Discrete spherical harmonics method for radiative transfer in scalar planar inhomogeneous atmosphere

Abstract

The radiative transfer problems in a participating inhomogeneous scalar planar atmosphere, subjected to diffuse or collimated incidence, are investigated using the discrete spherical harmonics method. In developing the method, the radiative intensity is expanded in a finite series of Legendre polynomials and the resulting first-order coupled differential equations of radiance moments are expressed in a set of discrete polar directions. The method is applied to homogeneous/inhomogeneous atmospheres of various anisotropic scattering degrees and thicknesses, and reflective boundary conditions. The discrete spherical harmonics method albedo, transmittance, and radiative intensity predictions agree well with benchmark literature results. Additionally, numerical predictions show that the discrete spherical harmonics method using Mark boundary conditions are more efficient than using Marshak boundary conditions.

High-order spherical harmonics–nodal collocation scheme for the numerical solution of the time-dependent radiative transfer equation

“…Higher order approximation methods have then been developed to deal with more complex situations: multi-dimensional geometry, anisotropic scattering, variable physical properties, etc. Among the most used, we can cite the discrete ordinate (S N ) method [4,5], the spherical harmonics (P N ) method [2,6,7] or the method of characteristics [8] for the angular dependence in the equation, and the finite-difference [9], finite-volume [5] or finite-element method [10,11] for the spatial discretization (see for example, [3], for a compilation of recent numerical studies), and more recently meshfree methods [12]. Also, the integral formulation of the radiative transfer equation has been used in transient problems [13].…”