THE EFFECT OF ORDERING ON PRECONDITIONED GMRES ALGORITHM, FOR SOLVING THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

We investigate the effect of the ordering of the blocks of unknowns on the rate of convergence of a preconditioned non-linear GMRES algorithm, for solving the Navier-Stokes equations for compressible flows, using finite element methods on unstructured grids. The GMRES algorithm is preconditioned by an incomplete LDU block factorization of the Jacobian matrix associated with the non-linear problem to solve. We examine a wide range of ordering methods including minimum degree, (reverse) Cuthill-McKee and snake, and consider preconditionings without fill-in. We show empirically that there can be a significant difference in the number of iterations required by the preconditioned non-linear GMRES method and suggest a criterion for choosing a good ordering algorithm, according to the problem to solve. We also consider the effect of orderings when an incomplete factorization which allows some fill-in is performed. We consider the effect of automatically controlling the sparsity of the incomplete factorization through the level of fill-in. Finally, following the principal ideas of non-linear GMRES algorithm, we suggest other inexact Newton methods.