  We introduce an efficient and accurate alternative to full hydrodynamic
simulations, Hydro-PM (HPM), for the study of the low column density
Lyman-alpha forest. It consists of a Particle-Mesh solver, modified to compute,
in addition to the gravitational potential, an effective potential due to the
gas pressure. Such an effective potential can be computed from the density
field because of a tight correlation between density and pressure in the low
density limit, which can be calculated for any photo-reionization history by a
method outlined in Hui & Gnedin (1997). Such a correlation exists, in part,
because of minimal shock-heating in the low density limit. We compare carefully
the density and velocity fields as well as absorption spectra, computed using
HPM versus hydrodynamic simulations, and find good agreement. We show that HPM
is capable of reproducing measurable quantities, such as the column density
distribution, computed from full hydrodynamic simulations, to a precision
comparable to that of observations. We discuss how, by virtue of its speed and
accuracy, HPM can enable us to use the Lyman-alpha forest as a cosmological
probe. We also discuss in detail the smoothing of the gas (or baryon)
fluctuation relative to that of the dark matter on small scales due to finite
gas pressure. It is shown the conventional wisdom that the linear gas
fluctuation is smoothed on the Jeans scale is incorrect for general
reionization (or reheating) history; the correct linear filtering scale is in
general smaller than the Jeans scale after reheating, but larger prior to it.
(abridged)
