Comparison of Grids and Difference Approximations for Numerical Weather Prediction Over a Sphere

Abstract

The accuracy of finite-diffrerence approximations to the shallow water equations on a sphere is examined for flow cases having an analytic solution. Approximations over grids with the longitudinal grid increment (??) increasing near the poles such that the distance between grid points is nearly constant have large errors near the poles. These large polar errors are caused by the large longitudinal grid increment used in the approximations and are reduced by using a grid with ?? constant. The normally severe limit on the time step caused by the small distance between grid points near the pole can be relaxed by removing the short-wave-length, fast-moving waves by Fourier analysis. With our test case, which contains only large scales, this filtering method produced a solution which is almost identical to that obtained over the uniform grid using a small time step. In comparing second- and fourth-order schemes applied to the above test case, we find that the fourth-order schemes offer more improvement per computer time than second-order Themes with mesh reduction.