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SUMMARY The reduction-to-periodicity method using the pseudospectral fast Fourier transform (FFT) technique is applied to the solution of non-periodic problems, including the two-dimensional incompressible Navier-Stokes equations. The accuracy of the method is explored by calculating the derivatives of given functions, one- and two-dimensional convective4iffusive problems, and by comparing the relative errors due to the FFT method with a second-order finite difference (FD) method. Finally, the two-dimensional Navier-Stokes equations are solved by a fractional step procedure using both the FFT and the FD methods for the driven cavity flow and the backward-facing step problems. Comparisons of these solutions provide a realistic assessment of the FFT method. In this paper we discuss a numerical technique for solving partial differential equations in nonperiodic domains by the use of the pseudospectral method. This work was motivated by the need to develop a time-dependent incompressible Navier-Stokes solver with non-periodic inflow/outflow boundary conditions involving wave propagation problems. In such problems the use of Chebyshev expansions along the streamwise direction provides an accurate solution procedure at the cost of imposing very stiff viscous time step restrictions. Note that along the streamwise direction, time-explicit methods can be efficiently used owing to the absence of any sharp gradients at inflow/outflow boundaries. In flows bounded by solid walls, however, it is expected that the spectral Chebychev method will be superior for the same number of grid points and the same operation count. The goal of this paper is to provide a direct comparison of the reduction-toperiodicity method with a widely used second-order finite difference method in an effort to isolate the advantages/shortcomings of the reduction-to-periodicity method. In periodic domains the use of the pseudospectral fast Fourier transform technique was instigated by the work of Or~zag,'-~ and since then the method has been extensively used in solving multidimensional fluid dynamics problems with periodic boundary conditions. The article by Orszag and Israeli4 provides other references and an introduction to the subject. The idea of polynomial subtraction for the purpose of satisfying non-periodic boundary constraints was introduced by Lanczos5 and later developed by Gottlieb and Orszag.6 In one-dimensional mode1 problems the well known Gibbs phenomenon that appears at the boundaries when such methods are used with non-periodic boundary conditions has been shown to be suppressed by the use of simple polynomials.'* * In Reference 8 the accuracy of the reduction-to-periodicity method is