In this work, we present an asymptotic-preserving semi-Lagrangian discontinuous Galerkin scheme for the Boltzmann equation that effectively handles multi-scale transport phenomena. The main challenge lies in designing appropriate moments update for penalization within the semi-Lagrangian framework. Inspired by [M. Ding, J. M. Qiu, and R. Shu, Multiscale Model. Simul. 21 (2023), no. 1, 143--167], the key ingredient is utilizing the Shu-Osher form of the scheme in the implicit-explicit Runge-Kutta (IMEX-RK) setting, which enables us to capture the correct limiting system by constructing an appropriate moments update procedure. Our theoretical analysis establishes accuracy order conditions for both the IMEX-RK time integration and the new moments update step. We also employ hypocoercivity techniques to establish stability for the linearized model. Numerical experiments for various test problems validate our proposed scheme's accuracy, asymptotic-preserving property, and robustness in various regimes, which demonstrates its effectiveness for multi-scale kinetic simulations.

翻译：本文提出了一种用于玻尔兹曼方程的渐近保持半拉格朗日间断伽辽金格式，该格式能有效处理多尺度输运现象。主要挑战在于在半拉格朗日框架内设计适用于惩罚项的恰当矩量更新策略。受[M. Ding, J. M. Qiu, and R. Shu, Multiscale Model. Simul. 21 (2023), no. 1, 143--167]启发，关键要素在于在隐式-显式龙格-库塔（IMEX-RK）框架下采用该格式的Shu-Osher形式，这使我们能够通过构建恰当的矩量更新过程来捕捉正确的极限系统。我们的理论分析为IMEX-RK时间积分和新矩量更新步骤建立了精度阶条件。同时，我们采用亚椭圆性技术为线性化模型建立了稳定性。针对各类测试问题的数值实验验证了所提格式的精度、渐近保持特性以及在多种状态下的鲁棒性，这证明了其在多尺度动理学模拟中的有效性。