JOURNAL OF COMPUTATIONAL PHYSICS 146, 664–690 (1998) ARTICLE NO. CP986076 A Fast, Matrix-free Implicit Method for Compressible Flows on Unstructured Grids Hong Luo, ∗, 1 Joseph D. Baum, ∗ and Rainald L ¨ ohner† ∗ Applied Physics Operations, Science Applications International Corporation, McLean, Virginia 22102; †Institute for Computational Sciences and Informatics, George Mason University, Fairfax, Virginia 22030 E-mail: luo@mclapo.saic.com Received October 21, 1997; revised July 31, 1998 A fast, matrix-free implicit method has been developed to solve the three-dimen- sional compressible Euler and Navier–Stokes equations on unstructured meshes. An approximate system of linear equations arising from the Newton linearization is solved by the GMRES (generalized minimum residual) algorithm with a LU- SGS (lower–upper symmetric Gauss–Seidel) preconditioner. A remarkable feature of the present GMRES+LU-SGS method is that the storage of the Jacobian matrix can be completely eliminated by approximating the Jacobian with numerical fluxes, resulting in a matrix-free implicit method. The method developed has been used to compute the compressible flows around 3D complex aerodynamic configurations for a wide range of flow conditions, from subsonic to supersonic. The numerical results obtained indicate that the use of the GMRES+LU-SGS method leads to a significant increase in performance over the best current implicit methods, GMRES+ILU and LU-SGS, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from eight to more than one order of magnitude for all test cases in comparison with the explicit method is demonstrated. c  1998 Academic Press 1. INTRODUCTION The use of unstructured meshes for computational fluid dynamics problems has become widespread due to their ability to discretize arbitrarily complex geometries and due to the ease of adaptation in enhancing the solution accuracy and efficiency through the use of adaptive refinement techniques. In recent years, significant progress has been made in 1 Corresponding author. 664 0021-9991/98 $25.00 Copyright c  1998 by Academic Press All rights of reproduction in any form reserved.