Analysis of the Turkel–Zwas Scheme for the Two-Dimensional Shallow Water Equations in Spherical Coordinates

In this paper we extend the linear transfer function analysis to the two-dimensional shallow water equations in A linear analysis of the shallow water equations in spherical coordinates for the Turkel–Zwas (T–Z) explicit large time-step scheme spherical coordinates for the Turkel–Zwas discretization. is presented. This paper complements the results of Schoenstadt, Actually, we show how to obtain the modal expansion for Neta and Navon, and others in 1-D, and of Neta and DeVito in 2-D, the shallow water equations in spherical coordinates and but applied to the spherical coordinate case of the T–Z scheme. for the Turkel–Zwas discretization of these equations. At This coordinate system is more realistic in meteorology and more this point, we should comment on the choice of the method. complicated to analyze, since the coefficients are no longer constant. The analysis suggests that the T–Z scheme must be staggered in Computationally efficient and accurate schemes for the a certain way in order to get eigenvalues and eigenfunctions apnumerical solution of the shallow water equations are of proaching those of the continuous case. The importance of such crucial importance in atmospheric and oceanographic an analysis is the fact that it is also valid for nonconstant coefficients models. Two different approaches have been taken, both and thereby applicable to any numerical scheme. Numerical experiof them dealing with the different time scales of advective ments comparing the original (unstaggered) and staggered versions of the T–Z scheme are presented. These experiments corroborate (Rossby) waves and gravity inertia waves separately. The the analysis by showing the improvements in accuracy gained by first of these was the split-explicit and semi-implicit staggering the Turkel–Zwas scheme. Q 1997 Academic Press schemes. The Turkel–Zwas scheme takes a different view by proposing a space (rather than time) splitting approach. This is based on the fact that the fast gravity inertia waves