Preconditioned Multigrid Methods for Compressible Flow Calculations on Stretched Meshes

whelming. On the other hand, the development of efficient numerical methods for solution of the Navier–Stokes equaEfficient preconditioned multigrid methods are developed for both inviscid and viscous flow applications. The work is motivated tions remains one of the ongoing challenges in the field by the mixed results obtained using the standard approach of scalar of computational fluid dynamics. Dramatic improvements preconditioning and full coarsened multigrid, which performs well over the performance of existing methods will be necessary for Euler calculations on moderately stretched meshes but is far before this area of research may be considered satisfactoless effective for turbulent Naiver–Stokes calculations, when the cell stretching becomes severe. In the inviscid case, numerical studies rily resolved. of the preconditioned Fourier footprints demonstrate that a blockThe difficulty for viscous calculations stems from the Jacobi matrix preconditioner substantially improves the damping need to use a computational mesh that is highly resolved and propagative efficiency of Runge–Kutta time-stepping schemes in the direction normal to the wall in order to accurately for use with full coarsened multigrid, yielding computational savrepresent the steep gradients in the boundary layer. The ings of approximately a factor of three over the standard approach. In the viscous case, determination of the analytic expressions for resulting high aspect ratio cells greatly reduce the efficiency the preconditioned Fourier footprints in an asymptotically stretched of existing numerical algorithms. The design of an approboundary layer cell reveals that all error modes can be effectively priate numerical approach must therefore be based on a damped using a combination of block-Jacobi preconditioning and careful assessment of the interaction between the discrete a J-coarsened multigrid strategy, in which coarsening is performed only in the direction normal to the wall. The computational savings method, the computational mesh, and the physics of the using this new approach are roughly a factor of 10 for turbulent viscous flow. Navier–Stokes calculations on highly stretched meshes. Q 1997 AcaSince the relevant problem size will continue to increase demic Press as fast as hardware constraints will allow, it is critical that the convergence rate of the method should be insensitive to the number of unknowns. The general solution strategy