  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.
