Implicit preconditioners for the 2-D multigrid Navier-Stokes solvers

Abstract

Implicit preconditioned multistage Runge-Kutta schemes are presented for the classical full-coarsening multigrid method to solve the steady 2-D compressible Navier-Stokes equations. A comparative study of the residual operators discretized with upwind difference schemes is performed in Fourier space, and it is shown that the second-order upwind scheme discretized with an enlarged stencil can be preconditioned by the implicit preconditioner based on the first-order upwind scheme. Characteristics of clustering eigenvalues by the classical ADI and alternating direction line Jacobi (ADLJ) preconditioners are examined using the convex hulls of the Fourier footprints. As a result, it is presented that the ADLJ preconditioned residual operator discretized by the second-order upwind scheme with enlarged stencil shows good eigenvalue clustering characteristics. The Baldwin-Lomax and the Spalart-Allmaras turbulence models are implemented in a loosely coupled manner. The resulting multigrid method is tested on three cases: an inviscid transonic channel flow and subsonic and transonic turbulent flows past an airfoil. The numerical tests show consistent results with the Fourier analysis.