EFFICIENT ITERATIVE METHODS APPLIED TO THE SOLUTION OF TRANSONIC-FLOWS

We investigate the use of an inexact Newton's method to solve the pote ntial equations in the transonic regime. As a test case, we solve the two-dimensional steady transonic small disturbance equation. Approxima te factorization/ADI techniques have traditionally been employed for i mplicit solutions of this nonlinear equation. Instead, we apply Newton 's method using an exact analytical determination of the Jacobian with preconditioned conjugate gradient-like iterative solvers for solution of the linear systems in each Newton iteration. Two iterative solvers are tested; a block s-step version of the classical Orthomin(k) algor ithm called orthogonal s-step Orthomin (OSOmin) and the well-known GMR ES method. The preconditioner is a vectorizable and parallelizable ver sion of incomplete LU (ILU) factorization. Efficiency of the Newton-it erative method on vector and parallel computer architectures is the ma in issue addressed. In vectorized tests on a single processor of the G ray C-90, the performance of Newton-OSOmin is superior to Newton-GMRES and a more traditional monotone AF/ADI method (MAF) for a variety of transonic Mach numbers and mesh sizes. Newton-GMRES is superior to MAF for some cases. The parallel performance of the Newton method is also found to be very good on multiple processors of the Gray C-90 and on the massively parallel thinking machine CM-5, where very fast executio n rates (up to 9 Gflops) are found for large problems. (C) 1996 Academ ic Press, Inc.