This study conducts a systematic evaluation of various numerical schemes in computational fluid dynamics (CFD), an area where a definitive best method is absent and user choices are often influenced by subjective experiences. Our focus is on high-fidelity simulations, particularly turbulent channel flow at a friction Reynolds number $Re_{\tau} \approx 550$, across a range of resolutions from DNS to LES. The evaluated schemes include finite difference methods (FDM) with 2nd, 4th, 6th, and mixed orders of accuracy, along with spectral element methods (SEM) with polynomial orders $p=5$, $7$, $9$, and $11$. Our analysis employs a base DNS resolution of $(\Delta x^{+}, y_1^{+}, \Delta z^{+}) = (12, 0.5, 5)$, and 216 grids derived by varying each directional resolution by factors of 1, 1.5, 2, 3, 4, 5, and their combinations. For the FDM method, highest accuracy is obtained at the highest order. A mixed order approach, combining a 4th-order convective term with a 2nd-order diffusive term, exhibits a performance similar to that of a standard 4th-order discretization. All FDM formulations exhibit noteworthy errors in some Reynolds stress components at resolutions of $(24,1.0, 10)$ or coarser. The 2nd-order FDM performs quite poorly at all resolutions. The spectral methods excel at finer resolutions of $(\Delta x^+, y_1^+, \Delta z^+)=(12,0.5,5)$ and $(18,0.75,7.5)$, marginally outperforming the finite differences. However, the SEM solution becomes sensitive to the polynomial order at $(\Delta x^+, y_1^+, \Delta z^+)=(24,1,10)$ and coarser. The $p=5$ formulation performs poorly, with an overall performance similar to a 2nd-order FDM. The highest accuracy and lowest sensitivity is obtained for $p=7$ at resolutions up to $(36,1.5,15)$, and for $p=11$ on the coarser grids; however, this emerging accuracy of $p=11$ seems to stem from error cancellation (similarly present in FDM). The best FDM solutions and the best SEM solutions have similar error levels on all grids, where FDM leads to more accurate mean velocity and SEM to better wall-normal and spanwise normal Reynolds stresses. The 6th-order FDM shows a better overall performance and versatility for the set of grids studied here. These initial findings may evolve with ongoing research. In particular, anisotropically coarsened grids will be utilized for additional assessments of sensitivity and robustness. Considerations regarding cost and efficiency will need to be taken into account as well.