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 ( $$\varpi_0\lt 1$$ ).