  The ionization of hydrogen in the solar chromosphere and transition region
does not obey LTE or instantaneous statistical equilibrium because the
timescale is long compared with important hydrodynamical timescales, especially
of magneto-acoustic shocks. We implement an algorithm to compute
non-equilibrium hydrogen ionization and its coupling into the MHD equations
within an existing radiation MHD code, and perform a two-dimensional simulation
of the solar atmosphere from the convection zone to the corona. Analysis of the
simulation results and comparison to a companion simulation assuming LTE shows
that: a) Non-equilibrium computation delivers much smaller variations of the
chromospheric hydrogen ionization than for LTE. The ionization is smaller
within shocks but subsequently remains high in the cool intershock phases. As a
result, the chromospheric temperature variations are much larger than for LTE
because in non-equilibrium, hydrogen ionization is a less effective internal
energy buffer. The actual shock temperatures are therefore higher and the
intershock temperatures lower. b) The chromospheric populations of the hydrogen
n = 2 level, which governs the opacity of Halpha, are coupled to the ion
populations. They are set by the high temperature in shocks and subsequently
remain high in the cool intershock phases. c) The temperature structure and the
hydrogen level populations differ much between the chromosphere above
photospheric magnetic elements and above quiet internetwork. d) The hydrogen n
= 2 population and column density are persistently high in dynamic fibrils,
suggesting that these obtain their visibility from being optically thick in
Halpha also at low temperature.
