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We consider multigrid methods for finite volume discretizations of the Reynolds averaged Navier–Stokes equations for both steady and unsteady flows. We analyze the effect of different smoothers based on pseudo time iterations, such as explicit and additive Runge–Kutta (AERK) methods. Furthermore, by deriving them from Rosenbrock smoothers, we identify some existing schemes as a class of additive W (AW) methods. This gives rise to two classes of preconditioned smoothers, preconditioned AERK and AW, which are implemented the exact same way, but have different parameters and properties. This derivation allows to choose some of these based on results for time integration methods. As preconditioners, we consider SGS preconditioners based on flux vector splitting discretizations with a cutoff function for small eigenvalues. We compare these methods based on a discrete Fourier analysis. Numerical results on pitching and plunging airfoils identify AW3 as the best smoother regarding overall efficiency. Specifically, for the NACA 64A010 airfoil steady-state convergence rates of as low as 0.85 were achieved, or a reduction of 6 orders of magnitude in approximately 25 pseudo-time iterations. Unsteady convergence rates of as low as 0.77 were achieved, or a reduction of 11 orders of magnitude in approximately 70 pseudo-time iterations.

Preconditioned Smoothers for the Full Approximation Scheme for the RANS Equations