Matrix riccati equation solution of the 1d radiative transfer equation

TL;DR

This paper introduces the Matrix Riccati Equation Method (MREM), a novel high-precision numerical approach for solving the 1D radiative transfer equation by addressing matrix Riccati ODEs, resulting in highly accurate benchmarks.

Contribution

The paper presents MREM, a new numerical method that solves two matrix Riccati ODEs for the 1D radiative transfer equation, achieving near seven-digit accuracy.

Findings

Achieved high-precision benchmarks with nearly seven decimal places.

Demonstrated the effectiveness of MREM in solving the 1D radiative transfer equation.

Combined MREM with Taylor series and doubling techniques for enhanced accuracy.

Abstract

In recent years, the first author has developed three successful numerical methods to solve the 1D radiative transport equation yielding highly precise benchmarks. The second author has shown a keen interest in novel solution methodologies and an ability for their implementation. Here, we combine talents to generate yet another high precision solution, the Matrix Riccati Equation Method (MREM). MREM features the solution to two of the four matrix Riccati ODEs that arise from the interaction principle of particle transport. Through interaction coefficients, the interaction principle describes how particles reflect from- and transmit through- a single slab. On combination with Taylor series and doubling, a high quality numerical benchmark, to nearly seven places, is established.