A well-balanced gas-kinetic scheme with adaptive mesh refinement for shallow water equations

This paper presents the development of a well-balanced gas-kinetic scheme (GKS) with space-time adaptive mesh refinement (STAMR) for the shallow water equations (SWE). While well-balanced GKS have been established on Cartesian and triangular meshes, the proposed STAMR framework utilizes arbitrary quadrilateral meshes with hanging nodes, introducing additional challenges for maintaining well-balanced properties. In addition to spatial adaptivity, temporal adaptivity is incorporated by assigning adaptive time steps to cells at different refinement levels, further enhancing computational efficiency. Furthermore, the numerical flux in the GKS adaptively transitions between equilibrium fluxes for smooth flows and non-equilibrium fluxes for discontinuities, providing the proposed GKS-based STAMR method with strong robustness, high accuracy, and high resolution. Standard benchmark tests and real-world case studies validate the effectiveness of the GKS-based STAMR and demonstrate its potential for interface capturing and the simulation of complex flows.