Adaptive Central-Upwind Scheme on Triangular Grids for the Shallow Water Model with variable density

TL;DR

This paper introduces a robust, adaptive central-upwind scheme on unstructured triangular grids for two-dimensional shallow water equations with variable density, ensuring well-balanced, positivity-preserving, and oscillation-free solutions at fluid interfaces.

Contribution

It extends previous schemes by incorporating adaptive methods, level set and volume fraction techniques, and specialized Riemann solvers to handle variable density and complex fluid interfaces.

Findings

The scheme accurately captures steady states and interfaces.

It demonstrates high resolution and efficiency in challenging tests.

The method effectively preserves positivity and avoids oscillations.

Abstract

In this paper, we construct a robust adaptive central-upwind scheme on unstructured triangular grids for two-dimensional shallow water equations with variable density. The method is well-balanced, positivity-preserving, and oscillation-free at the curve where two types of fluid merge. The proposed approach is an extension of the adaptive well-balanced, positivity-preserving scheme developed in Epshteyn and Nguyen (arXiv preprint arXiv:2011.06143, 2020). In particular, to preserve "lake-at-rest" steady states, we utilize the Riemann Solver with appropriately rotated coordinates to obtain the point values in the neighborhood of the fluid interface. In addition, to improve the efficiency of an adaptive method in the multifluid flow, the curve of density discontinuity is reconstructed by using the level set method and volume fraction method. To demonstrate the accuracy, high resolution, and efficiency of the new adaptive central-upwind scheme, several challenging tests for Shallow water models with variable density are performed.