Adaptive mesh refinement algorithm for CESE schemes on unstructured quadrilateral meshes

This study introduces the development of space-time Conservation Element and Solution Element (CESE) methods tailored for adaptive unstructured quadrilateral meshes. An efficient algorithm is then proposed to manage the mesh adaptation process for these staggered schemes, utilizing a unique cell-tree-vertex data structure. This structure accelerates the construction of conservation elements and simplifies the interconnection of computational cells, enabling a flexible approach for handling adaptive mesh refinement in complex computational domains. The integration of second-order a-α, Courant number-insensitive, and upwind CESE schemes with this adaptation algorithm is demonstrated. Numerical simulations of compressible inviscid flows are conducted to validate the global conservation property, ensure second-order accuracy across interfaces at different refinement levels, and evaluate the effectiveness of the extended schemes and adaptation algorithm.