Efficient acceleration techniques typical of explicit steady-state solvers are extended to tune-accurate calcu- lations. Stability restrictions are greatly reduced by means of a fully implicit time discretization. A four-stage Runge-Kutta scheme with local tune stepping, residual smoothing, and multigridding is used instead of traditional computationally expensive factorizations. Two applications to natural unsteady viscous flows are presented to check for the capability of the procedure. ECENT progress in computational fluid dynamics along with the evolution of computer performance encourages scientists to look in to details of flow physics. There are practical applications where the unsteadiness of the problem can not be neglected (i.e., vortex shedding, natural unsteadiness, forced unsteadiness, aeroe- lasticity, turbomachinery rotor-stator interaction). Up to now, most of the analysis and designing tools are based on a steady or quasi- steady assumption, even if the flow is known to be unsteady. Today, due to the improvement in computer resources, there is a strong interest in developing methodologies for efficient and reliable sim- ulation of unsteady flow features. It is a common experience, while using time-accurate explicit schemes, to be forced to choose the time step on the basis of stability restrictions. As a consequence, unless the problem is a very high frequency one, the number of time steps to be performed is much higher than the one required for time accuracy. By means of some implicit factorization, stability restrictions can be relaxed, but the work required at each time step grows rapidly with grid dimension and complexity of the flow equations. In addition, application of boundary conditions in a fully implicit manner is difficult. In viscous flow calculations, the grid is clustered close to the shear layer and the characteristic time step varies several orders of mag- nitude inside the computational domain. Even if in several practical applications the characteristic time step of the core-flow region is comparable with the one required by time accuracy, close to the walls the time step restrictions become extremely severe. There- fore, highly vectorizable schemes with less stability restrictions on the allowable time step would be an interesting combination. Explicit schemes combined with acceleration techniques have proven to be very effective for solving steady problems.13 Unfortu- nately, the computational efficiency of those time-marching solvers is achieved by sacrificing the accuracy in time. In this paper, it is shown that the conventional steady-state acceleration techniques, specifically the multigrid techniques, can still be applied to unsteady viscous problems. The basic idea is to introduce a dual time stepping and to reformulate the governing equations so that they can be han- dled by an explicit accelerated scheme.4 If the time discretization is made implicit, stability restrictions are removed and accelera- tion techniques can be used instead of traditional time-consuming factorizations (i.e., alternate directional implicit and lower/upper).