AI 中文总结
本文提出一种基于微观-宏观分解和Grad 13矩系统的双曲松弛系统，用于低马赫数极限动力学方程模拟，并开发了高阶IMEX-WENO渐近保持格式，在保证结构稳定性的同时显著降低计算成本。
AI 中文摘要
本文提出了一种双曲松弛系统，用于模拟低马赫数极限下的动力学方程。该方法基于缩放BGK模型的微观-宏观分解，将动力学分布函数重构为一个耦合系统，该系统由宏观平衡部分和微观非平衡余项组成。通过将微观偏差投影到一组正交多项式上，我们推导出一个封闭的矩松弛系统。所得系统是Grad 13矩系统的一个版本，具有线性双曲部分和适应不可压缩极限的松弛项。我们证明了该模型在不可压缩Navier-Stokes极限下的结构稳定性。此外，我们开发了一种高阶渐近保持（AP）数值框架，采用隐式-显式（IMEX）Runge-Kutta格式实现时间精度，并使用有限差分WENO重构以及中心差分近似实现高阶空间分辨率。所提出的格式确保了不同物理区域间的均匀稳定性和一致性，当缩放参数趋于零时，自动退化为不可压缩热Navier-Stokes极限的一致高阶离散格式。一维和二维数值实验证实了理论发现，展示了计算成本的显著降低以及在广泛Knudsen数和Mach数范围内的稳健性能。
英文摘要
This work introduces a hyperbolic relaxation system designed for the simulation of kinetic equations in the low-Mach number limit. The methodology is built upon a micro-macro decomposition of the scaled BGK model, which reformulates the kinetic distribution function into a coupled system consisting of a macroscopic equilibrium part and a microscopic non-equilibrium remainder. By projecting the microscopic deviations onto a set of orthogonal polynomials, we derive a closed moment relaxation system. The resulting system is a version of Grad's 13 moment system with a linear hyperbolic part and a relaxation adapted to the incompressible limit. We prove the model's structural stability in the incompressible Navier-Stokes limit. Moreover, we develop a high order Asymptotic-Preserving (AP) numerical framework using Implicit-Explicit (IMEX) Runge Kutta schemes for temporal accuracy and finite difference WENO reconstructions as well as central difference approximations for high order spatial resolution. The proposed scheme ensures uniform stability and consistency across different physical regimes, automatically degenerating into a consistent high order discretization of the incompressible thermal Navier-Stokes limit as the scaling parameter vanishes. Numerical experiments in one and two dimensions corroborate the theoretical findings, demonstrating significant reductions in computational cost and robust performance across a wide range of Knudsen and Mach numbers.
Comments34 figures, 4 tables
