An adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations

TL;DR

This paper introduces an adaptive, well-balanced, positivity-preserving central-upwind scheme on quadtree grids for shallow water equations, enhancing accuracy and efficiency in modeling flows over complex bottom topographies.

Contribution

It develops a novel adaptive quadtree-based scheme that preserves steady states and positivity, improving numerical modeling of shallow water flows over discontinuous terrains.

Findings

Accurately preserves lake-at-rest steady states.

Capable of modeling flows over discontinuous bottom topography.

Demonstrates promising performance in numerical tests.

Abstract

We present an adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-upwind scheme proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), pp. 133--160]. The designed scheme is well-balanced in the sense that it is capable of exactly preserving "lake-at-rest" steady states. In order to achieve this as well as to preserve positivity of water depth, a continuous piecewise bilinear interpolation of the bottom topography function is utilized. This makes the proposed scheme capable of modelling flows over discontinuous bottom topography. Local gradients are examined to determine new seeding points in grid refinement for the next timestep. Numerical examples demonstrate the promising performance of the central-upwind quadtree scheme.