Parallel Implicit Methods for Aerodynamics

Domain decomposition (Krylov-Schwarz) iterative methods are natural for the parallel implicit solution of multidimensional systems of boundary value problems that arise, for instance, in aerodynamics. They provide good data locality so that even a high-latency workstation network can be employed as a parallel machine. Matrix-free (Newton-Krylov) methods are natural when it is unreasonable to compute or store a true Jacobian. We call their combinationNewton-Krylov-Schwarz and report experimental progress on two algorithmic aspects: the use of a coarse grid in additive Schwarz preconditioning and the use of mixed discretization schemes in the (implicitly de ned) Jacobian and its preconditioner. Two model problems in two-dimensional compressible ow are considered: the full potential equation, and the Euler equations. 1. Krylov-Schwarz Algorithms Fully implicit linear solvers in aerodynamics allow more rapid asymptotic approach to steady states than time-explicit, approximate factorization, or relaxation solvers that hold the outer nonlinear iteration to small time steps. Nevertheless, the all-to-all data dependencies between the unknown elds in a fully implicit method have led to a resurgence of interest in less rapidly convergent methods in highlatency parallel environments. Resisting, we brie y overview two related e orts that lie along the route to parallel implicit computational aerodynamics. Though the governing equation formulations are mathematically very di erent { elliptic subsonic full potential and hyperbolic transonic Euler { a common implicit software core allows them to be treated together. Our ultimate interest is in applying Schwarzian domain decomposition techniques to industrial computations still being carried out in these (physically) primitive potential and Euler formulations, and in extending them to Navier-Stokes, for which implicit solvers are even more important. For a variety of reasons, industrial CFD groups are inclining towards the distributed network computing environment characterized by coarse to medium 1991 Mathematics Subject Classi cation. Primary 65N55, 65N22, 76G25. The work was supported in part by theNational ScienceFoundationand the KentuckyEPSCoR Program under grant STI-9108764 (XCC); by the O ce of Scienti c Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38 (WDG); by the National Science Foundation under contract number ECS-8957475, the State of Connecticut and the United Technologies Research Center (DEK); and by the National Aeronautics and Space Administration under NASA contract NAS1-19480 while three of the authors (XCC,DEK,MDT) were in residence at the Institute for Computer Applications in Science and Engineering. This paper is in nal form and no version of it will be submitted elsewhere.