The system Eq. 27 for is non-linear with
respect to the
and ne because the coefficients of the
and
-operators are quadratic in
and the dependence of the Saha-Boltzmann factors and the collisional
rates on the electron density, respectively. The system is closed by the
abundance and charge conservation equations. To simplify the iteration
scheme, and to take advantage of the fact that not all levels strongly
influence all radiative transitions, we use a linearized and splitted
iteration scheme for the solution of Eq. 27. This scheme has
the further advantage that many different elements in different ionization
stages and even molecules can be treated consistently. A problem where
this is important is the modeling of nova and supernova atmospheres,
where there are typically very large temperature gradients within the
line forming region of the atmosphere.
To linearize Eq. 27, we follow [33] and replace
terms of the form
in Eq. 27 by
:
The most important advantage of this method is that it requires the solution of large linear systems and low-dimensional non-linear system (for the electron density). Thus, its solution is more stable and uses much less computer resources (time and memory) than the direct solution of the original non-linear equations. This allows us to treat many more levels with this method then with more conventional algorithms. Using a nested iteration scheme like the one described here will slow down the convergence of the iterations, but this is more than offset for by the possibility of calculating much larger models with less memory. Since we are able to solve a separate equation for each group of elements, we can trivially parallelize the solution by distributing the groups among the available processors. Convergence acceleration methods can in principle be used, but they frequently lead to convergence instabilities in the nested iterations for the solution of the statistical equilibrium equations.
We have so far assumed that the electron density ne is
given. Although this is a good assumption if only trace elements are
considered, the electron density may be sensitive to non-LTE
effects. This can be taken into account by using either a fixed point
iteration scheme for the electron density or, if many species or
molecules are included in the non-LTE equation of state, by a
modification of the LTE partition functions to include the effects of
non-LTE in the ionization equilibrium. The latter method replaces the
partition function, , with its non-LTE
generalization,
, and uses
in the solution of the ionization/dissociation
equilibrium equation. We use this method because of the
large number of elements with various ionization stages as well
as molecules and condensation of dust grains included in
statistical equilibrium calculations (and not all of them in non-LTE).
Our iteration scheme for the solution of the multi-level non-LTE problem can be summarized as follows: (1) for given ni and ne, solve the radiative transfer equation at each wavelength point and update the radiative rates and the approximate rate operator, (2) solve the linear system Eq. 29 for each group for a given electron density, (3) compute new electron densities (by either fixed point iteration or the generalized partition function method), (4) if the electron density has not converged to the prescribed accuracy, go back to step 2, otherwise go to step 1. The iterations are repeated until a prescribed accuracy for the ne and the ni is reached. It is important to account for coherent scattering processes during the solution of the wavelength dependent radiative transfer equation, it explicitly removes a global coupling from the iterations.