无热化正则化格子玻尔兹曼方法的线性稳定性
Linear stability of athermal regularized lattice Boltzmann methods
摘要：
本工作致力于更好地理解正则化格子Boltzmann（LB）格式的稳定性。为此，提出了二维模型的线性稳定性分析：标准的Bhatnagar-Gross-Krook（BGK）碰撞模型，原始的预碰撞正则化和递归正则化模型，其中非平衡分布部分计算由于递归公式。通过分析线性系统的特征向量，系统地识别每个LB模式所携带的物理内容。稳定性的结果，然后进行数值模拟的剪切波和声波确认。这项工作允许得出公平的结论，每个模型的稳定性。特别地，递归正则化被证明是D2 Q9晶格的最稳定的模型，特别是在零粘度极限下。强调了每个正则化模型共有的两个主要属性：（1）模式过滤属性，以及（2）在欠分辨条件下携带物理波的模式的不正确且广泛各向异性的耗散率。第一个属性是增加稳定性的主要来源，特别是对于递归正则化。这是在每次碰撞过程之前重建非平衡态粒子数的直接结果，降低了离散方程系统的秩。第二个属性似乎与高阶矩的平衡直接引起的数值误差有关。在这种情况下，遵循这种稳定方法的任何碰撞模型都可能出现这种特性。
英文摘要：
The present work is dedicated to a better understanding of the stability properties of regularized lattice Boltzmann (LB) schemes. To this extent, linear stability analyses of two-dimensional models are proposed: the standard Bhatnagar-Gross-Krook (BGK) collision model, the original pre-collision regularization and the recursive regularized model, where off-equilibrium distributions are partially computed thanks to a recursive formula. A systematic identification of the physical content carried by each LB mode is done by analyzing the eigenvectors of the linear systems. Stability results are then numerically confirmed by performing simulations of shear and acoustic waves. This work allows drawing fair conclusions on the stability properties of each model. In particular, recursive regularization turns out to be the most stable model for the D2Q9 lattice, especially in the zero-viscosity limit. Two major properties shared by every regularized model are highlighted: (1) a mode filtering property, and (2) an incorrect, and broadly anisotropic, dissipation rate of the modes carrying physical waves in under-resolved conditions. The first property is the main source of increased stability, especially for the recursive regularization. It is a direct consequence of the reconstruction of off-equilibrium populations before each collision process, decreasing the rank of the system of discrete equations. The second property seems to be related to numerical errors directly induced by the equilibration of high-order moments. In such a case, this property is likely to occur with any collision model that follows such a stabilization methodology.
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