Discontinuous finite element solution of 2-D radiative transfer with and without axisymmetry

A discontinuous Galerkin finite element methodology is presented for the computation of two-dimensional (2-D) radiative transfer in participating media with and without axisymmetry. The central idea of the discontinuous formulation is that the variables are allowed to be discontinuous across the inter-element boundaries. Consequently, the formulation should be particularly useful for radiative transfer calculations where the radiation intensity experiences a jump across the boundary. Being able to compute an axisymmetric problem over a 2-D mesh rather than using one three-dimensional (3-D) scenario results in reduced computing time and lower memory storage requirements. Mathematical formulations and numerical implementations using the discontinuous finite element (DFE) method for radiative transfer are given. The procedures for incorporating the mapping of the discontinuous formulation of radiative transfer develop a unified approach to both 2-D and 2-D axisymmetric problems are discussed in detail. The computed results are given, and they compare well with the solutions reported in the literature that were obtained using other methods. Examples include both non-scattering and scattering cases. The effects of the solid-angular and spatial discretization on the accuracy of results are discussed. Numerical simulations show that an even discretization of solid angle is important to ensure an adequate numerical accuracy.