Chebyshev series solution for radiative transport in a medium with a linearly anisotropic scattering phase function

Abstract

One‐dimensional radiative heat transfer is considered in a plane‐parallel geometry for an absorbing, emitting, and linearly anisotropic scattering medium subjected to azimuthally symmetric incident radiation at the boundaries. The integral form of the transport equation is used throughout the analysis. This formulation leads to a system of weakly‐singular Fredholm integral equations of the second kind. The resulting unknown functions are then formally expanded in Chebyshev series. These series representations are truncated at a specified number of terms, leaving residual functions as a result of the approximation. The collocation and the Ritz‐Galerkin methods are formulated, and are expressed in terms of general orthogonality conditions applied to the residual functions. The major contribution of the present work lies in developing quantitative error estimates. Error bounds are obtained for the approximating functions by developing equations relating the residuals to the errors and applying functional norms to the resulting set of equations. The collocation and Ritz‐Galerkin methods are each applied in turn to determine the expansion coefficients of the approximating functions. The effectiveness of each method is interpreted by analyzing the errors which result from the approximations.