Improvements to a Newton-Krylov Solver for Aerodynamic Flows

In this paper, we present and justify the strategies and parameters used in PROBE, a Newton-Krylov solver for steady aerodynamic flows. The Krylov solver GMRES is used in matrix-free form. An approximate Jacobian matrix with some modifications to increase the magnitude of the diagonal entries is used to form a preconditioner based on an incomplete lower-upper factorization with two levels of fill. The resulting solver is very efficient, generally reducing the residual twelve orders of magnitude with a CPU expense equivalent to between 500 and 1000 function evaluations. For all cases studied, PROBE converged significantly faster than an approximately-factored multigrid algorithm used for comparison. We propose a convergence rate based on the reduction in the residual obtained in the CPU time required for one function evaluation. With this definition, PROBE achieves a convergence rate per function evaluation between 0.945 and 0.972 for the cases studied, which include inviscid, laminar, and turbulent flows. This is substantially faster than many current algorithms applied to similar flows