Step-by-step verification of Particle-in-Cell codes

The Particle-in-Cell (PIC) method with Monte Carlo collisions (MCC) is widely used in the simulation of plasmas for electric propulsion and laboratory applications. Due to the simplicity of the basic PIC algorithm and the specific modeling needs of the different research groups, many codes have been independently developed. Verification of these codes, i.e. ensuring that the computational code correctly implements the intended mathematical models and algorithms, is of fundamental importance. Different benchmark cases, such as one from Turner et al., Charoy et al., and Villafana et al., have been published in recent years. These have consisted of a complex physical setup, in which many computation modules interact to yield the final result. Although this approach has the advantage of testing the code in a realistic case, it may hide some implementation errors. Moreover, in case of disagreement, the previous works do not provide an easy way to identify the faulty code modules. In this work, we propose a step-by-step approach for the verification of PIC/MCC codes in a 2D-3V electrostatic setup. The criteria for the test cases are (i) they should highlight possible implementation errors by testing the modules separately, whenever possible (ii) they should be free from physical instabilities to avoid chaotic behavior, and (iii) the numerical result should be accompanied by analytical calculations, for confirmation purposes. The 5 test cases identified all show excellent agreement between the authors' codes.