  Our understanding of the Lyman-$\alpha$ forest has received a great boost
with the advent of the Keck Telescope and large 3D hydrodynamical simulations.
We present new simulations using the SPH technique with a P^3MG
(Particle-Particle Particle-MultiGrid) non-periodic gravity solver. Our method
employs a high resolution (1 kpc) inner volume, essential for capturing the
complex gas physics, a larger low resolution volume, essential for correct
larger scale tidal fields and a self-consistently applied, uniform tidal field
to model the influence of ultra long waves. We include a photoionizing UV flux
and relevant atomic cooling processes. We use constrained field realisations to
probe a selection of environments and construct a statistical sample
representative of the wider universe. We generate artificial Lyman-$\alpha$
spectra and fit Voigt profiles. We examine the importance of (1) the
photoionizing flux level and history, (2) tidal environment and (3) differing
cosmologies, including CDM and CDM+{$\Lambda$}. With an appropriate choice for
the UV flux, we find that the data is fit quite well if the rms density
contrast is $\sim$ 1 at $z\sim 3$ on galaxy scales.
