A Comparison of Angular Discretization Schemes of the Hybrid Finite Volume/finite Element Method for the Solution of the Radiative Transfer Equation

A hybrid finite volume / finite element method was developed a few years ago to solve the radiative transfer equation. In this method, the radiation intensity is approximated as a linear combination of basis functions, dependent only on the angular direction. The coefficients of the approximation are unknown functions of the spatial coordinates. This Galerkin-like approximation is introduced into the radiative transfer equation. Then, this is multiplied by the nth basis function and integrated over all directions, i.e., over a solid angle of 4π, yielding a set of differential equations. The spatial discretization is carried out using the finite volume method, like in the discrete ordinates and finite volume methods, transforming the differential equations into algebraic equations. The angular discretization is accomplished using a methodology similar to that employed in the finite element method. In previous works, the basis functions were taken as the bilinear basis functions used in the finite element method. In the present work, spherical triangular basis functions are employed, and the results are compared with those computed using bilinear basis functions, as well as with results reported in the literature using other methods. Two-dimensional enclosures containing an emittingabsorbing, non-scattering, grey medium with prescribed temperature or in radiative equilibrium are considered. It is shown that the spherical triangular elements yield results slightly better than those calculated using bilinear elements, except in the case of optically thick media, where both elements perform similarly. The results of the hybrid method are less sensitive to the angular discretization than those obtained using the discrete ordinates method.