On Nonlinear Preconditioners in Newton-Krylov-Methods for Unsteady Flows Philipp Birken 1 and Antony Jameson 2 October 14, 2008 1 Department of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany. email: birken@mathematik.uni-kassel.de 2 Department of Aeronautics & Astronautics, Stanford University, Stanford, CA 94305, USA. Abstract The application of nonlinear schemes like dual time stepping as preconditioners in matrix-free Newton-Krylov-solvers is considered and analyzed. We provide a novel formulation of the left preconditioned operator that says it is in fact linear in the matrix-free sense, but changes the Newton scheme. This allows to get some insight in the convergence properties of these schemes which are demonstrated through numerical results. Keywords: Unsteady flows, Preconditioning, Newton-Krylov 1 Introduction As was shown by Jameson and Caughey in [2], the solution of steady Euler flows is today possible in three to five multigrid steps. Thus, two dimensional flows around airfoils can be solved on a PC in a matter of seconds. The solution of the steady RANS equations is more difficult and takes about fifty steps. Nevertheless this means that adequate methods for steady flows exist and the next big challenge for computational fluid dynamics is the compu- tation of unsteady problems. Now, for a lot of applications, the interesting flow phenomena are not on the scale of the fast acoustic eigenvalues, but on the scale of the convective eigenvalues. This makes implicit schemes for time integration much more interesting than explicit schemes, which are then severely restrained by the CFL condition. Usually, A-stable methods like BDF-2 are employed. For implicit schemes, their applicability is determined by the availability of fast solvers for the arising large nonlinear equation systems. Using dual time stepping, the above mentioned multigrid method can be used for unsteady flows. This results in a good method for Euler flows, but for the Navier-Stokes equations, 1