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

Abstract: This paper is about the construction of a well-balanced gas-kinetic scheme (GKS) with space-time adaptive mesh refinement (STAMR) for the shallow water equations (SWE). While the well-balanced GKS has been established on Cartesian and triangular meshes, the present STAMR framework employs arbitrary quadrilateral meshes with hanging nodes, posing additional challenges for constructing well-balanced schemes. Complementing spatial adaptivity, temporal adaptivity is achieved by selecting adaptive time steps for cells of different refinement levels; the resulting STAMR further improves computational efficiency. Moreover, the numerical flux of the GKS adaptively makes transitions between equilibrium fluxes for smooth flows and non-equilibrium fluxes for discontinuities, endowing the proposed GKS-based AMR method with strong robustness, high accuracy, and high resolution. Standard benchmarks and realistic cases validate the GKS-based STAMR and highlight its potential for interface capturing and complex flow simulations.