Simulation of plasma accelerators with the Particle-In-Cell method

We present the standard electromagnetic Particle-in-Cell method, starting from the discrete approximation of derivatives on a uniform grid. The application to second-order, centered, finite-difference discretization of the equations of motion and of Maxwells equations is then described in one dimension, followed by two and three dimensions. Various algorithms are presented, for which we discuss the stability and accuracy, introducing and elucidating concepts like numerical stochastic heating, CFL limit and numerical dispersion. The coupling of the particles and field quantities via interpolation at various orders is detailed, together with its implication on energy and momentum conserving. Special topics of relevance to the modeling of plasma accelerators are discussed, such as moving window, optimal Lorentz boosted frame, the numerical Cherenkov instability and its mitigation. Examples of simulations of laser-driven and particle beam-driven accelerators are given, including with mesh refinement. We conclude with a discussion on high-performance computing and a brief outlook.