High-order scheme implementation using Newton-Krylov solution methods

Implementation of high-order discretization for the convective transport terms in the inexact Newton method for a benchmark fluid flow and heat transfer problem using various solution configurations at two Reynolds numbers has been investigated. These configurations include fully consistent discretization of the Jacobian, preconditioner and residual of the Newton method, low-order preconditioning using a matrix-free method to approximate the action of the Jacobian, and defect correction or low-order Jacobian and preconditioning. The residual in each case employs high-order discretization to preserve the high-order solution. Two preconditioners, point incomplete lower-upper factorization ILU(k) and block incomplete lower-upper factorization BILU(k) for k = 0,1,2 were applied. Also, one-way multigriding and capping the inner iterations was applied to determine the behavior of the solution performance. It was determined that, overall, the configuration using low-order preconditioning with ILU(1), BILU(1), or BILU(2), mesh sequencing, and inner linear solve iterations capped at the same value of the dimension n, used with the GMRES(n) iterative solver (i.e., no restarts), performed best for time, memory, and robustness considerations.