On the application of the ADO method to the solution of two-dimensional radiative transfer problems in anisotropic scattering media

Abstract

In this work, the Analytical Discrete Ordinates (ADO) method is applied to the solution of the discrete ordinates approximation of the two-dimensional radiative transfer equation in Cartesian geometry. The phase function is expanded in terms of Legendre polynomials up to arbitrary order L to model higher-order anisotropy. The discrete ordinates equations are transversally integrated over regions of the domain, yielding one-dimensional equations for average angular intensities in x and y-directions. The ADO method is then applied to the one-dimensional equations, with approximations for the unknown intensities on the contours of the regions, to derive explicit solutions for the spatial variables. This approach, based on the use of the ADO method along with the nodal technique, is usually referred to as ADO-Nodal. The scheme does not use sweeping. Linear systems have to be solved to fully establish the general solutions. Here, using a recent rearrangement of those linear systems, numerical results are provided for more refined domain partitions, into regions, than previously. Higher-order quadrature schemes are also explored. Comparisons with few results available in the literature reinforce the good performance of the method. A detailed analysis is carried out to contribute in the direction of obtaining reference solutions for the class of problems investigated. Furthermore, numerical results for different optical thicknesses and section average intensities are included, that according to our knowledge it has not been presented before in the literature.