  In the stellar chromospheres, radiative energy transport is dominated by only
the strongest spectral lines. For these lines, the approximation of local
thermodynamic equilibrium (LTE) is known to be very inaccurate, and a state of
equilibrium cannot be assumed in general. To calculate the radiative energy
transport under these conditions, the population evolution equation must be
evaluated explicitly, including all time-dependent terms. We develop a
numerical method to solve the evolution equation for the atomic-level
populations in a time-implicit way, keeping all time-dependent terms to first
order. We show that the linear approximation of the time dependence of the
populations can handle very large time steps without losing the accuracy. We
reproduce the benchmark solutions from earlier, well-established works in terms
of non-LTE kinetic equilibrium solution and typical ionization/recombination
time-scales in the solar chromosphere.
