In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times when the time step is proportional to the space step and tends to zero. We do this by a combination of theory and numerical experiment. The classical formal analysis when all the relaxation parameters are fixed and the time step tends to zero shows that the numerical solution converges to solutions of the heat equation, with a constraint connecting the diffusivity, the space step and the coefficient of relaxation of the momentum. If the diffusivity is fixed and the space step tends to zero, the relaxation parameter for the momentum is very small, causing a discrepency between the previous analysis and the numerical results. We propose a new analysis of the method for this specific situation of evanescent relaxation, based on the dispersion equation of the lattice Boltzmann scheme. A new asymptotic partial differential equation, the damped acoustic system, is emergent as a result of this formal analysis. Complementary numerical experiments establish the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times and acoustic scaling in this specific case of evanescent relaxation towards the numerical solution of the damped acoustic system.
Convergence Analysis of the Grad's Hermite Approximation to the Boltzmann Equation
在（Commun Pure Appl Math 2（4）：331-407，1949）中，Grad提出了一种Hermite级数展开，用于近似具有无界速度空间的动力学方程的解。然而，对于初边值问题，不恰当地施加边界条件会导致Grad的Hermite展开的不稳定性，这可能导致非收敛解。对于线性动力学方程，最近提出了一种（形式上）任意阶Hermite逼近的稳定边界条件的构造方法。在目前的工作中，我们研究了这些稳定的Hermite逼近的$L^2$ -收敛性，并证明了明确的收敛速度在适当的正则性假设的精确解。通过对稀薄气体动力学线性化BGK方程的数值实验，我们证实了所提出的收敛速度。
Curious convergence properties of lattice Boltzmann schemes for diffusion with acoustic scaling
Tufts University（塔夫茨大学）；University Paris-Sud（巴黎第十一大学）；Conservatoire National des Arts et Métiers（国立工艺院）；Beijing Computational Science Research Center（北京计算科学研究中心）；University Tunis El Manar（突尼斯艾曼努尔大学）
Convergence of a two-relaxation-times kinetic approximation towards the solution of a scalar conservation law
Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（波尔多大学）；Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C & Fédération de Mathématiques de CentraleSupélec（巴黎萨克雷大学）
