Grid instability growth rates of electrostatic particle-in-cell simulation algorithms

An exhaustive study of grid-instability behavior in uniform drifting Maxwellian plasmas is presented for several explicit electrostatic particle-in-cell algorithms. Studied are the standard momentum conserving algorithm (MPIC), an energy conserving algorithm (EPIC), the Precise-Particle-in-Fourier method (PPIF), and a new cubic-spline based method (Particle-in-Cubic-Spline, PICS). The new PICS method has computational effort with ``PIC scaling'', while PPIF does not. MPIC and EPIC are studied for the three lowest-order shape functions. These methods are compared on the basis of accuracy, region of instability in the drift-velocity/temperature plane, instability growth rate, and particle noise. It is found that PPIF is stable throughout the plane and that MPIC is unstable near the origin and for vanishing temperature (i.e. Debye length is less than cell size), regardless of shape function. EPIC and PICS are stable for vanishing drift velocity. PICS is shown to have error that is fourth-order in the cell size, while MPIC and EPIC have second-order error, regardless of the shape function. Thermal noise is also lower for PICS, compared with MPIC and EPIC, but all methods have Monte Carlo noise scaling, $1/\sqrt{N_p}$, with particle number, $N_p$.