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

TL;DR

This paper introduces a hybrid discrete approximation for thermal radiative transfer equations that combines advantages of existing methods, ensuring hyperbolicity and efficiency in stiff regimes through a semi-implicit discretization.

Contribution

The paper proposes a novel hybrid discrete (H^T_N) approximation that unifies P_N and S_N methods, and develops a semi-implicit discontinuous Galerkin scheme for efficient numerical solutions.

Findings

H^T_N$ approximation reduces to P_N and S_N in specific limits.

The proposed method is hyperbolic for all T ≥ 1 and N ≥ 0.

Numerical experiments demonstrate accuracy, efficiency, and robustness.

Abstract

The thermal radiative transfer (TRT) equations form an integro-differential system that describes the propagation and collisional interactions of photons. Computing accurate and efficient numerical solutions TRT are challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase. In order to reduce the dimensionality of the phase space, 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 to the radiative thermal transfer equations. This approach acquires desirable properties of both P$_N$ and S$_N$, and indeed reduces to each of these approximations in various limits: H$^1_N$ $\equiv$ P$_N$ and H$^T_0$ $\equiv$ S$_T$. We prove that H$^T_N$ results in a system of hyperbolic equations for all $T\ge 1$ and $N\ge 0$. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally 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, making such schemes prohibitively expensive. In this work we develop a numerical method that is based on a nodal discontinuous Galerkin discretization in space, coupled with a semi-implicit discretization in time. We conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the H$^T_N$ ansatz and the numerical discretizations.