Efficient Iterative Methods Applied to the Solution of Transonic Flows

When powerful machines such as the Cray-2 became available in the late 1980s, a number of researchers investiWe investigate the use of an inexact Newton’s method to solve the potential equations in the transonic regime. As a test case, gated use of exact Newton’s method for solving steady we solve the two-dimensional steady transonic small disturbance state CFD problems. Direct sparse matrix solvers (e.g., equation. Approximate factorization/ADI techniques have traditionGaussian elimination) were used for exact solution of the ally been employed for implicit solutions of this nonlinear equation. linear systems in each Newton iteration. Results using this Instead, we apply Newton’s method using an exact analytical deterapproach were obtained for transonic flows using the pomination of the Jacobian with preconditioned conjugate gradientlike iterative solvers for solution of the linear systems in each Newtential equations [1, 2], Euler equations [3, 4], and Navier ton iteration. Two iterative solvers are tested; a block s-step version Stokes equations [5, 6]. While the exact Newton’s method of the classical Orthomin(k) algorithm called orthogonal s-step Orwas found to be robust and have quadratic convergence thomin (OSOmin) and the well-known GMRES method. The precon(with a good initial guess), the CPU time was not competiditioner is a vectorizable and parallelizable version of incomplete tive with existing iterative implicit methods. This was due LU (ILU) factorization. Efficiency of the Newton–Iterative method mainly to the time required for exact solution of the large on vector and parallel computer architectures is the main issue addressed. In vectorized tests on a single processor of the Cray linear systems. C-90, the performance of Newton–OSOmin is superior to Newton– An approach that has shown promising results recently GMRES and a more traditional monotone AF/ADI method (MAF) is the inexact Newton’s method. In this approach, an for a variety of transonic Mach numbers and mesh sizes. Newton– inexact solution using an iterative solver is performed GMRES is superior to MAF for some cases. The parallel performance in each Newton iteration. There are two advantages of of the Newton method is also found to be very good on multiple processors of the Cray C-90 and on the massively parallel thinking using a conjugate gradient-like iterative solver over a machine CM-5, where very fast execution rates (up to 9 Gflops) are direct solver for a problem with n unknowns. First, found for large problems. Q 1996 Academic Press, Inc. direct methods tend to be memory intensive and costly, requiring O(n2) operations to generate an exact solution for banded systems with bandwidth Ïn (arising in two