DG-IMEX 方法用于 $\mathcal{O}(v/c)$ 极限下辐射输运的双矩模型
DG-IMEX Method for a Two-Moment Model for Radiation Transport in the $\mathcal{O}(v/c)$ Limit
AI总结：
本文提出一种保持可实现性的 DG-IMEX 数值方法，用于演化带有 $\mathcal{O}(v/c)$ 相对论修正的粒子双矩模型，通过定制求解器与限制器保证矩可实现性及 Eulerian 参考系能量守恒。
摘要：
我们考虑由相空间密度矩描述的粒子系统，并提出一种可实现性保持的数值方法，用于演化与非相对论速度运动的背景流体相互作用的粒子的谱双矩模型。带有 $\mathcal{O}(v/c)$ 特殊相对论修正的非线性矩方程组表达了相空间平流与碰撞之间的平衡，并包含解释空间平流、Doppler 频移和角度像差的依赖于速度的项。该模型与 Lowrie 等人 (2001; JQSRT, 69, 291-304) 提出的模型密切相关，并且类似于当前用于研究天体物理环境大规模模拟中输运现象的模型。该方法旨在保持矩可实现性，以保证矩对应于非负的相空间密度。该可实现性保持格式包含以下关键组件：(i) 强稳定保持的隐式-显式 (IMEX) 时间积分方法；(ii) 带有精心构建的数值通量的间断 Galerkin (DG) 相空间离散；(iii) 可实现性保持的隐式碰撞更新；以及 (iv) 可实现性强制限制器。在时间积分中，矩模型的非线性需要求解非线性方程，我们将其表述为不动点问题，并使用定制的迭代求解器进行求解，这些求解器在保证收敛性的同时保持矩可实现性。我们还分析了半离散 DG 格式的同步 Eulerian 参考系数和能量守恒性质，并提出了一个促进 Eulerian 参考系能量守恒的“能量限制器”。通过数值实验，我们展示了此 DG-IMEX 方法的准确性和鲁棒性，并研究了其 Eulerian 参考系能量守恒性质。
英文摘要：
We consider particle systems described by moments of a phase-space density and propose a realizability-preserving numerical method to evolve a spectral two-moment model for particles interacting with a background fluid moving with nonrelativistic velocities. The system of nonlinear moment equations, with special relativistic corrections to $\mathcal{O}(v/c)$, expresses a balance between phase-space advection and collisions and includes velocity-dependent terms that account for spatial advection, Doppler shift, and angular aberration. This model is closely related to the one promoted by Lowrie et al. (2001; JQSRT, 69, 291-304) and similar to models currently used to study transport phenomena in large-scale simulations of astrophysical environments. The method is designed to preserve moment realizability, which guarantees that the moments correspond to a nonnegative phase-space density. The realizability-preserving scheme consists of the following key components: (i) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (ii) a discontinuous Galerkin (DG) phase-space discretization with carefully constructed numerical fluxes; (iii) a realizability-preserving implicit collision update; and (iv) a realizability-enforcing limiter. In time integration, nonlinearity of the moment model necessitates solution of nonlinear equations, which we formulate as fixed-point problems and solve with tailored iterative solvers that preserve moment realizability with guaranteed convergence. We also analyze the simultaneous Eulerian-frame number and energy conservation properties of the semi-discrete DG scheme and propose an "energy limiter" that promotes Eulerian-frame energy conservation. Through numerical experiments, we demonstrate the accuracy and robustness of this DG-IMEX method and investigate its Eulerian-frame energy conservation properties.
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