Jacobian-Free Newton-Krylov Methods: Issues and Solutions

A Newton-Krylov method computes the solution of a system of nonlinear algebraic equations, often arising from a discretization of a system of partial differential equations, using an inexact-Newton method combined with a Krylov subspace method for linear systems. Such methods are very efficient in a variety of applications, as discussed in the review paper by Knoll and Keyes [1]. However, they have not gained widespread acceptance in computational fluid dynamics (CFD), i.e. in the numerical solution of the Reynolds-averaged compressible or incompressible Navier-Stokes equations. Considerable interest was generated during the 1990’s [2, 3, 4, 5, 6, 7], primarily using the Krylov method GMRES [8], but current interest is more limited [9, 10, 11, 12, 13, 14, 15].