The Meshless Lattice Boltzmann Method (MLBM) is a numerical tool that relieves the standard Lattice Boltzmann Method (LBM) from regular lattices and, at the same time, decouples space and velocity discretizations. In this study, we investigate the numerical convergence of MLBM in two benchmark tests: the Taylor-Green vortex and annular (bent) channel flow. We compare our MLBM results to LBM and to the analytical solution of the Navier-Stokes equation. We investigate the method's convergence in terms of the discretization parameter, the interpolation order, and the LBM streaming distance refinement. We observe that MLBM outperforms LBM in terms of the error value for the same number of nodes discretizing the domain. We find that LBM errors at a given streaming distance $\delta x$ and timestep length $\delta t$ are the asymptotic lower bounds of MLBM errors with the same streaming distance and timestep length. Finally, we suggest an expression for the MLBM error that consists of the LBM error and other terms related to the semi-Lagrangian nature of the discussed method itself.

翻译：无网格格子玻尔兹曼方法（MLBM）是一种数值工具，它解除了标准格子玻尔兹曼方法（LBM）对规则格子的依赖，同时将空间离散与速度离散解耦。本研究通过两个基准测试——Taylor-Green涡和环形（弯曲）通道流——考察了MLBM的数值收敛性。我们将MLBM结果与LBM结果及纳维-斯托克斯方程解析解进行对比，分析了该方法在离散参数、插值阶数以及LBM迁移距离细化方面的收敛特性。观察发现，在相同节点数离散域的情况下，MLBM的误差值优于LBM。我们进一步发现，当给定迁移距离δx和时间步长δt时，LBM的误差构成MLBM误差的渐近下界。最后，我们提出了MLBM误差的表达式，该表达式由LBM误差及其他与所讨论方法本身的半拉格朗日特性相关的项共同组成。