A hybrid Maxwell solver for fully relativistic and electromagnetic (EM) particle-in-cell (PIC) codes is described. In this solver, the EM fields are solved in $k$ space by performing an FFT in one direction, while using finite difference operators in
the other direction(s). This solver eliminates the numerical Cerenkov radiation for particles moving in the preferred direction. Moreover, the numerical Cerenkov instability (NCI) induced by the relativistically drifting plasma and beam can be eliminated using this hybrid solver by applying strategies that are similar to those recently developed for pure FFT solvers. A current correction is applied for the charge conserving current deposit to correctly account for the EM calculation in hybrid Yee-FFT solver. A theoretical analysis of the dispersion properties in vacuum and in a drifting plasma for the hybrid solver is presented, and compared with PIC simulations with good agreement obtained. This hybrid solver is applied to both 2D and 3D Cartesian and quasi-3D (in which the fields and current are decomposed into azimuthal harmonics) geometries. Illustrative results for laser wakefield accelerator simulation in a Lorentz boosted frame using the hybrid solver in the 2D Cartesian geometry are presented, and compared against results from 2D UPIC-EMMA simulation which uses a pure spectral Maxwell solver, and from OSIRIS 2D lab frame simulation using the standard Yee solver. Very good agreement is obtained which demonstrates the feasibility of using the hybrid solver for high fidelity simulation of relativistically drifting plasma with no evidence of the numerical Cerenkov instability.
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 electr
omagnetic waves on a grid to wave-particle resonances, including aliased resonances, i.e., $omega + 2pimu/Delta t=(k_1+ 2pi u_1/Delta x_1)v_0$, where $mu$ and $ u_1$ refer to the time and space aliases and the plasma is drifting relativistically at velocity $v_0$ in the $hat{1}$-direction. Recent studies have shown that an EM-PIC code which uses a spectral field solver and a low pass filter can eliminate the fastest growing modes of the NCI. Based on these studies a new spectral PIC code for studying laser wakefield acceleration (LWFA) in the Lorentz boosted frame was developed. However, we show that for parameters of relevance for LWFA simulations in the boosted frame, a relativistically drifting plasma is susceptible to a host of additional unstable modes with lower growth rates, and that these modes appear when the fastest growing unstable modes are filtered out. We show that these modes are most easily identified as the coupling between modes which are purely transverse (EM) and purely longitudinal (Langmuir) in the rest frame of the plasma for specific time and space aliases. We rewrite the dispersion relation of the drifting plasma for a general field solver and obtain analytic expressions for the location and growth rate for each unstable mode, i.e, for each time and space aliased resonances. We show for the spectral solver that when the fastest growing mode is eliminated a new mode at the fundamental resonance ($mu= u_1=0$) can be seen. (Please check the whole abstract in the paper).
