Formal solutions for polarized radiative transfer. III. Stiffness and instability

TL;DR

This paper analyzes the stability of numerical methods for polarized radiative transfer equations, identifies instability issues caused by stiffness, and proposes practical solutions to improve accuracy and stability in simulations.

Contribution

It provides a detailed stability analysis of formal solvers for polarized radiative transfer and introduces a pragmatic solver addressing identified instability problems.

Findings

Runge-Kutta methods have limitations in stability for stiff equations

A new formal solver improves stability and accuracy

Comparison with scalar equations highlights polarization-specific challenges

Abstract

Efficient numerical approximation of the polarized radiative transfer equation is challenging because this system of ordinary differential equations exhibits stiff behavior, which potentially results in numerical instability. This negatively impacts the accuracy of formal solvers, and small step-sizes are often necessary to retrieve physical solutions. This work presents stability analyses of formal solvers for the radiative transfer equation of polarized light, identifies instability issues, and suggests practical remedies. In particular, the assumptions and the limitations of the stability analysis of Runge-Kutta methods play a crucial role. On this basis, a suitable and pragmatic formal solver is outlined and tested. An insightful comparison to the scalar radiative transfer equation is also presented.