Grid instability growth rates for explicit, electrostatic momentum- and energy-conserving particle-in-cell algorithms

When the Debye length is not resolved in a simulation using the most common particle-in-cell (PIC) algorithm, the plasma will unphysically heat until the Debye length becomes resolved via a phenomenon known as grid heating. This paper presents detailed numerical measurements of grid heating for several explicit PIC algorithms, including the first systematic (covering the Debye length resolution and drift-velocity parameter space) study of grid-heating growth rates for the most common electrostatic momentum-conserving PIC algorithm. Additionally, we derive and test a cubic-spline-based PIC algorithm that ensures that the interpolated electric field has a continuous first derivative but find that a differentiable electric field has minimal impact on grid-heating stability. We also considered energy-conserving PIC algorithms with linear and quadratic interpolation functions. In all cases, we find that unphysical heating can occur for some combinations of Debye under-resolution and plasma drift. We demonstrate analytically and numerically that grid heating cannot be eliminated by using a higher-order field solve and give an analytical expression for the cold-beam stability limits of some energy-conserving algorithms.