A Jacobian-free Newton–Krylov algorithm for compressible turbulent fluid flows

Despite becoming increasingly popular in many branches of computational physics, Jacobian-free Newton–Krylov (JFNK) methods have not become the approach of choice in the solution of the compressible Navier–Stokes equations for turbulent aerodynamic flows. To a degree, this is related to some subtle aspects of JFNK methods that are not well understood, and, if poorly handled, can lead to inefficient and unreliable performance. These are described here, along with strategies for addressing them, leading to an efficient JFNK algorithm for turbulent aerodynamic flows applicable to multi-block structured grids and a one-equation turbulence model. Development of globalization strategies for field-equation turbulence models represents one of the key contributions of the paper. Numerous examples of subsonic and transonic flows over single and multi-element airfoils are presented in order to demonstrate the efficiency and reliability of the algorithm. In addition, a number of guidelines are presented to aid in diagnosing problems with JFNK algorithms.