“…In the current study with UCNS3D [51,52] the least-square reconstruction approaches of Tsoutsanis et al [9,10,23,53] and Titarev et al [54] are adopted, that have been applied to laminar, transitional, and fully turbulent flows at the a wide-range of Mach numbers [9,10,23,45, . Herein the most key details of this approach are presented, and the reader is pointed to previous work for a detailed description of the implementations For an arbitrary cell i obtaining a high-order r + 1 order, requires the formation of an r order polynomial p i (x, y, z) which has to provide the volume-averaged value W i as follows:…”