The objective is to improve the stability and accuracy of finite volume spatial discretization on unstructured grids. The interest lies in the growing use of finite volumes for large eddy simulation (LES) that requires accurate discretization methods. Another goal is the design of algorithms capable of reconstructing polynomials of higher degree on unstructured grids using only small and compact stencils. The study starts with a general analysis of the reconstruction of polynomials of degree k on unstructured grids, completed by numerical measurements of the convergence rate of the reconstruction error for polynomials of degree 2 and 3. The study presents algorithms for the reconstruction of polynomials on small stencils. Numerical experiments confirm the order of the approximation of these reconstruction methods for quadratic polynomials in 2 dimensions. A theoretical stability analysis exhibits general principles for the design of stable reconstruction methods. A theoretical accuracy analysis, based on the modified equation approach, highlights the errors induced by unstructured grids. The theoretical investigations are completed and confirmed by numerical experiments. The study of slope limiters on unstructured grids formulates algorithms based on a geometric approach.Large eddy simulations of a subsonic flow over a cavity and of a supersonic jet allow the validation and comparison of several discretization features implemented in the code Cedre of Onera.The results of the theoretical stability analysis make it possible to obtain better results for the jet computation on tetrahedral grids.
