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We study the statistics of the Lyman-$\alpha$ forest in a flat LCDM cosmologywith the N-body + Eulerian hydrodynamics code Nyx. We produce a suite ofsimulations, covering the observationally relevant redshift range $2 \leq z\leq 4$. We find that a grid resolution of 20 kpc/h is required to produce onepercent convergence of Lyman-$\alpha$ flux statistics, up to k = 10 h/Mpc. Inaddition to establishing resolution requirements, we study the effects ofmissing modes in these simulations, and find that box sizes of L > 40 Mpc/h areneeded to suppress numerical errors to a sub-percent level. Our optically-thinsimulations with the ionizing background prescription of Haardt & Madau (2012)reproduce an IGM equation of state with $T_0 \approx 10^4 K$ and $\gamma\approx 1.55$ at z=2, with a mean transmitted flux close to the observedvalues. When using the ionizing background prescription of Faucher-Giguere etal. (2009), the mean flux is 10-15 per cent below observed values at z=2, and afactor of 2 too small at z = 4. We show the effects of the common practice ofrescaling optical depths to the observed mean flux and how it affectsconvergence rates. We also investigate the common practice of `splicing'results from a number of different simulations to estimate the 1D flux powerspectrum and show it is accurate at the 10 percent level. Finally, we find thatcollisional heating of the gas from dark matter particles is negligible inmodern cosmological simulations.

The Lyman-$\alpha$ forest in optically-thin hydrodynamical simulations