The relativistic implicit Particle-in-Cell method

Relativistic Particle-in-Cell (PiC) methods are among the most reliable methods for the investigation of plasma phenomena at the particle scale. Standard explicit and semi-implicit PiC methods are affected by numerical instabilities that restrain the range of admissible simulation parameters, and prevent their application to large domains over long time scales. Here, we present a three-dimensional, fully-implicit algorithm for relativistic PiC simulations that conserves energy exactly (to machine precision) and eliminates numerical instabilities, allowing long-term calculations. We describe the full numerical solution procedure and test the algorithm with several one- and two-dimensional case-studies of kinetic instabilities, comparing the results with theoretical predictions.