A Set of Benchmark Tests for Validation of 3-D Particle in Cell Methods

While the particle-in-cell (PIC) method is quite mature, verification and validation of both newly developed methods and individual codes have largely focused on an idiosyncratic choice of a few test cases. Many of these test cases involve either 1-D or 2-D simulations. This is either due to the availability of (quasi-) analytic solutions or historical reasons. In addition, tests often focus on the investigation of particular physics problems, such as particle emission or collisions, and do not necessarily study the combined impact of the suite of algorithms necessary for a full-featured PIC code. As 3-D codes become the norm, there is a lack of benchmark tests that can establish the validity of these codes; existing papers either do not delve into the details of the numerical experiment or provide other measurable numeric metrics (such as noise) that are outcomes of the simulation. This article seeks to provide several test cases that can be used for validation and bench-marking of PIC codes in 3-D. We focus on examples that are collisionless and can be run with reasonable computational resources. Three test cases are presented in significant detail; these include basic particle motion, beam expansion, and adiabatic expansion of plasma. All presented cases are compared either against existing analytical data or other codes. We anticipate that these cases should help fill the void of bench-marking and validation problems and help the development of new PIC codes.