The Cubic-Spline Transform Method: Basic Definitions and Tests in a 1D Single Domain

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 1D domain. The actual procedures for two-way interactive nesting will be presented in a subsequent paper.