High-order compact gas-kinetic scheme for two-layer shallow water equations on unstructured mesh

For the two-layer shallow water equations, a high-order compact gas-kinetic scheme (GKS) on triangular mesh is proposed. The two-layer shallow water equations have complex source terms in comparison with the single layer equations. The main focus of this study is to construct a time-accurate evolution solution at a cell interface and to design a well-balanced scheme. The evolution model at a cell interface provides not only the numerical fluxes, but also the flow variables. In the development of the well-balanced scheme, the time-dependent flow variables at the cell interfaces surrounding the cell can be utilized to update the cell-averaged gradients. These time-evolving gradients are subsequently applied in the discretization of source terms inside each control volume. Based on the cell-averaged flow variables and their gradients, high-order initial data reconstruction can be achieved with compact stencils. The compact high-order GKS has advantages to simulate the flow evolution in complex domain covered by unstructured mesh. Many test cases are used to validate the accuracy and robustness of the scheme for the two-layer shallow water equations.