On essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation

A few years ago, the class of Essentially Non-Oscillatory Schemes for the numerical simulation of hyperbolic equations and systems was constructed. Since then, some extensions have been made to multidimensional simulations of compressible flows, mainly in the context of very regular structured meshes. In this paper, we first recall and improve the results of an earlier paper about non-oscillatory reconstruction on unstructured meshes, emphasizing the effective calculation of the reconstruction. Then we describe a class of numerical schemes on unstructured meshes and give some applications for its third order version. This demonstrates that a higher order of accuracy is indeed obtained, even on very irregular meshes.