A new compact third-order discretization for the solution of the steady/unsteady flow equations on unstructured grids (tetrahedral cells) is proposed. The proposed algorithm, which belongs to the class of Multidimensional Upwind schemes, is a cell-based algorithm which uses a finite element like reconstruction procedure to achieve its third order (spatial) accuracy. The derivation of the scheme will be presented. The asymptotic accuracy, both for steady and unsteady inviscid and viscous flow simulations has been proved using numerous numerical experiments. Some representative results will be presented. These results will be compared with the performances of the second order Multidimensional Upwind schemes. The parallel scalability and the computational cost of the new discretization are also addressed. Our studies show clearly the advantages of the new compact third order scheme, as compared with the classical second order Multidimensional Upwind schemes. Application of this new scheme target LES of complex turbulent flows"