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Two different categories of high-order numerical methods are adopted for the electro-hydrodynamic modeling of the streamer: the finite-element method (FEM) with flux-corrected-transport (FCT) technique and the finite-volume method (FVM) with monotonic upwind-centered scheme for conservation-law (MUSCL) algorithm. Specific numerical tests on the transport equation are used to investigate the efficiency of the two methods to propagate sharp density gradients in stationary uniform and nonuniform velocity fields. The streamer simulations are performed in a positive point-to-plane electrodes filled with air at room temperature and atmospheric pressure. The influence of the numerical schemes on the 2-D streamer modeling is analyzed. Both FVM-MUSCL and FEM-FCT accurately describe the streamer propagation and the morphology of the discharge channel, thereby giving similar axial and radial density profiles. Furthermore, the computational cost is higher for the FEM-FCT solver as compared to the FVM-MUSCL one, which is at least twice as fast. However, the unstructured-grid approach adopted by FEM-FCT proves to be very efficient in describing nonuniform geometries.