwell's equations. In FEM, the system is divided into subsystems of simple ...

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