In this paper, a high-order/low-order (HOLO) method is combined with a micro-macro (MM) decomposition to accelerate iterative solvers in fully implicit time-stepping of the BGK equation for gas dynamics. The MM formulation represents a kinetic distribution as the sum of a local Maxwellian and a perturbation. In highly collisional regimes, the perturbation away from initial and boundary layers is small and can be compressed to reduce the overall storage cost of the distribution. The convergence behavior of the MM methods, the usual HOLO method, and the standard source iteration method is analyzed on a linear BGK model. Both the HOLO and MM methods are implemented using a discontinuous Galerkin (DG) discretization in phase space, which naturally preserves the consistency between high- and low-order models required by the HOLO approach. The accuracy and performance of these methods are compared on the Sod shock tube problem and a sudden wall heating boundary layer problem. Overall, the results demonstrate the robustness of the MM and HOLO approaches and illustrate the compression benefits enabled by the MM formulation when the kinetic distribution is near equilibrium.

翻译：本文结合高阶/低阶方法与微观-宏观分解方法，以加速气体动力学BGK方程全隐式时间步进中的迭代求解器。微观-宏观表述将动理学分布函数表示为局部麦克斯韦分布与扰动项之和。在高碰撞区域中，远离初始层和边界层的扰动项幅值较小，可通过压缩技术降低分布函数的总体存储成本。在线性BGK模型上，本文分析了微观-宏观方法、常规高阶/低阶方法以及标准源迭代法的收敛特性。高阶/低阶方法与微观-宏观方法均采用相空间中的间断伽辽金离散格式实现，该格式天然保持了高阶/低阶方法所需的高低阶模型间的一致性。通过Sod激波管问题和壁面突然加热边界层问题，对比了这些方法的精度与计算性能。总体而言，研究结果证明了微观-宏观方法与高阶/低阶方法的鲁棒性，并展示了当动理学分布接近平衡态时，微观-宏观表述通过压缩技术带来的存储优势。