A lattice Boltzmann model for shallow water equations on the smoothed quadtree grid

Abstract

AbstractIn hydraulic simulation, the fine land‐water boundary often means more accurate result but more lattices and time cost. In this study, we developed a smoothed quadtree grid and introduced it into the lattice Boltzmann model for the shallow water equations (LABSWE) to balance this problem. The recommended grid is meticulously designed to own a minimum width of advantage over a uniform grid or multi‐block grid, and can describe the zigzag boundary via smaller size lattices with acceptable total number of lattice. As an improved multi‐block grid, the lattices are numbered in serialization for facilitating data storage and parallel computation, and a standardized process is developed to employ this grid in the LABSWE. The spatial interpolation is modified based on the water surface and fluxes instead of the water depth and velocities, in order to satisfy the necessary property (N‐property: the numerical scheme can replicate the exact solution to a stationary case in which there is a nonzero force or source term). The proposed model is verified by two benchmark tests and then applied to the long‐term simulation of unsteady flows in large aquatorium. The agreement between predictions and analytical results or other solutions are satisfactory. Three popular collision schemes, the Bhatnagar–Gross–Krook (BGK) scheme, the multiple relaxation time (MRT) scheme and the central moments (CMs) scheme, have also been tested, showing that the stability of the MRT and CMs schemes are better than the BGK scheme for the flows with higher Reynolds number, but cost only 4%–6% more computational load.