Application of the Newton-Krylov method to geophysical flows

An implicit nonlinear algorithm, the Newton-Krylov method, for the efficien t and accurate simulation of the Navier-Stokes equations, is presented. Thi s method is a combination of a nonlinear outer Newton-based iteration and a linear inner conjugate residual (Krylov) iteration but does not require th e explicit formation of the Jacobian matrix. This is referred to here as Ja cobian-free Newton-Krylov (JFNK). The mechanics of the method are quite sim ple and the method has been previously used to solve a variety of complex c oupled nonlinear equations. Like most Krylov-based schemes, the key to the efficiency of the method is preconditioning. Details concerning how precond itioning 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 a gainst 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 th is paper are presented. This solver employs both the JFNK and MOA approache s, and it is reasonably efficient and accurate over a large parameter space . As an illustration of the robustness of this fluid solver two different f low regimes will be shown: two-dimensional hydrostatic mountain-wave flow e mploying a broad mountain and two-dimensional nonhydrostatic flow employing a steep mountain and high spatial resolution.