In this paper, the trailing edge noise generated by a 2D airfoil around the critical angle of attack for vortex shedding is numerically investigated using an in-house code with high accuracy and efficiency. In the present method, a fourth-order upwind compact finite-difference scheme with dispersion relation preserving (DRP) property is applied for the convection terms, and a fourth-order Runge-Kutta scheme is used for temporal discretization. The reflection of sound on the boundary is suppressed with Navier-Stokes characteristics boundary condition (NSCBC). To improve computational efficiency, a novel parallel computing strategy for the high-order compact schemes is employed. Thus, direct numerical simulation (DNS) can be realized for the flows of low Reynolds number (Re), while implicit large eddy simulation (ILES) would be carried for the flows of high Reynolds number. The present numerical method is validated by comparing the lift coefficient, drag coefficient and Strouhal number (St) to the previous publications. Based on the high accuracy and high-fidelity method, the flow field and sound field of a two-dimensional NACA0012 airfoil around critical angle of attack (AoA) at Re = 1000 are simultaneously solved. The results indicate that sound source is dipole centered at the surface of the airfoil at vortex shedding frequency, and is dipole, quadrupole or more complex sources located at the wake close to the trailing edge at higher order frequencies. These findings will help to improve understanding about the generation and propagation mechanisms of trailing edge noises at low Reynolds number.@en