Chebyshev spectral methods for radiative transfer

Abstract

We study the performance of Chebyshev spectral methods for time-dependent radiative transfer equations. Starting with a method for one-dimensional problems in homogeneous media, we show that the modifications needed to consider more general problems such as inhomogeneous media, polarization, and higher dimensions are straightforward. In this method, we approximate the spatial dependence of the intensity by an expansion of Chebyshev polynomials. This yields a coupled system of integro-differential equations for the expansion coefficients that depend on angle and time. Next, we approximate the integral operation on the angle variables using a Gaussian quadrature rule resulting in a coupled system of differential equations with respect to time. Using a second-order finite difference approximation, we discretize the time variable. We solve the resultant system of equations with an efficient algorithm that makes Chebyshev spectral methods competitive with other methods for radiative transfer equations.