Convergence of Chandrasekhar's Method for the Problem of Diffuse Reflection

Abstract

Let |$I(\tau, \mu), \enspace J(\tau)$| and |$\Im(\tau)$| denote the intensity, average intensity and source function for radiation in a semi-infinite, plane-parallel, isotropically scattering atmosphere, with albedo |$\varpi_0\leqq 1$| and the only external source due to an incident parallel beam. Let |$I_m(\tau, \mu),\enspace J_m(\tau)$| and |$\Im_m(\tau)$| denote the corresponding Chandrasekhar approximations, derived with the aid of either the Gauss or double Gauss quadrature formula. It is proved that |$I_m\rightarrow I, J_m\rightarrow J$| , and |$\Im_m\rightarrow \Im$| uniformly as |$m\rightarrow \infty$| . Error bounds are obtained in the non-conservative case ( |$\varpi_0\lt 1$| ).