Macroscopic Lattice Boltzmann Method for Shallow Water Equations

The lattice Boltzmann method (LBM) is characterised by its simplicity, parallel processing and easy treatment of boundary conditions. It has become an alternative powerful numerical method in computational physics, playing a more and more important role in solving challenging problems in science and engineering. In particular, the lattice Boltzmann method with the single relaxation time (SLBM) is the simplest and most popular form of the LBM that is used in research and applications. However, there are two long-term unresolved problems that prevent the SLBM from being an automatic simulator for any flows: (1) stability problem associated with the single relaxation time and (2) no method of direct implementation of physical variables as boundary conditions. Recently, the author has proposed the macroscopic lattice Boltzmann method (MacLAB) to solve the Navier-Stokes equations for fluid flows, resolving the aforementioned problems; it is unconditionally stable and uses physical variables as boundary conditions at lower computational cost compared to conventional LBMs. The MacLAB relies on one fundamental parameter of lattice size delta x, and is a minimal version of the lattice Boltzmann method. In this paper, the idea of the MacLAB is further developed to formulate a macroscopic lattice Boltzmann method for shallow water equations (MacLABSWE). It inherits all the advantages from both the MacLAB and the conventional LBM. The MacLABSWE is developed regardless of the single relaxation time tau. Physical variables such as water depth and velocity can directly be used as boundary conditions, retaining their initial values for Dirichlet's boundary conditions without updating them at each time step. This makes not only the model to achieve the exact no-slip boundary condition but also the model's efficiency superior to the most efficient bounce-back scheme for approximate no-slip boundary condition in the LBMs, although the scheme can similarly be implemented in the proposed model when it is necessary. The MacLABSWE is applied to simulate a 1D unsteady tidal flow, a 2D steady wind-driven flow in a dish-shaped lake and a 2D steady 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.