Comparing Momentum-Conserving and Energy-Conserving Electrostatic Particle-In-Cell Schemes in VSim

The Particle-in-Cell (PIC) method has been a useful tool for understanding basic plasma processes for over 50 years. In this paper we revisit an energy conserving (EPIC) scheme [1], [2] and demonstrate how useful it can be (particularly in 2D and 3D) in a large class of problems such as ion sources and capacitively coupled plasmas. The key to PIC performance is computing particle-particle interactions via discretized fields. The most straightforward PIC scheme uses the same interpolation scheme both to interpolate the charges to the grid and the field back to the charged particle location. This is known as a “momentum-conserving scheme” (MPIC). The severe limitation to this scheme is that the cell size must be smaller than the electron Debye length in order to avoid the so-called grid heating instability. Recently we have implemented an explicit energy-conserving (in the limit of infinitesimal timestep) particle deposition scheme (EPIC) [1], [2] in VSim [3], [4] in which the Debye length restriction can be relaxed. We follow previous work in 1D [1], [2] and extend the results to 2D and 3D. We find that EPIC is stable in the regime that the electron drift velocity is less than the thermal velocity and unstable for some cold beams. In this paper we first discuss the basic theory behind the implementation of EPIC. We then discuss the limitations of its application. Finally, we show simulations which benchmark results between MPIC and EPIC. We find that for a large class of problems EPIC can decrease the computational cost by allowing for grid sizes larger than the electron Debye length.