Stokes方程的半程反弹格子玻尔兹曼方法的最优误差估计
Optimal error estimates for the half-way bounce-back lattice Boltzmann method for the Stokes equations
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中文总结 AI 辅助
本文证明了平板通道不可压缩Stokes方程的半程反弹D2Q9 BGK格子玻尔兹曼方法的最优收敛速率，速度二阶、压力一阶，解决了现有严格定理仅得速度O(h^(1/2))界的问题，通过分解边界误差并结合校正子得到改进预测函数，结合稳定性估计完成证明。
中文摘要 AI 辅助
本文针对平板通道中不可压缩Stokes方程，对采用半程反弹规则的D2Q9 BGK格子玻尔兹曼方法，给出了最优收敛速率的数学证明。当网格间距h趋于0时，速度的收敛速率为二阶，压力的收敛速率为一阶，这与形式分析和数值实验结果一致，而现有严格收敛定理仅能给出速度误差的O(h^(1/2))界。证明的关键步骤是将主导边界相容性误差分解为宏观分量和动理学分量，这些分量分别被适当构造的Stokes校正子和离散Knudsen层校正子吸收。将这些校正子融入现有严格分析所用的预测函数，得到具有足够高阶相容性误差的改进预测函数，结合已知的加权L²稳定性估计，即可得到最优收敛速率。
英文摘要
We give a mathematical proof of the optimal convergence rates for the D2Q9 BGK lattice Boltzmann method with the half-way bounce-back rule for the incompressible Stokes equations in a flat channel. The convergence rates are second-order for the velocity and first-order for the pressure as the lattice spacing $h$ tends to zero, in agreement with formal analyses and numerical experiments, whereas the available rigorous convergence theorems only yield an $O(h^{1/2})$ bound for the velocity error. A key step in the proof is a decomposition of the leading boundary consistency error into macroscopic and kinetic components. These components are absorbed by suitably constructed Stokes and discrete Knudsen layer correctors, respectively. Incorporating these correctors into the prediction function used in previous rigorous analyses, we obtain a refined prediction function with consistency errors of sufficiently high order. Combined with the known weighted $L^2$-stability estimate, this gives the optimal convergence rates.
