A methodology for the rigorous verification of Particle-in-Cell simulations

A methodology to perform a rigorous verification of Particle-in-Cell (PIC) simulations is presented, both for assessing the correct implementation of the model equations (code verification), and evaluating the numerical uncertainty affecting the simulation results (solution verification). The proposed code verification methodology is a generalization of the procedure developed for plasma simulation codes based on finite difference schemes that was described by Riva et al. [Phys. Plasmas 21, 062301 (2014)] and consists of an order-of-accuracy test using the method of manufactured solutions. The generalization of the methodology for PIC codes consists of accounting for numerical schemes intrinsically affected by statistical noise and providing a suitable measure of the distance between continuous, analytical distribution functions and finite samples of computational particles. The solution verification consists of quantifying both the statistical and discretization uncertainties. The statistical uncertainty is estimated by repeating the simulation with different pseudorandom number generator seeds. For the discretization uncertainty, the Richardson extrapolation is used to provide an approximation of the analytical solution and the grid convergence index is used as an estimate of the relative discretization uncertainty. The code verification methodology is successfully applied to a PIC code that numerically solves the one-dimensional, electrostatic, collisionless Vlasov-Poisson system. The solution verification methodology is applied to quantify the numerical uncertainty affecting the two-stream instability growth rate, which is numerically evaluated thanks to a PIC simulation.