  When modeling laser wakefield acceleration (LWFA) using the particle-in-cell
(PIC) algorithm in a Lorentz boosted frame, the plasma is drifting
relativistically at $\beta_b c$ towards the laser, which can lead to a
computational speedup of $\sim \gamma_b^2=(1-\beta_b^2)^{-1}$. Meanwhile, when
LWFA is modeled in the quasi-3D geometry in which the electromagnetic fields
and current are decomposed into a limited number of azimuthal harmonics,
speedups are achieved by modeling three dimensional problems with the
computation load on the order of two dimensional $r-z$ simulations. Here, we
describe how to combine the speed ups from the Lorentz boosted frame and
quasi-3D algorithms. The key to the combination is the use of a hybrid Yee-FFT
solver in the quasi-3D geometry that can be used to effectively eliminate the
Numerical Cerenkov Instability (NCI) that inevitably arises in a Lorentz
boosted frame due to the unphysical coupling of Langmuir modes and EM modes of
the relativistically drifting plasma in these simulations. In addition, based
on the space-time distribution of the LWFA data in the lab and boosted frame,
we propose to use a moving window to follow the drifting plasma to further
reduce the computational load. We describe the details of how the NCI is
eliminated for the quasi-3D geometry, the setups for simulations which combine
the Lorentz boosted frame and quasi-3D geometry, the use of a moving window,
and compare the results from these simulations against their corresponding lab
frame cases. Good agreement is obtained, particularly when there is no
self-trapping, which demonstrates it is possible to combine the Lorentz boosted
frame and the quasi-3D algorithms when modeling LWFA to achieve unprecedented
speedups.
