Time dependent non-LTE calculations of ionisation in the early universe

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.