Examination of Solution Methods for the Second Order Discrete Ordinates Formulation

Abstract

The radiative transfer equation is cast into a second-order formulation and various solution schemes are examined critically. The second-order formulation is valid for any type of scattering, and the numerical schemes are compatible with conventional control-volume grids. Number of ordinates is reduced to half when compared with the conventional (first-order) discrete ordinate method, while the resulting equation for the radiation transfer is a familiar second-order elliptic type. Methods of discretization are discussed: Taylor series, control-volume, and control-line approaches. The last method is considered most promising and is investigated in detail; i.e., two types of dependent variable evaluation along he line of sight are discussed-numerical differentiation and exact exponential schemes. Sample computation for a square enclosure with a cold black wall and gas of uniform temperature is made to check the characteristics such as false scattering, ray effect, etc. The control-line approach with a higher-order interpolation scheme gives better accuracy in general. Numerical differentiation along the line of sight gives the same result as the exact exponential schemes when the intergrid optical depth is small. However, the latter is more accurate when the medium is optically thick.