Achieving Textbook Multigrid Efficiency (TME) in CFD

Achi Brandt
Weizmann Institute of Science, Israel

Abstract

``Textbook multigrid efficiency'' means solving a discrete PDE problem in a computational work which is only a small (less than 10) multiple of the operation count in the discretized system of equations itself. As a road map for attaining this optimal performance for general CFD problems, we list in a table every foreseen kind of computational difficulty for achieving that goal, together with the possible ways for resolving that difficulty, their current state of development, and references.

Included in the table are staggered and nonstaggered, conservative and nonconservative discretizations of viscous and inviscid, incompressible and compressible flows at various Mach numbers, as well as a simple (algebraic) turbulence model and comments on chemically reacting flows. The listing of associated computational barriers involves: non-alignment of streamlines or sonic characteristics with the grids; recirculating flows; stagnation points; discretization and relaxation on and near shocks and boundaries; far-field artificial boundary conditions; small-scale singularities; large grid aspect ratios; boundary layer resolution; and grid adaption.