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

TL;DR

This paper introduces an adaptive central-upwind scheme on quadtree grids for efficiently simulating variable density shallow water equations, ensuring stability, positivity, and accurate steady-state preservation.

Contribution

It develops a novel adaptive scheme combining well-balanced central-upwind methods with quadtree grids for variable density shallow water modeling.

Findings

Exact preservation of lake-at-rest steady states

Higher-order spatial accuracy with bi-linear interpolation

Adaptive grid refinement based on local gradients

Abstract

Minimizing computational cost is one of the major challenges in the modelling and numerical analysis of hydrodynamics, and one of the ways to achieve this is by the use of quadtree grids. In this paper, 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 well-balanced positivity-preserving central-upwind scheme proposed in [18]. 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.