Response Matrix Benchmark for the 1D Transport Equation with Matrix Scaling

TL;DR

This paper introduces a novel response matrix approach for the 1D transport equation using matrix scaling, simplifying the solution process compared to existing methods.

Contribution

The paper proposes a new response matrix method based on exponential solutions with matrix scaling for the 1D transport equation.

Findings

Simplifies the response matrix solution process.

Provides an alternative to traditional methods.

Potentially improves computational efficiency.

Abstract

The linear 1D transport equation is likely the most solved transport equation in radiative transfer and neutron transport investigations. Nearly every method imaginable has been applied to establish solutions, including Laplace and Fourier transforms, singular eigenfunctions, solutions of singular integral equation, PN expansions, double PN expansions, Chebychev expansions, Lagrange polynomial expansions, numerical discrete ordinates with finite difference, analytical discrete ordinates, finite elements, solutions to integral equations, adding and doubling, invariant imbedding, solution of Ricatti equations and response matrix methods -- and probably more methods of which the authors are unaware. Of those listed, the response matrix solution to the discrete ordinates form of the 1D transport equation is arguably the simplest and most straightforward. Here, we propose another response of exponential solutions but to the first order equation enabled by matrix scaling.