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In the stellar chromospheres, radiative energy transport is dominated by onlythe strongest spectral lines. For these lines, the approximation of localthermodynamic equilibrium (LTE) is known to be very inaccurate, and a state ofequilibrium cannot be assumed in general. To calculate the radiative energytransport under these conditions, the population evolution equation must beevaluated explicitly, including all time-dependent terms. We develop anumerical method to solve the evolution equation for the atomic-levelpopulations in a time-implicit way, keeping all time-dependent terms to firstorder. We show that the linear approximation of the time dependence of thepopulations can handle very large time steps without losing the accuracy. Wereproduce the benchmark solutions from earlier, well-established works in termsof non-LTE kinetic equilibrium solution and typical ionization/recombinationtime-scales in the solar chromosphere.

Non-equilibrium Equation of State in stellar atmospheres