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.

翻译：在本文中, 我们为两维浅水方程式构建一个强大的适应性中上风系统, 用于具有可变密度的两维浅水方程式。 这种方法在两种液体合并类型的曲线上是平衡的、 正振- 保存和无振荡的。 提议的方法是扩展Epshteyn 和 Nguyen (ar Xiv printprint arXiv: 2011. 06.143, 2020) 开发的适应性良好、 正振- 保护计划( ar Xiv preprint arXiv: 2011. 06.143, 2020) 。 特别是为了维护“ lac- at- rest” 稳定状态, 我们使用带适当旋转坐标的Riemann 溶液解器在液体界面附近获取点值。 此外, 为了提高多流流流流流中适应方法的效率, 密度不连续的曲线通过水平设定方法和体积分法重建。 为了显示新的适应性中风方办法的准确性、 高分辨率和效率, 进行了几项具有挑战性的浅水模型试验。