Journal of Scientific Computing (2020) 84:36 https://doi.org/10.1007/s10915-020-01291-0 Numerical Analysis of the Lattice Boltzmann Method for the Boussinesq Equations Wen-An Yong 1 · Weifeng Zhao 2 Received: 6 January 2019 / Revised: 5 June 2020 / Accepted: 26 July 2020 / Published online: 3 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract 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. Keywords Lattice Boltzmann method · Boussinesq equations · Stability structure · Asymptotic analysis · Convergence Mathematics Subject Classification 65M12 · 76M28 · 35Q30 1 Introduction During the last two decades, the lattice Boltzmann method (LBM) has been developed into an effective and viable tool for simulating various flow problems. It has been proved to be quite successful in simulating complex fluid flows such as turbulent flows, micro-flows, multi-phase and multi-component flows, particulate suspensions, and interfacial dynamics. Consequently, the LBM is becoming a serious alternative to traditional computational fluid methods. We refer to [1–4] for a comprehensive account of the method and its applications. B Weifeng Zhao wfzhao@ustb.edu.cn Wen-An Yong wayong@tsinghua.edu.cn 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China 2 Department of Applied Mathematics, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China 123