Semi-implicit Hybrid Discrete $\left(\text{H}^T_N\right)$ Approximation of Thermal Radiative Transfer

The thermal radiative transfer (TRT) equations form a system that describes the propagation and collisional interactions of photons. Computing accurate and efficient numerical solutions to TRT is challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space. In order to reduce the dimensionality, classical approaches such as the P$_N$ (spherical harmonics) or the S$_N$ (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H$^T_N$) approximation. This approach acquires desirable properties of both P$_N$ and S$_N$, and indeed reduces to each of these approximations in various limits. We prove that H$^T_N$ results in a system of hyperbolic equations. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions. This can be partially overcome via implicit time integration, although fully implicit methods may become expensive due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions. We develop a method that is based on a discontinuous Galerkin scheme in space, coupled with a semi-implicit scheme in time. In particular, we make use of an explicit Runge-Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each step, we linearize the temperature term; this avoids the need for an iterative procedure. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the method.