Multigrid and defect correction for the steady Navier-Stokes equations : application to aerodynamics

Theoretical and expcrimental convergence results are presented for nonlinear multigrid and iterative defect correction applied to finite volume discretizations of the full, steady, 2D, compressible NavierStokes equations. lterative defect correction is introduced for circumventing the difficulty in solving Navier Stokes equations discretized with a second- or higher-order accurate convective part. By Fourier analysis applied to a model equation, an optimal choice is made for the operator to be inverted in the defect correction iteration. As a smoothing technique for thc multigrid method, collective symmetric point Gauss-Seidel relaxation is applied with as the basic solution technique: exact Newton iteration applied to a continuously differentiable, first-order upwind discretization of the full Navier Stokes equations. For nonsmooth flow problems, the convergence results obtained are already competitive with those of well-established Navier-Stokes methods. For smooth flow problems, the present method performs better than any standard method. Here, first-order discretization error accuracy is attained in a single multigrid cycle, and second-order accuracy in only one defect correction cycle. The method contributes to the state of the art in efficiently computing compressible viscous flows.