Efficient GMRES algorithm in time spectral method

In this paper, the computational efficiency of the time-spectral method for solving the periodic unsteady flow field is studied, and the implicit method of time spectral method for solving the periodic unsteady flow is discussed. When the number of sampling points increases or the reduced frequency magnifies, the diagonal dominant property of the Jacobian matrix corresponding to the time spectral method deteriorates rapidly, resulting in the failure of many traditional iterative methods. In order to solve the problems above, the generalized minimum residual (GMRES) algorithm with preprocessing is used to improve the computational convergence of the Jacobian matrix. The time spectral method is used to compute the NACA0012 airfoil forced oscillation, and the computational efficiency and accuracy is compared with that of the time-domain difference method. The results show that the time spectral method can generally improve the computational efficiency an order of magnitude with saturated computational accuracy. For the transonic periodic flow, the GMRES algorithm is superior to SGS iterative algorithm both in stability and computational convergence.