Canonical-Variables Multigrid Method for Steady-State Euler Equations

In this paper we describe a novel approach for the solution of inviscid ow problems for subsonic compressible ows. The approach is based on canonical forms of the equations, in which subsystems governed by hyperbolic operators are separated from those governed by elliptic ones. The discretizations used as well as the iterative techniques for the di erent subsystems, are inherently di erent. Hyperbolic parts, which describe, in general, propagation phenomena, are discretized using upwind schemes and are solved by marching techniques. Elliptic parts, which are directionally unbiased, are discretized using h-elliptic central discretizations, and are solved by pointwise relaxations together with coarse grid acceleration. The resulting discretization schemes introduce arti cial viscosity only for the hyperbolic parts of the system; thus a smaller total arti cial viscosity is used, while the multigrid solvers used are much more e cient. Solutions of the subsonic compressible Euler equations are achieved at the same e ciency as the full potential equation. This research was supported in part under the Incumbent of the Lilian and George Lyttle Career Development Chair and in part by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, Va 23681.