A projection-based reduced order model (ROM) based on the Fourier collocation method is proposed for compressible flows. The incorporation of localized artificial viscosity model and filtering is pursued to enhance the robustness and accuracy of the ROM for shock-dominated flows. Furthermore, for Euler systems, ROMs built on the conservative and the skew-symmetric forms of the governing equation are compared. To ensure efficiency, the discrete empirical interpolation method (DEIM) is employed. An alternative reduction approach, exploring the sparsity of viscosity is also investigated for the viscous terms. A number of one- and two-dimensional benchmark cases are considered to test the performance of the proposed models. Results show that stable computations for shock-dominated cases can be achieved with ROMs built on both the conservative and the skew-symmetric forms without additional stabilization components other than the viscosity model and filtering. Under the same parameters, the skew-symmetric form shows better robustness and accuracy than its conservative counterpart, while the conservative form is superior in terms of efficiency.