Adaptive Central-Upwind Scheme on Triangular Grids for the Saint-Venant System

TL;DR

This paper introduces an adaptive, well-balanced central-upwind scheme on unstructured triangular grids for shallow water equations, improving accuracy and efficiency through local error estimation and mesh refinement.

Contribution

It extends previous schemes by incorporating adaptivity and positivity-preservation, with a novel a posteriori error estimator for mesh refinement.

Findings

Demonstrates high accuracy and resolution on challenging shallow water tests.

Shows improved computational efficiency with adaptive mesh refinement.

Validates robustness and stability of the scheme in various scenarios.

Abstract

In this work, we develop a robust adaptive well-balanced and positivity-preserving central-upwind scheme on unstructured triangular grids for shallow water equations. The numerical method is an extension of the scheme from [{\sc Liu {\em et al.}},J. of Comp. Phys, 374 (2018), pp. 213 - 236]. As a part of the adaptive central-upwind algorithm, we obtain local a posteriori error estimator for the efficient mesh refinement strategy. The accuracy, high-resolution, and efficiency of the new adaptive central-upwind scheme are demonstrated on a number of challenging tests for shallow water models.