Enslaved finite difference approximations for quasigeostrophic shallow flows

We describe a procedure to improve both the accuracy and computational efficiency of finite difference schemes used to simulate the nonlinear PDEs that govern barotropic two-dimensional geophysical fluid dynamics. Our underlying strategy is to reduce the truncation error of a given difference scheme in such a way that the time step restrictions are not changed. To accomplish this reduction, we use information from the governing equations in the case of vanishing time derivatives. Our method is based on a change of variables from fine to coarse grids, which allows us to order the various terms that appear and justify further approximations. These approximations lead to algebraic closures for the new higher-order variables, and finally to a new, enslaved scheme. We demonstrate the utility of the procedure for the shallow water equations in both periodic and closed basins. In the latter case we present results that demonstrate the ability of the enslaved scheme to capture dynamics on scales smaller than those resolved by the original scheme.