AbstractThis paper discusses the different sources of non‐physical entropy production which occur in the numerical resolution of the Euler equations for compressible inviscid flow and proposes several ways of correcting these effects. In particular a hybrid corrected centred, augmented by an accurate upwind scheme near singular boundaries is proposed which satisfies the mathematical entropy condition, and which solves the flow accurately within regions near non‐smooth boundaries of the computational domain. A coupled new dynamic auto‐adaptive mesh algorithm which produces highly accurate solutions is also introduced. This algorithm is non‐hierarchical. i.e. it does not depend on a fixed background mesh, which allows structural and geometrical changes and generates extremely precise discretizations for steady and unstea