A matrix-free preconditioned Newton/GMRES method for unsteady Navier-Stokes solutions

The unsteady compressible Reynolds-averaged Navier-Stokes equations are dis cretized using the Osher approximate Riemann solver with fully implicit tim e stepping. The resulting non-linear system at each time step is solved ite ratively using a Newton/GMRES method. In the solution process, the Jacobian matrix-vector products are replaced by directional derivatives so that the evaluation and storage of the Jacobian matrix is removed from the procedur e. An effective matrix-free preconditioner is proposed to fully avoid matri x storage. Convergence rates, computational costs and computer memory requi rements of the present method are compared with those of a matrix Newton/GM RES method, a four stage Runge-Kutta explicit method, and an approximate fa ctorization sub-iteration method. Effects of convergence tolerances for the GMRES linear solver on the convergence and the efficiency of the Newton it eration for the non-linear system at each time step are analysed for both m atrix-free and matrix methods. Differences in the performance of the matrix -free method for laminar and turbulent flows are highlighted and analysed. Unsteady turbulent Navier-Stokes solutions of pitching and combined transla tion-pitching aerofoil oscillations are presented for unsteady shock-induce d separation problems associated with the rotor blade flows of forward flyi ng helicopters. Copyright (C) 2000 John Wiley & Sons, Ltd.