On the Spectral Convergence of the Supercompact Finite-Difference Schemes for the f-Plane Shallow-Water Equations

For the f-plane shallow-water equations, the convergence properties of the supercompact finite-difference method (SCFDM) are examined during the evolution of complex, nonlinear flows spawned by an unstable jet. The second-, fourth-, sixth-, and eighth-order SCFDMs are compared with a standard pseudospectral (PS) method. To control the buildup of small-scale activity and thus the potential for numerical instability, the vorticity field is damped explicitly by the application of a triharmonic hyperdiffusion operator acting on the vorticity field. The global distribution of mass between isolevels of potential vorticity, called mass error, and the representation of the balance and imbalance are used to assess numerical accuracy. In each of the quantitative measures, a clear convergence of the SCFDM to the PS method is observed. There is no saturation in accuracy up to the eighth order examined. Taking the PS solution as the reference, for the fundamental quantity of potential vorticity the rate of convergence to PS turns out to be algebraic and near-quadratic.