Macroscopic Lattice Boltzmann Method for Shallow Water Equations (MacLABSWE)

Abstract: It is well known that there are two integral steps of streaming and collision in the lattice Boltzmann method (LBM). This concept has been changed by the author's recently proposed macroscopic lattice Boltzmann method (MacLAB) to solve the Navier-Stokes equations for fluid flows. The MacLAB contains streaming step only and relies on one fundamental parameter of lattice size dx, which leads to a revolutionary and precise minimal "Lattice" Boltzmann method, where physical variables such as velocity and density can be retained as boundary conditions with less required storage for more accurate and efficient simulations in modelling flows using boundary condition such as Dirichlet's one. Here, the idea for the MacLAB is further developed for solving the shallow water flow equations (MacLABSWE). This new model has all the advantages of the conventional LBM but without calculation of the particle distribution functions for determination of velocity and depth, e.g., the most efficient bounce-back scheme for no-slip boundary condition can be implemented in the similar way to the standard LBM. The model is applied to simulate a 1D unsteady tidal flow, a 2D wind-driven flow in a dish-shaped lake and a 2D complex flow over a bump. The results are compared with available analytical solutions and other numerical studies, demonstrating the potential and accuracy of the model.