Preconditioning Techniques for a Newton-Krylov Algorithm for the Compressible Navier-Stokes Equations

An investigation of preconditioning techniques is presented for a Newton--Krylov algorithm that is used for the computation of steady, compressible, high Reynolds number flows about airfoils. A second-order centred-difference method is used to discretize the compressible Navier--Stokes (NS) equations that govern the fluid flow. The one-equation Spalart--Allmaras turbulence model is used. The discretized equations are solved using Newton's method and the generalized minimal residual (GMRES) Krylov subspace method is used to approximately solve the linear system. These preconditioning techniques are first applied to the solution of the discretized steady convection-diffusion equation. Various orderings, iterative block incomplete LU (BILU) preconditioning and multigrid preconditioning are explored. The baseline preconditioner is a BILU factorization of a lower-order discretization of the system matrix in the Newton linearization. An ordering based on the minimum discarded fill (MDF) ordering is developed and compared to the widely popular reverse Cuthill--McKee ordering. An evolutionary algorithm is used to investigate and enhance this ordering. For the convection-diffusion equation, the MDF-based ordering performs well and RCM is superior for the NS equations. Experiments for inviscid, laminar and turbulent cases are presented to show the effectiveness of iterative BILU preconditioning in terms of reducing the number of GMRES iterations, and hence the memory requirements of the Newton--Krylov algorithm. Multigrid preconditioning also reduces the number of GMRES iterations. The framework for the iterative BILU and BILU-smoothed multigrid preconditioning algorithms is presented in detail.Ph