  We present a suite of cosmological N-body simulations with cold dark matter
and baryons aiming at modeling the low-density regions of the IGM as probed by
the Lyman-$\alpha$ forests at high redshift. The simulations are designed to
match the requirements imposed by the quality of BOSS and eBOSS data. They are
made using either 2x768$^3$ or 2x192$^3$ particles, spanning volumes ranging
from (25 Mpc.h$^{-1})^3$ for high-resolution simulations to (100
Mpc.h$^{-1})^3$ for large-volume ones. Using a splicing technique, the
resolution is further enhanced to reach the equivalent of simulations with
2x3072$^3$= 58 billion particles in a (100 Mpc.h$^{-1}$)^3 box size, i.e. a
mean mass per gas particle of 1.2x10$^5$M_sun.h$^{-1}$. We show that the
resulting power spectrum is accurate at the 2% level over the full range from a
few Mpc to several tens of Mpc. We explore the effect on the one-dimensional
transmitted-flux power spectrum of 4 cosmological parameters ($n_s, \sigma_8,
\Omega_m, H_0$) and 2 astrophysical parameters ($T_0, \gamma$) related to the
heating rate of the IGM. By varying the input parameters around a central model
chosen to be in agreement with the latest Planck results, we built a grid of
simulations that allows the study of the impact on the flux power spectrum of
these six relevant parameters. We improve upon previous studies by not only
measuring the effect of each parameter individually, but also probing the
impact of the simultaneous variation of each pair of parameters. We thus
provide a full second-order expansion, including cross-terms, around our
central model. We check the validity of the second-order expansion with
independent simulations obtained either with different cosmological parameters
or different seeds. Finally, a comparison to the one-dimensional Ly-$\alpha$
forest power spectrum obtained with BOSS shows an excellent agreement.
