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.
翻译:热离散传输( TRT) 方程式形成一个描述光子的传播和碰撞互动的直径分流系统。 计算准确和高效的数字解决方案 TRT 具有挑战性, 原因有几种, 其中第一个是高维阶段定义 TRT 。 为了降低阶段空间的维度, 经典方法, 如 P$_ N美元( 球调) 或 S$_ N美元( 分解坐标) 。 文献中经常使用一个描述光子相传和碰撞互动的直径系统。 在这项工作中, 我们引入了一个新颖的方法: 混合的离散( H$_ T_ N$) 预知和高效的数字解决方案 。 这个方法可以让超离子的电离子系统( P$_ N美元) 和 S_ N美元 元( 分解) 等离子系统( 等离子系统) 等离子( 等离子) 等离子( 等离子) 等离子系统( 等离子) 等离子( 等) 等离子系统( 等离子) 、 等离子( 等离子( 等) 等离子) 等离子( 等离子) 等离子) 等离子( 等)