Numerical solutions to linear transfer problems of polarized radiation

Abstract

Context. The numerical modeling of the generation and transfer of polarized radiation is a key task in solar and stellar physics research and has led to a relevant class of discrete problems that can be reframed as linear systems. In order to solve such problems, it is common to rely on efficient stationary iterative methods. However, the convergence properties of these methods are problemdependent, and a rigorous investigation of their convergence conditions, when applied to transfer problems of polarized radiation, is still lacking. Aims. After summarizing the most widely employed iterative methods used in the numerical transfer of polarized radiation, this article aims to clarify how the convergence of these methods depends on different design elements, such as the choice of the formal solver, the discretization of the problem, or the use of damping factors. The main goal is to highlight advantages and disadvantages of the different iterative methods in terms of stability and rate of convergence. Methods. We first introduce an algebraic formulation of the radiative transfer problem. This formulation allows us to explicitly assemble the iteration matrices arising from different stationary iterative methods, compute their spectral radii and derive their convergence rates, and test the impact of different discretization settings, problem parameters, and damping factors. Results. Numerical analysis shows that the choice of the formal solver significantly affects, and can even prevent, the convergence of an iterative method. Moreover, the use of a suitable damping factor can both enforce stability and increase the convergence rate. Conclusions. The general methodology used in this article, based on a fully algebraic formulation of linear transfer problems of polarized radiation, provides useful estimates of the convergence rates of various iterative schemes. Additionally, it can lead to novel solution approaches as well as analyses for a wider range of settings, including the unpolarized case.

Modeling the scattering polarization of the solar Ca I 4227 Å line with angle-dependent partial frequency redistribution

“…Step 2 was carried out by means of a Matlab code that solves the linear problem (12) in the general AD case with a Richardson iterative method (see Janett et al 2021), using the unpolarized radiation field calculated at step 1 as an initial guess. In step 2, a two-level atomic model was considered.…”

“…Unlike the original version of the code, this modified version uses a damped Jacobi iterative method 3 and the DELO-parabolic formal solver. The damped Jacobi method is used because the Jacobi iterative method does not guarantee convergence when used in combination with the DELO-parabolic formal solver (see Janett et al 2021). A detailed analysis on the convergence properties of various iterative methods applied to transfer problems of polarized radiation can be found in Janett et al (2021).…”

“…After a suitable discretization of the continuous problem (see, e.g., Janett et al 2021), the solution of the transfer Eq. ( 1) and the calculation of the emission coefficients (3) can be expressed in the compact matrix form…”

“…Although this problem can in principle be solved in a single step with a direct method, this task is beyond current computational capabilities. A stationary or Krylov iterative method can therefore be applied (see, e.g., Janett et al 2021;Benedusi et al 2021), using the unpolarized radiation field obtained in the first step as an initial guess. Although it is not fully consistent, this two-step approach permits one to obtain an accurate estimate of the lower level population from a multilevel atomic model, while retaining the simplicity of a two-level atom when accounting for polarization.…”

“…The idea of damping consists in multiplying the correction in the Jacobi iteration by a damping parameter (or relaxation factor) ω, which typically satisfies 0 < ω < 1 (see, e.g.,Hackbusch 1994;Janett et al 2021). …”