  In this work we present a new hybrid method to simulate the thermal effects
of the reionization in cosmological hydrodynamical simulations. The method
improves upon the standard approach used in simulations of the intergalactic
medium (IGM) and galaxy formation without a significant increase of the
computational cost allowing for efficient exploration of the parameter space.
The method uses a small set of phenomenological input parameters and combines a
semi-numerical reionization model to solve for the topology of reionization and
an approximate model of how reionization heats the IGM, with the massively
parallel \texttt{Nyx} hydrodynamics code, specifically designed to solve for
the structure of diffuse IGM gas. We have produced several large-scale high
resolution cosmological hydrodynamical simulations ($2048^3$, $L_{\rm box} =
40$ Mpc/h) with different instantaneous and inhomogeneous HI reionization
models that use this new methodology. We study the IGM thermal properties of
these models and find that large scale temperature fluctuations extend well
beyond the end of reionization. Analyzing the 1D flux power spectrum of these
models, we find up to $\sim 50\%$ differences in the large scale properties
(low modes, $k\lesssim0.01$ s/km) of the post-reionization power spectrum due
to the thermal fluctuations. We show that these differences could allow one to
distinguish between different reionization scenarios already with existing
Ly$\alpha$ forest measurements. Finally, we explore the differences in the
small-scale cutoff of the power spectrum and we find that, for the same heat
input, models show very good agreement provided that the reionization redshift
of the instantaneous reionization model happens at the midpoint of the
inhomogeneous model.
