Elliptic PDE formulation of general, three-dimensional high-order [formula omitted]-approximations for radiative transfer

Abstract

A new methodology is presented for the application of the spherical harmonics method ( P N ) to decompose the radiative transfer equation (RTE) into a set of coupled second-order partial differential equations, allowing for variable properties and arbitrary three-dimensional geometries. The proposed methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P N -approximations ( = ( N + 1 ) 2 ). The result is a relatively small set ( = N ( N + 1 ) / 2 ) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. Moreover, the general boundary conditions and supplementary conditions to be used in the P N -approximation are formulated for arbitrary three-dimensional geometries. Numerical computations are carried out with the P 3 -approximation for several one-dimensional and two-dimensional problems with emitting, absorbing and scattering media, using both constant and variable properties. Results are compared to analytical solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.