Preconditioned Multigrid Methods for Compressible Flow Calculations on Stretched Meshes

Abstract

Efficient preconditioned multigrid methods are developed for both inviscid and viscous flow applications. The work is motivated by the mixed results obtained using the standard approach of scalar preconditioning and full coarsened multigrid, which performs well for Euler calculations on moderately stretched meshes but is far less effective for turbulent Naiver–Stokes calculations, when the cell stretching becomes severe. In the inviscid case, numerical studies of the preconditioned Fourier footprints demonstrate that a block-Jacobi matrix preconditioner substantially improves the damping and propagative efficiency of Runge–Kutta time-stepping schemes for use with full coarsened multigrid, yielding computational savings of approximately a factor of three over the standard approach. In the viscous case, determination of the analytic expressions for the preconditioned Fourier footprints in an asymptotically stretched boundary layer cell reveals that all error modes can be effectively damped using a combination of block-Jacobi preconditioning and a J-coarsened multigrid strategy, in which coarsening is performed only in the direction normal to the wall. The computational savings using this new approach are roughly a factor of 10 for turbulent Navier–Stokes calculations on highly stretched meshes.