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.

翻译：我们考虑由相空间密度矩描述的粒子系统，并提出一种保持可实现性的数值方法，用于演化与以非相对论速度运动的背景流体相互作用的谱双矩模型。该非线性矩方程组包含$\mathcal{O}(v/c)$阶的特殊相对论修正，体现了相空间平流与碰撞之间的平衡，并包括考虑空间平流、多普勒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，且类似于目前用于天体物理环境大规模模拟中输运现象研究的模型。该方法旨在保持矩的可实现性，即保证矩对应非负的相空间密度。保持可实现性的方案包含以下关键组成部分：(i) 强稳定性保持的隐式-显式（IMEX）时间积分方法；(ii) 带精心构造数值通量的间断Galerkin（DG）相空间离散；(iii) 保持可实现性的隐式碰撞更新；(iv) 强制可实现性的限制器。在时间积分过程中，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，并采用定制的迭代求解器求解，该求解器在保证收敛的同时保持矩的可实现性。我们还分析了半离散DG格式的欧拉框架下粒子数与能量守恒特性，并提出了一种促进欧拉框架下能量守恒的“能量限制器”。通过数值实验，我们展示了该DG-IMEX方法的精度和鲁棒性，并研究了其欧拉框架下的能量守恒特性。