Lattice Boltzmann Method based on Body-fitted Grids using General Orthogonal Coordinates for Shallow Water Equations

Shallow water equations (SWE) arise in the modeling of geophysical fluid dynamics applications. They represent the depth-averaged Navier-Stokes equations with a nonlinear equation of state for pressure. In order to simulate them efficiently, we construct a novel lattice Boltzmann method (LBM) based on body-fitted grids using general orthogonal coordinates that naturally accommodate the clustering of grids around flow features with finer scales and their stretching elsewhere. This represents a significant improvement over the standard LBM which is restricted to using uniform Cartesian grids. The general orthogonal coordinates-based LBM is developed via using a Chapman-Enskog analysis and uses central moments for the collision step for improved stability. This involves constructing the equilibria based on metric factors together with additional source terms to recover the SWE. The resulting approach maintains the simplicity and locality of the collide-and-stream steps of the standard LBM while enabling the use of flexible gridding structures. We demonstrate the utility of our formulation for various benchmark case studies based on SWE.