An efficient correction storage scheme on a structured grid is applied to a sequence of approximate Jacobian systems arising at each time step from a linearization of the discrete nonlenear system of equations, obtained by the implicit time discretization of the conservation laws for unsteady fluid flows. The contribution of freezing the Jacobian matrix to computing costs is investigated within the correction storage scheme. The performance of the procedure is exhibited by measuring CPU time required to obtain a fully developed laminar vortex shedding flow past a circular cylinder, and is compared with that of a collective iterative method on a single grid. In addition, some computed results of the flow are presented in terms of some functionals along with measured data. The computational test shows that the computing costs may be saved in favor of the correction storage scheme with the frozen Jacobian matrix, to a great extent.