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
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.