# Discontinuities in numerical radiative transfer

## TL;DR

This paper investigates the challenges and limitations of numerical methods in solving the radiative transfer equation in media with steep gradients or discontinuities, revealing reduced accuracy and convergence issues.

## Contribution

It exposes the limitations of standard convergence analyses for discontinuous media and evaluates existing numerical schemes for their applicability to radiative transfer problems.

## Key findings

- Discontinuities induce first-order errors in radiative transfer solutions.
- High-order convergence is hindered by steep gradients in physical parameters.
- Existing numerical schemes are assessed for their suitability in discontinuous media.

## Abstract

Observations and magnetohydrodynamic simulations of solar and stellar atmospheres reveal an intermittent behavior or steep gradients in physical parameters, such as magnetic field, temperature, and bulk velocities. The numerical solution of the stationary radiative transfer equation is particularly challenging in such situations, because standard numerical methods may perform very inefficiently in the absence of local smoothness. However, a rigorous investigation of the numerical treatment of the radiative transfer equation in discontinuous media is still lacking. The aim of this work is to expose the limitations of standard convergence analyses for this problem and to identify the relevant issues. Moreover, specific numerical tests are performed. These show that discontinuities in the atmospheric physical parameters effectively induce first-order discontinuities in the radiative transfer equation, reducing the accuracy of the solution and thwarting high-order convergence. In addition, a survey of the existing numerical schemes for discontinuous ordinary differential systems and interpolation techniques for discontinuous discrete data is given, evaluating their applicability to the radiative transfer problem.