晶格Boltzmann方案在扩散问题中具有好奇的收敛性质
Curious convergence properties of lattice Boltzmann schemes for diffusion with acoustic scaling
AI总结：
研究晶格Boltzmann方案在扩散模拟中的收敛特性，发现网格细化时收敛性异常，通过渐近分析得到声学型偏微分方程，并揭示其非平凡复特征值与异常收敛性质。
摘要：
我们考虑带有声学尺度的D1Q3晶格Boltzmann方案用于扩散过程的模拟。当网格细化而保持扩散系数恒定时，首先获得渐近收敛。然而，当网格尺寸趋于零时，这种收敛却以奇特的方式破裂，观察到与热方程解析解的定性差异。本文通过Taylor展开方法推导了新的渐近分析，得到渐近极限下的声学型偏微分方程。我们证明D1Q3数值解与该声学型偏微分方程的有限差分近似在渐近极限下的误差趋于零。此外，对该渐近 regime 的波矢分析显示，色散方程具有非平凡复特征值，表明底层传播现象的存在，并预示上述异常收敛性质。
英文摘要：
We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the simulation of diffusive processes. When the mesh is refined while holding the diffusivity constant, we first obtain asymptotic convergence. When the mesh size tends to zero, however, this convergence breaks down in a curious fashion, and we observe qualitative discrepancies from analytical solutions of the heat equation. In this work, a new asymptotic analysis is derived to explain this phenomenon using the Taylor expansion method, and a partial differential equation of acoustic type is obtained in the asymptotic limit. We show that the error between the D1Q3 numerical solution and a finite-difference approximation of this acoustic-type partial differential equation tends to zero in the asymptotic limit. In addition, a wave vector analysis of this asymptotic regime demonstrates that the dispersion equation has nontrivial complex eigenvalues, a sign of underlying propagation phenomena, and a portent of the unusual convergence properties mentioned above.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
