In this paper, we develop and implement an efficient asymptotic-preserving (AP) scheme to solve the gas mixture of Boltzmann equations under the disparate mass scaling relevant to the so-called "epochal relaxation" phenomenon. The disparity in molecular masses, ranging across several orders of magnitude, leads to significant challenges in both the evaluation of collision operators and the designing of time-stepping schemes to capture the multi-scale nature of the dynamics. A direct implementation of the spectral method faces prohibitive computational costs as the mass ratio increases due to the need to resolve vastly different thermal velocities. Unlike [I. M. Gamba, S. Jin, and L. Liu, Commun. Math. Sci., 17 (2019), pp. 1257-1289], we propose an alternative approach based on proper truncation of asymptotic expansions of the collision operators, which significantly reduces the computational complexity and works well for small $\varepsilon$. By incorporating the separation of three time scales in the model's relaxation process [P. Degond and B. Lucquin-Desreux, Math. Models Methods Appl. Sci., 6 (1996), pp. 405-436], we design an AP scheme that captures the specific dynamics of the disparate mass model while maintaining computational efficiency. Numerical experiments demonstrate the effectiveness of the proposed scheme in handling large mass ratios of heavy and light species, as well as capturing the epochal relaxation phenomenon.

翻译：本文针对与所谓"时代弛豫"现象相关的质量悬殊尺度下的玻尔兹曼方程气体混合问题，开发并实现了一种高效的渐近保持（AP）格式。分子质量跨越数个数量级的悬殊差异，给碰撞算子的计算以及捕捉动力学多尺度特性的时间步进方案设计带来了重大挑战。随着质量比的增加，由于需要解析差异巨大的热速度，谱方法的直接实现面临难以承受的计算成本。不同于[I. M. Gamba, S. Jin, and L. Liu, Commun. Math. Sci., 17 (2019), pp. 1257-1289]的工作，我们提出了一种基于碰撞算子渐近展开适当截断的替代方法，该方法显著降低了计算复杂度，并在小$\varepsilon$情况下表现良好。通过结合模型弛豫过程中三个时间尺度的分离[P. Degond and B. Lucquin-Desreux, Math. Models Methods Appl. Sci., 6 (1996), pp. 405-436]，我们设计了一种AP格式，该格式在保持计算效率的同时，能够捕捉质量悬殊模型特有的动力学行为。数值实验证明了所提格式在处理轻重组分大质量比以及捕捉时代弛豫现象方面的有效性。