NA Seminar: Philipp Birken, "Additive W smoothers for multigrid methods for the RANS equations"
Abstract: We consider multigrid methods for compressible turbulent flow problems. For the Reynolds averaged Navier-Stokes equations (RANS), important progress has been achieved for finite volume discretizations through a new class of preconditioned Runge-Kutta (RK) smoothers in the last ten years. We show that properties of these schemes can be better understood if derived from additive Runge-Kutta (RK) methods. This gives rise to two classes of preconditioners: Preconditioned additive explicit RK and additive W methods. The latter class can be implemented exactly as the first one, with a suitably chosen preconditioner.
As a preconditioner we look at an SGS preconditioner based on a simplified discretization developed by Jameson. A crucial part is a cutoff function for zero eigenvalues, where the cutoff value has to be chosen. We perform a local Fourier analysis of an SGS preconditioner for the Euler equations. The results can be easily understood from the theory of time integration methods and give guidance on how to choose the various parameters of the scheme.
Finally, numerical results for the RANS equations and pitching airfoils are presented that show that with the new schemes, convergence rates below 0.8 can be acchieved.