Angular-spatial discontinuous Galerkin method for radiative heat transfer with a participating medium in complex three-dimensional geometries

The angular-spatial discontinuous Galerkin method (ASDGM) is a numerical technique developed to solve the radiative transfer equation (RTE) in complex three-dimensional geometries with absorbing, emitting, and scat-tering media. The ASDGM discretizes both the angular and spatial domains of the RTE using structured quad-rilateral elements for the angular domain and unstructured tetrahedral elements for the spatial domain. In the solution process, the Riemann upwind solver separates the input and output information at the tetrahedral element boundary. Moreover, the discontinuous Galerkin (DG) sweep algorithm optimizes the order of the un-structured tetrahedral elements and reduces memory usage caused by the coupling of the discretization in the angular and spatial domains. The performance of the ASDGM has been evaluated using several radiative transfer cases in complex three-dimensional geometries, which demonstrate its good accuracy compared to available data from the literature. Additionally, the computational time in the angular domain for high-order DG is lower than that of the finite element method for the same discrete angles. Furthermore, the DG sweep algorithm provides a faster convergence rate in the spatial domain compared to the original sweep algorithm. Therefore, the ASDGM with the DG sweep algorithm is a preferable method for solving radiative transfer problems in complex three-dimensional geometries.