Angular-spatial hp-adaptivity for radiative transfer with discontinuous Galerkin spectral element methods

Abstract

Radiative transfer is important for many science and engineering applications, and numerical simulations of radiative transfer can be challenging. For instance, the radiation field is seven-dimensional – three spatial, two angular, one wavelength, and one temporal – and often features steep gradients. Therefore, memory usage is a key issue. To reduce memory, some past work has investigated the use of adaptive mesh refinement (AMR), typically for either the spatial or angular coordinate, and typically for only $h$-adaptivity. Here, we propose the use of AMR for the spatial and angular coordinates together, and the use of $h$- and $p$-adaptivity together as $hp$-AMR for the potential for further memory savings. We implemented the proposed method for several test cases in two spatial and one angular dimension, with the discontinuous Galerkin spectral element method. These test cases featured highly anisotropic angular radiation, with or without steep spatial gradients. Our primary findings from these test cases were: (1) Angular $hp$-adaptivity can deliver the radiation solution with the same accuracy as, and with much less computational memory than, uniform angular $h$- or $p$-refinements, or angular $h$-adaptivity alone. This is most obvious when the incoming radiation is highly anisotropic, in which case the savings can be orders of magnitude. (2) Full spatial-angular $hp$-adaptivity is more efficient in solution representation, compared to solely spatial or solely angular $hp$-adaptivity. This is most evident when steep gradients are present in both the spatial and angular distribution. These results suggest that adaptive spatial-angular $hp$-refinement may perform well in large-scale seven-dimensional applications.