Block-Jacobi Implicit Algorithms for the Time Spectral Method

Many industrial applications involve flows periodic in time. Such flows are not simulated with enough efficiency when using classical unsteady techniques, as a transient regime must be bypassed. New techniques, dedicated to time-periodic flows and based on Fourier analysis, have been developed recently. Among these, the time spectral method casts a time-periodic flow computation in several coupled steady computations, corresponding to a uniform sampling of the period. Up until now, the steady states were reached using an explicit pseudotime algorithm. Thus, very small time steps were needed and the convergence rate slowed down when increasing the number of harmonics. In this paper, a block-Jacobi approach is presented to solve the stationary problems with an implicit algorithm. Numerical simulations show, on one hand, the good quality of the results and, on the other hand, the interest of the proposed method to reduce the sensitivity of computations to a large number of harmonics.