Nowadays, Computational Fluid Dynamics (CFD) is widely used in the aviation and aerospace industry to develop new designs. Despite the improvement in numerical simulation over the past decades, current results remain inaccurate due to multiple sources of error, which can appear at every step of the numerical simulation pipeline. In this work, we will focus on the mesh discretization error and the numerical scheme discretization error. The finite volume method discretizes partial differential equations and is well suited to conservative laws. In this study, we focus on the cell-vertex finite volume approach initially developed by A. Dervieux, P. Rostand and B. Soufflet. Recent results obtained by the use of this class of numerical schemes show a strong robustness on very anisotropic meshes. In combination with anisotropic mesh adaptation, there is a real need in the industry for robust, high-order numerical schemes. In our work, we propose two approaches for improving cell-vertex schemes. On the one hand, we look at the geometric error made when calculating convective flows. We propose a new correction based on the derivative of the numerical flux and the mesh metrics. On the other, we tackle the problems associated with deconvolution. Several deconvolution approaches are studied on stationary cases. In order to measure the errors of the various scheme corrections, we carried out a numerical error analysis. This new analysis makes it possible to numerically quantify the error of the various numerical schemes.