Impact of particle number and cell size in fully implicit charge- and energy-conserving particle-in-cell schemes

Fully implicit charge- and energy-conserving electrostatic particle-in-cell codes have recently been investigated for their potential to model plasmas at temporal and spatial scales larger than the plasma period and Debye length, respectively. Recent literature on the topic emphasizes the accuracy of these codes, and more importantly for many researchers, the prospect of orders of magnitude speedup of plasma simulations in 3D (three dimensions). In this paper, we further examine previous case studies on 1D fully implicit charge- and energy-conserving electrostatic codes by varying numerical parameters to determine whether this scalability is achievable. We first apply the scheme on an ion acoustic shockwave with periodic conditions, and then to a series of benchmarks for bounded capacitively coupled radio-frequency plasmas. Our findings show that to reproduce highly resolved convergent solutions, a higher amount of particles per cell need to be used in the implicit scheme for both periodic and bounded simulations when the cell size exceeds the Debye length. The addition of non-uniform grids and collisions is found to exacerbate the errors in the final solutions for the bounded plasma case using the implicit scheme. Combined with the higher computational cost of particle calculations vs field solvers, we found that the implicit scheme leads to an increased runtime in 1D (one dimension) compared to the explicit momentum-conserving algorithm when accuracy relative to a well-resolved converged solution is required.