Numerics of the lattice boltzmann method on nonuniform grids: Standard LBM and finite-difference LBM

The present study is focused on the comparison between the standard “collide-and-stream” lattice Boltzmann method (LBM) and the Lax-Wendroff-based finite-difference LBM (FDLBM) on block-structured nonuniform grids with an adaptive mesh refinement (AMR) strategy. While the standard LBM (SLBM) is found to be slightly faster than the FDLBM, the latter is shown to be more stable at higher Reynolds numbers. Although both approaches are as accurate in simulation of fluid flow problems, the SLBM has a more complicated algorithm and its implementation is more involved; this is mainly because, in applying SLBM, the AMR blocks at different refinement levels do not advance in time simultaneously. On the other hand, the underlying differences between the cell-center and cell-vertex data structures are explained and their advantages and disadvantages are highlighted. In general, the cell-center data structure is favorable because it is more efficient in terms of computational time and memory. The effect of the interpolation schemes on the order of accuracy of the LBM is also investigated. It is reestablished that the popular linear interpolation degrades the order of accuracy of LBM to first order. A variety of benchmark studies, including Taylor-Green decaying vortex, gravity-driven Poiseuille flow, thin shear layer instability, and unsteady flow past a square cylinder, are carried out to assess SLMB and FDLBM with a multiple-relaxation-time collision operator.