Title:Asymptotic-Preserving schemes for the Boltzmann mixture model with disparate mass

Abstract:In this paper, we develop and implement an efficient asymptotic-preserving (AP) scheme to solve the gas mixture of Boltzmann equations, under the so-called "relaxation time scale" relevant to the 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 designing of efficient numerical schemes in order to capture the multi-scale nature of the dynamics. A direct implementation by using the spectral method faces prohibitive computational costs as the mass ratio decreases due to the need to resolve vastly different thermal velocities. Different from [I. M. Gamba, S. Jin, and L. Liu, Commun. Math. Sci., 17 (2019), pp. 1257-1289], we propose an alternative approach by conducting asymptotic expansions for the collision operators, which can significantly reduce the computational complexity and works well for uniformly 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 is able to capture the epochal relaxation phenomenon of disparage mass mixtures while maintaining the computational efficiency. Numerical experiments will demonstrate the effectiveness of our proposed scheme in handling large mass ratios of heavy and light species, in addition to validating the AP properties.