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

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.