A fast, matrix-free implicit method for compressible flows on unstructured grids

Abstract

A fast, matrix-free implicit method has been developed to solve the three dimensional compressible flow problems 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 developed method has been used to compute compressible flows around 3D complex aerodynamic configurations for a wide range of flow conditions, from subsonic to supersonic. The numerical results 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 competitive to its explicit counterpart.