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Context. The numerical modeling of the generation and transfer of polarizedradiation is a key task in solar and stellar physics research and has led to arelevant class of discrete problems that can be reframed as linear systems. Inorder to solve such problems, it is common to rely on efficient stationaryiterative methods. However, the convergence properties of these methods areproblem-dependent, and a rigorous investigation of their convergenceconditions, when applied to transfer problems of polarized radiation, is stilllacking. Aims. After summarizing the most widely employed iterative methodsused in the numerical transfer of polarized radiation, this article aims toclarify how the convergence of these methods depends on different designelements, such as the choice of the formal solver, the discretization of theproblem, or the use of damping factors. The main goal is to highlightadvantages and disadvantages of the different iterative methods in terms ofstability and rate of convergence. Methods. We first introduce an algebraicformulation of the radiative transfer problem. This formulation allows us toexplicitly assemble the iteration matrices arising from different stationaryiterative methods, compute their spectral radii and derive their convergencerates, and test the impact of different discretization settings, problemparameters, and damping factors. Conclusions. The general methodology used inthis article, based on a fully algebraic formulation of linear transferproblems of polarized radiation, provides useful estimates of the convergencerates of various iterative schemes. Additionally, it can lead to novel solutionapproaches as well as analyses for a wider range of settings, including theunpolarized case.

Numerical solutions to linear transfer problems of polarized radiation I. Algebraic formulation and stationary iterative methods