We investigate the performance of a class of hp-adaptive discontinuous Galerkin methods for the numerical solution of simplified PN approximations of radiative transfer in non-grey semitransparent media. By introducing an optical scale and using asymptotic expansions in the radiative transfer equation we formulate the simplified PN approximations. The optical spectrum is decomposed in frequency bands and the simplified PN equations are solved for each frequency band. As a numerical solver for the simplified PN equations we consider a high-order discontinuous Galerkin method. The solver belongs to a class of finite element methods whose approximate solutions are discontinuous across inter-element boundaries; this property renders the method ideally suited for the hp-adaptivity. An error estimator is shown to provide reliable and practically useful upper bounds for the numerical errors independent of the optical scales used in the simulations. The proposed method is simple, fast and highly accurate. The performance of the method is analyzed on several applications in frequency-dependent radiative transfer. The aim of such a method compared to the conventional finite element methods is to solve the simplified PN equations efficiently and with a high level of accuracy on unstructured meshes with different elements. The obtained results demonstrate the ability of the proposed method to capture the main radiative features.
Giani, S., & Seaid, M. (2016). hp-adaptive discontinuous Galerkin methods for simplified PN approximations of frequency-dependent radiative transfer. Computer Methods in Applied Mechanics and Engineering, 301, 52-79. https://doi.org/10.1016/j.cma.2015.12.013