Numerical instabilities of standard Particle-In-Cell (PIC) codes were observed and explained very early in their development. In the zero-time step limit, these instabilities arise from the interaction between the spatial grid and the (artificial) particle shape functions. δf PIC codes, which have recently been especially popular in gyrokinetic simulations, suffer from similar instabilities. In the zero time step limit, the numerical stabilities of standard and δf methods are equivalent. Numerical instabilities arise when the simulation grid does not “resolve the Debye length,” but many modern PIC codes use relatively high order shape functions, and as a result, the worst-case numerical growth rates are undetectably small; in addition, some codes use energy-conserving methods which usually prevent this numerical instability from arising. Similarly, a numerical instability was found in a gyrokinetic δf code using a first-order shape function; we show that this is related to the usual PIC numerical instability. In the gyrokinetic case, where waves have acoustic dispersion at a large wavenumber, increasing the grid resolution actually increases the growth rate of the numerical instability, and the prescription of resolving an effective Debye length is not applicable. However, using higher order shape functions is still an effective remedy.

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