Fast Newton-Krylov method for unstructured grids

Newton-Krylov Algorithm with a Loosely Coupled Turbulence Model for Aerodynamic Flows

“…Equations (1-3) constitute the algorithm to which many in this community refer as matrix-or Jacobian-free. The present authors [14] found that a permutation due to Cuthill and McKee [15] with modifications by Liu and Sherman [16] and Gibbs et al [17] (see [17] Sec. 4.1, Algorithm I) implemented by Balay et al [18] performs well when combined with the particular preconditioner.…”

“…Once the GMRES, or "inner," iterations labeled l terminate (see our previous article [14] for implementation details), we update the solution vector Q k1 Q k Q k . Equations (1-3) constitute the algorithm to which many in this community refer as matrix-or Jacobian-free.…”

Iterative Solvers for Discretized Stationary Euler Equations

“…Many studies on numbering techniques for ILU preconditioners appear in the literature, cf., e.g., [18,36] and references therein. For ILU methods, in many applications, a reverse Cuthill-McKee ordering algorithm [17] provides good results [6,29,34,35]. The PBGS preconditioner can be significantly improved by reordering techniques that should be such that one approximately follows the directions in which information is propagated.…”

“…In our numerical experiments we therefore usually switch at a certain threshold ν for the density residual between the different methods. A similar approach is followed in [6,29]. In the actual implementation we do not switch back from the higher order method to the lower-order method if the residual increases again during the computation.…”

“…Therein, the choice of the preconditioner is crucial for the convergence process. Our main focus is on the incomplete LU-factorization (ILU) and Gauss-Seidel (GS) preconditioners that are widely-used in solvers in the numerical simulation of compressible flows [1,6,18,31,35,40].…”