  Hydrogen at high temperature and pressure undergoes a phase transition from a
liquid molecular phase to a conductive atomic state, or liquid metallic
hydrogen, sometimes referred to as the plasma phase transition (PPT). The PPT
phase line was observed in a recent experiment studying laser-pulse heated
hydrogen in a diamond anvil cell in the pressure range $\sim 100 - 170
\text{GPa}$ for temperatures up to $\sim 2000 \text{K}$. The experimental
signatures of the transition are (i) a negative pressure-temperature slope,
(ii) a plateau in the heating curve, assumed to be related to the latent heat
of transformation, and (iii) an abrupt increase in the reflectance of the
sample. We present a finite element simulation that accurately takes into
account the position and time dependence of the heat deposited by the laser
pulse. We calculate the heating curves and the sample reflectance and
transmittance. This simulation confirms that the observed plateaus are related
to the phase transition, however we find that large values of latent heat are
needed and may indicate that dynamics at the transition are more complex than
considered in current models. Finally, experiments are proposed that can
distinguish between a change in optical properties due to a transition to a
metallic state or due to closure of the bandgap in molecular hydrogen.
