The basis of the method is a pseudo-spectral approach to recast a non-linear unsteady system of equations in the temporal domain into a stationary system in the frequency domain. The NLFD method, in principle, provides the rapid convergence of a spectral method with increasing numbers of modes, and, in this sense, it is an optimal scheme for time-periodic problems. In practice it can also be effectively used as a reduced order method in which users deliberately choose not to resolve temporal modes in the solution.

The method is easily applied to problems where the time period of the unsteadiness is known a priori. A method is proposed that iteratively calculates the time period when it is not known a priori. Convergence acceleration techniques like local time-stepping, implicit residual averaging and multigrid are used in the solution of the frequency-domain equations. A new method, spectral viscosity is also introduced. In conjunction with modifications to the established techniques this produces convergence rates equivalent to state-of-the-art steady-flow solvers.

Results from the NLFD solver have been compared to experimental data from two different problems.

1. Vortex shedding in low Reynolds number flows past cylinders. Numerical results demonstrate the efficiency of the NLFD method in representing complex flow field physics with a limited number of temporal modes. The shedding frequency is unknown a priori, which serves to test the application of the proposed variable-time period method.
2. Airfoil undergoing a forced pitching motion in transonic flow. Comparisons with experimental results demonstrate that a limited number of temporal modes can accurately represent a non-linear unsteady solution. Comparisons with time-accurate codes also demonstrate the efficiency gains realized by the NLFD method.