  We present a new implicit numerical algorithm for the calculation of the time
dependent non-Local Thermodynamic Equilibrium of a gas in an external radiation
field that is accurate, fast and unconditionally stable for all spatial and
temporal increments. The method is presented as a backward difference scheme in
1-D but can be readily generalised to 3-D. We apply the method for calculating
the evolution of ionisation domains in a hydrogen plasma with plane-parallel
Gaussian density enhancements illuminated by sources of UV radiation. We
calculate the speed of propagation of ionising fronts through different ambient
densities and the interaction of such ionising fronts with density
enhancements. We show that for a typical UV source that may be present in the
early universe, the introduction of a density enhancement of a factor ~10 above
an ambient density 10^{-4} atoms/cm^3 could delay the outward propagation of an
ionisation front by millions of years. Our calculations show that within the
lifetime of a single source (~a few million years), and for ambient
intergalactic densities appropriate to redshifts z~6-20, neutral fractions of
of ~10^{-3}-10^{-5} can be achieved within its zone of influence. We also
present calculations which demonstrate that once started, ionisation will
proceed very efficiently as multiple sources are subsequently introduced, even
if the time between the appearence of such sources may be much longer than
their lifetimes.
