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Jacobian‐free Newton–Krylov method for implicit time‐spectral solution of the compressible Navier‐Stokes equations

We introduce a Jacobian‐free Newton–Krylov method for the implicit time‐spectral solution of the compressible Navier‐Stokes equations. A new type of preconditioner is presented, which is based upon an approximate factorization of an approximation to the exact time‐spectral Jacobian. The choice of this type of preconditioner is particularly useful when the time‐spectral scheme is to be implemented into a computational code that already contains an implicit time‐marching solver. The spatial discretization of the Navier–Stokes equation consists of a sixth‐order compact scheme with a high‐order, low‐pass filter. Numerical simulation of the laminar flow over a circular cylinder at two post‐critical Reynolds numbers (Re=50,100) is used to characterize the performance of the method. In general, the time‐spectral solution is two to ten times faster, for equivalent accuracy in the lift coefficient, than a time‐marching implicit Beam‐Warming solution with the upper end of this range noted when the Reynolds number is closer to the dynamic instability bifurcation Reynolds number. Other numerical and analytical results are presented, which demonstrate various aspects of the preconditioner performance. In addition, results and discussion are given for some characteristics of time‐spectral solutions for autonomous problems where the fundamental frequency is unknown. Copyright © 2015 John Wiley & Sons, Ltd.