An efficient Newton-Krylov method for the Euler and Navier-Stokes equations

An efficient inexact-Newton-Krylov algorithm is presented for the computation of steady compressible aerodynamic flows on structured grids. The spatial discretization consists of a second-order centered-difference operator with the second and fourth-difference dissipation model of Jameson et al. The Baldwin-Lomax algebraic model is used for turbulent flows. The thin-layer Navier-Stokes equations are linearized using Newton's method. Preconditioned restarted GMRES in matrix-free form is used to solve the linear system arising at each Newton iteration. The preconditioner is formed using an incomplete factorization of an approximate-Jacobian matrix after applying a reordering technique.; An optimization study is presented to obtain an efficient parameter-free solver for a wide range of flows. An inexact-Newton strategy that avoids oversolving is established. Comparison between different preconditioners of the incomplete-lower-upper factorization family is presented. The best performance/memory ratio was obtained for the Block-Fill ILU(2) preconditioner. A parametric optimization of the approximate-Jacobian used to produce well-conditioned LU factors is also shown. Different reordering techniques are considered; results show that the Reverse Cuthill-McKee is the most efficient technique.; The algorithm has been successfully applied to a wide range of test cases which include inviscid, laminar, and turbulent aerodynamic flows. In all cases except one, convergence of the residual to 10−12 is achieved with a CPU cost equivalent to fewer than 1000 function evaluations. The sole exception is a low Mach number case where some form of local preconditioning is needed. Several other efficient implicit solvers have been applied to the same test cases, and the matrix-free inexact-Newton-GMRES algorithm is seen to be the fastest and most robust of the methods studied.