A Finite Element Treatment of the Angular Dependency of the Even-Parity Equation of Radiative Transfer

Abstract

The present article introduces a new method to solve the radiative transfer equation (RTE). First, a finite element discretization of the solid angle dependence is derived, wherein the coefficients of the finite element approximation are functions of the spatial coordinates. The angular basis functions are defined according to finite element principles on subdivisions of the octahedron. In a second step, these spatially dependent coefficients are discretized by spatial finite elements. This approach is very attractive, since it provides a concise derivation for approximations of the angular dependence with an arbitrary number of angular nodes. In addition, the usage of high-order angular basis functions is straightforward. In the current paper, the governing equations are first derived independently of the actual angular approximation. Then, the design principles for the angular mesh are discussed and the parameterization of the piecewise angular basis functions is derived. In the following, the method is applied to one-dimensional and two-dimensional test cases, which are commonly used for the validation of approximation methods of the RTE. The results reveal that the proposed method is a promising alternative to the well-established practices like the discrete ordinates method (DOM) and provides highly accurate approximations. A test case, which is known to exhibit the ray effect in the DOM, verifies the ability of the new method to avoid ray effects.