An adaptive central‐upwind scheme on quadtree grids for variable density shallow water equations

Minimizing computational cost is one of the major challenges in the modeling and numerical analysis of hydrodynamics, and one of the ways to achieve this is by the use of quadtree grids. In this article, we present an adaptive scheme on quadtree grids for variable density shallow water equations. A scheme for the coupled system is developed based on the work of [M.A. Ghazizadeh, A. Mohammadian, and A. Kurganov, Computers & Fluids, 208 (2020)]. The scheme is capable of exactly preserving “lake‐at‐rest” steady states. A continuous piecewise bi‐linear interpolation of the bottom topography function is used to achieve higher‐order in space in order to preserve the positivity of water depth for the point values of each computational cell. Necessary conditions are checked to be able to preserve the positivity of water depth and density, and to ensure the achievement of a stable numerical scheme. At each timestep, local gradients are examined to find new seeding points to locally refine/coarsen the computational grid. The presented scheme is well‐balanced and is able to hold the positivity‐preserving property for the scalar variables. The well‐balanced scheme is computationally less expensive than the nonwell‐balanced one. The presented scheme is robust and accurate.