Multigrid and defect correction for the steady Navier-Stokes equations

Abstract

Theoretical and experimental convergence results are presented for nonlinear multigrid and iterative defect correction applied to finite volume discretizations of the full, steady, 2D, compressible Navier-Stokes equations. Iterative 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 the 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 non-smooth 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.