Stability and Convergence of an Upwind Finite Difference Scheme for the Radiative Transport Equation

TL;DR

This paper introduces an explicit upwind finite difference scheme for the radiative transport equation, demonstrating its positivity, stability, convergence, and effectiveness through numerical examples.

Contribution

It presents a novel explicit scheme combining finite difference and quadrature methods with proven stability and convergence for radiative transport problems.

Findings

The scheme is positive, stable, and convergent.

Numerical examples confirm the scheme's effectiveness.

It can be used as an iterative method for stationary solutions.

Abstract

An explicit numerical scheme is proposed for solving the initial-boundary value problem for the radiative transport equation in a rectangular domain with completely absorbing boundary condition. An upwind finite difference approximation is applied to the differential terms of the equation, and the composite trapezoidal rule to the scattering integral. The main results are positivity, stability, and convergence of the scheme. It is also shown that the scheme can be regarded as an iterative method for finding numerical solutions to the stationary transport equation. Some numerical examples for the two-dimensional problems are given.