This paper discusses the potential of the application of a high-order cell-centered finite difference method (CCFDM) to problems in aeroacoustics. Based on an in-house code, a class of low-dissipation and low-dispersion compact schemes adopting optimization techniques is proposed for direct noise calculation of long-time long-distance sound propagation problems. These schemes are further validated with benchmark cases from a CAA workshop and the results indicate these schemes outperform the widely-used 5th-order upwind scheme. The second part of this paper investigates the capability of combining CCFDM with the FW-H acoustic analogy for the far-field noise prediction. Two example cases, including a tonal noise case (laminar flow past a 2D NACA0012 airfoil) and a broadband case (turbulent flow past a 3D rod-airfoil configuration), are both analyzed in detail. Good agreement with experimental data has been achieved, which indicates that the CCFDM coupled with the FW-H acoustic analogy has great potential in aeroacoustics.