Elimination of the numerical Cerenkov instability for spectral EM-PIC codes

Abstract

When using an electromagnetic particle-in-cell (EM-PIC) code to simulate a relativistically drifting plasma, a violent numerical instability known as the numerical Cerenkov instability (NCI) occurs. The NCI is due to the unphysical coupling of electromagnetic waves on a grid to wave-particle resonances, including aliased resonances, i.e., omega + 2 pi mu/Delta t = (k(1) + 2 pi v(1)/Delta x(1))nu(0), where mu and v(1) refer to the time and space aliases and the plasma is drifting relativistically at velocity nu(0) in the (1) over cap -direction. We extend our previous work Xu et al. (2013) by recasting the numerical dispersion relation of a relativistically drifting plasma into a form which shows explicitly how the instability results from the coupling modes which are-purely-transverse electromagnetic (EM) modes-and purely longitudinal modes in-the rest frame of the plasma for each time and space aliasing. The dispersion relation for each mu and v(1) is the product of the dispersion relation of these two modes set equal to a coupling term that vanishes in the continuous limit. The new form of the numerical dispersion relation provides an accurate method of systematically calculating the growth rate and location of the mode in the fundamental Brillouin zone for any Maxwell solver for each mu, and v(1). We then focus on the spectral Maxwell solver and systematically discuss its NCI modes. We show that the second fastest growing NCI mode for the spectral solver corresponds to mu = v(1) = 0, that it has a growth rate approximately one order of magnitude smaller than the fastest growing mu = 0 and v(1) = 1 mode, and that its location in the k space fundamental Brillouin zone is sensitive to the grid size and time step. Based on these studies, strategies to systematically eliminate the NCI modes for a spectral solver are developed. We apply these strategies to both relativistic collisionless shock and LWFA simulations, and demonstrate that high-fidelity multi-dimensional simulations of drifting plasmas can be carried out with a spectral Maxwell solver with no evidence of numerical Cerenkov instability.