  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. The time-dependent flow variables at the closed
cell interfaces can be used to update the cell-averaged gradients for the
discretization of the the source terms inside each control volume in the
development of the well-balanced scheme. Based on the cell-averaged flow
variable 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.
