This paper is concerned with the lattice Boltzmann method (LBM) with BGK collision models for the two-dimensional Bousinessq equations on periodic domains. We show the numerical stability of the LBM linearized at a quiescent state and conduct a formal asymptotic analysis which indicates the consistency of the method to the Boussinesq equations. With these, we establish the convergence of the LBM for the nonlinear equations. Moreover, our present analysis provides some important hints on how to construct initial data and how to add force terms in the LBM. It can be straightforwardly extended to other LB models or three-dimensional cases.

中文翻译：

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Boussinesq方程的格子Boltzmann方法的数值分析。

本文涉及具有周期域的二维Bousinessq方程的BGK碰撞模型的格子Boltzmann方法（LBM）。我们显示了在静止状态下线性化的LBM的数值稳定性，并进行了正式的渐近分析，这表明该方法与Boussinesq方程的一致性。利用这些，我们建立了非线性方程的LBM的收敛性。此外，我们目前的分析为如何构造初始数据以及如何在LBM中添加力项提供了一些重要的提示。它可以直接扩展到其他LB模型或三维案例。