Application of the Newton–Krylov Method to Geophysical Flows

Abstract

An implicit nonlinear algorithm, the Newton?Krylov method, for the efficient and accurate simulation of the Navier?Stokes equations, is presented. This method is a combination of a nonlinear outer Newton-based iteration and a linear inner conjugate residual (Krylov) iteration but does not require the explicit formation of the Jacobian matrix. This is referred to here as Jacobian-free Newton?Krylov (JFNK). The mechanics of the method are quite simple and the method has been previously used to solve a variety of complex coupled nonlinear equations. Like most Krylov-based schemes, the key to the efficiency of the method is preconditioning. Details concerning how preconditioning is implemented into this algorithm will be illustrated in a simple one-dimensional shallow-water framework. Another important aspect of this work is examining the accuracy and efficiency of the Newton?Krylov method against an explicit method of averaging (MOA) approach. This will aid in the determination of regimes for which implicit techniques are accurate and/or efficient. Finally, results from the Navier?Stokes fluid solver used in this paper are presented. This solver employs both the JFNK and MOA approaches, and it is reasonably efficient and accurate over a large parameter space. As an illustration of the robustness of this fluid solver two different flow regimes will be shown: two-dimensional hydrostatic mountain-wave flow employing a broad mountain and two-dimensional nonhydrostatic flow employing a steep mountain and high spatial resolution.