This paper uses a spectral algorithm for the numerical solution of unsteady periodic problems. This algorithm which is based on discrete Fourier transforms uses a Fourier representation for periodic conditions of problems and hence solves the periodic state directly, without resolving transients because the discrete Fourier transforms have the periodic property to confirm physics of flow. The algorithm has been proposed for the fast and efficient computation of periodic flows. The algorithm has been validated with Stokes’ second problem as a linear problem and Burgers' equation as a nonlinear problem. The same numerical results are compared with an analytical solution, second-order Backward Difference Formula (BDF) and finite difference method (FDM) results. By enforcing periodicity by using Fourier representation that has a spectral accuracy, a tremendous increase of accuracy has been obtained compared to using the conventional numerical methods like BDF and FDM. Results verify the small number of time intervals per oscillating cycle required to capture the flow physics in Stokes’ second problem. Moreover, they show that a small number of points in a computational grid are required to capture the flow physics in Burgers' equation. Furthermore, this algorithm is more able than a finite difference method to capture shock.