The Response Matrix Discrete Ordinates Solution to the 1D Radiative Transfer Equation

TL;DR

This paper introduces a new, highly accurate analytical solution to the 1D radiative transfer equation using the discrete ordinates method, enhanced by convergence acceleration and extended to heterogeneous media.

Contribution

It proposes a novel, consistent analytical solution form for the discrete ordinates method, improving accuracy and extending applicability to heterogeneous media with benchmark comparisons.

Findings

Achieves extreme accuracy with Wynn-epsilon acceleration.

Extends solution to heterogeneous media using star product formulation.

Provides a new benchmark for Henyey-Greenstein scattering.

Abstract

The discrete ordinates method (DOM) of solution to the 1D radiative transfer equation has been an effective method of solution for nearly 70 years. During that time, the method has experienced numerous improvements as numerical and computational techniques have become more powerful and efficient. Here, we again consider the analytical solution to the discrete radiative transfer equation in a homogeneous medium by proposing a new, and consistent, form of solution that improves upon previous forms. Aided by a Wynn-epsilon convergence acceleration, its numerical evaluation can achieve extreme accuracy as demonstrated by comparison with published benchmarks. Finally, we readily extend the solution to a heterogeneous medium through the star product formulation producing a novel benchmark for closed form Henyey-Greenstein scattering as an example.