Fully Implicit Chebyshev Time-Spectral Method for General Unsteady Flows

Abstract

An efficient fully implicit Chebyshev time-spectral method (TSM) is developed to solve general unsteady flows with or without periodicity. High accuracy of the Chebyshev time-spectral operator is demonstrated by simulating the one-dimensional shock motion at constant speed. A space-time lower–upper symmetric Gauss–Seidel (LU-SGS) fully implicit scheme with multigrid acceleration is used to efficiently solve the equations resulting from the Chebyshev TSM. The implicit temporal coupling term is properly split so that the LU-SGS sweeps are extended to the time domain. For cases in which the physical time step is small, a modification to the implicit scheme is suggested to enhance numerical stability. Computational results of nonperiodic flows over a pitching airfoil are presented for different physical time steps and various numbers of Chebyshev collocation points. The fully implicit Chebyshev TSM is also tested to predict the natural vortex shedding frequency for a stationary circular cylinder. This efficient approach can be easily applied to the time-accurate computation of general nonperiodic unsteady flows with a given initial condition.