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ABSTRACT
The purpose of the paper is to describe the technical details of a numerical method that combines the cubic-- spline representation of spatial variables in a finite domain with the logistics of the spectral transform method for the time integration of nonlinear meteorological equations. The reason for developing the method lies in its application to two-way interacting nested models of the atmosphere. When compared with the gridpoint representation, the cubic-spline representation allows direct evaluation of derivatives in the model equations, and leads to a substantial reduction of shortwave dispersion of advecting and propagating waves. When compared with the Fourier spectral representation, the cubic B-splines as basis functions provide simple but exact means of implementing a variety of boundary conditions that are needed at the domain interfaces, as well as at natural boundaries. A sharp (sixth order) low-pass filter, which is built into the cubic-spline transform, effectively eliminates adverse nonlinear accumulation of small-scale errors near the resolution limit. These features, critically important to noise-free nesting, are defined and analyzed in this paper in the simpler context of a single ID domain. The actual procedures for two-way interactive nesting will be presented in a subsequent paper.
1. Introduction
The purpose of the present paper is to describe the basic definitions and fundamental properties of a numerical method that combines the spatial representation of dependent variables by cubic splines with the logistics of the spectral transform method (Orszag 1970; Machenhauer 1979) for the time integration of nonlinear meteorological equations. While the essence of the method was described earlier in DeMaria et al. (1992), this paper offers a more refined and detailed account, including an error analysis.
Acknowledging its conceptual indebtedness to the spectral transform method, DeMaria et al. called the method the Spectral Application of Finite-Element Representation (SAFER), in which cubic B-splines on regularly spaced nodes are the finite elements. Cubic-spline functions are of second-order continuity, and the primitive-form meteorological equations are, at most, of second order. Thus, evaluation of the equations in physical space parallels the corresponding inverse leg of the spectral transform method. On the other hand, the projection of a given spatial function on a spline function is not a true spectral transform, since the B-spline basis is neither analytic nor orthogonal....