Full multigrid (FMG) algorithms are the fastest solvers for elliptic problems. These algorithms can solve a general discretized elliptic problem to the discretization accuracy in a computational work that is a small (less than 10) multiple of the operation count in one target-grid residual evaluation. Such efficiency is known as textbook multigrid efficiency (TME). The difficulties associated with extending TME for solution of the Reynolds-averaged Navier-Stokes (RANS) equations relate to the fact that the RANS equations are a system of coupled nonlinear equations that is not, even for subsonic Mach numbers, fully elliptic, but contain hyperbolic partitions. TME for the RANS simulations can be achieved if the different factors contributing to the system could be separated and treated optimally, e.g., by multigrid for elliptic factors and by downstream marching for hyperbolic factors. One of the ways to separate the factors is the distributed relaxation approach proposed by A. Brandt. Earlier demonstrations of TME solvers with distributed relaxation has been performed for the incompressible free-stream inviscid and viscous flows without boundary layers.

In this talk, I am going to outline a general framework for achieving TME in solution of the high-Reynolds-number Navier-Stokes equations. TME distributed-relaxation solvers will be demonstrated for the viscous incompressible and subsonic compressible flows with boundary layers and for the inviscid compressible flow with shock.