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.

翻译：针对双层浅水方程，本文提出了一种基于三角形网格的高阶紧致气体动理学格式（GKS）。与单层方程相比，双层浅水方程具有更复杂的源项。本研究的主要目标是构建单元界面处的时间精确演化解，并设计一种平衡格式。单元界面处的演化模型不仅提供数值通量，还提供流动变量。在闭单元界面处的时间相关流动变量可用于更新单元平均梯度，进而在平衡格式的构建过程中离散每个控制体内部的源项。基于单元平均流动变量及其梯度，可采用紧致模板实现高阶初始数据重构。该紧致高阶GKS在模拟非结构化网格覆盖的复杂域内流动演化方面具有优势。通过多个测试算例验证了该格式求解双层浅水方程的精度和鲁棒性。