(untitled) (Communications in Computational Physics, 2019)
In order to overcome the strong limitation of the computational cost for classic explicit schemes in quasi-geostrophic limit typically involved in oceanographic flows, in this study, a highly efficient numerical method for rotating oceanographic flows modeled by Saint-Venant system with Coriolis forces is developed. To efficiently deal with the rotating flows in the low Rossby and Froude number regime, the core idea is splitting fast varying flux terms into stiff and non-stiff parts, and implicitly approximating the fast dynamic waves using central difference method with an iteration algorithm and explicitly approximating slow dynamic waves using a finite-volume hyperbolic solver with minmod limiter. The proposed approach has a second order convergence rate in the quasi-geostrophic limit. The temporal evolution is approximated using a high-order implicit-explicit Runge-Kutta method. The proposed semi-implicit scheme is proved to be uniformly asymptotically consistent in the quasi-geostrophic limit when Froude and Rossby numbers→0. The proposed numerical methods are finally verified by numerical experiments of rotating shallow flows. The tests show that the proposed numerical scheme is stable and accurate with any grid size with low Rossby and Froude numbers, which leads to significant reduction of the computational cost comparing with classic explicit schemes.