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 \sim 10 above an ambient density 10 ^ { -4 } 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 ( \sim a few million years ) , and for ambient intergalactic densities appropriate to redshifts z \sim 6 - 20 , degrees of ionisation of \sim 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 .