Time Dependent Monochromatic Scattering of Radiation in One Dimensional Media: Analytical and Numerical Solutions

TL;DR

This paper derives an exact analytical solution and evaluates a numerical method for modeling time-dependent monochromatic radiation scattering in a one-dimensional medium, with applications to early universe radiation.

Contribution

It provides an exact solution for radiative transport in 1D media and assesses the Lax-Wendroff numerical method's effectiveness for this problem.

Findings

Lax-Wendroff method is suitable for this problem

Series expansion helps overcome triangular domain difficulty

Numerical and exact solutions agree within estimated accuracy

Abstract

In order to choose a numerical method for solving the time dependent equations of radiative transport, we obtain an exact solution for the time dependent radiation field in a one dimensional infinite medium with monochromatic, isotropic scattering for sources with an arbitrary spatial distribution and an arbitrary time variation of their power. The Lax-Wendroff method seems to be the most suitable. Because it is assumed that radiation delay is caused by the finite speed of light, the following difficulty arises when the numerical method is used: the region of variation of the variables (dimensionless coordinate \tau and time t) is triangular (the inequality \tau< t). This difficulty is overcome by expanding the unknown functions in series in terms of small values of the time and coordinate. By comparing the numerical and exact solutions for a point source with a given time dependence for its power and with pure scattering, the steps in the variables required to obtain a desired accuracy are estimated. This numerical method can be used to calculate the intensity and polarization of the radiation from sources in the early universe during epochs close to the recombination epoch.