Boundary Effects in Regional Spectral Models

Abstract

Abstract The choice of an appropriate spectral spatial discretization is governed by considerations of accuracy and efficiency. The purpose of this article is to discuss the boundary effects on regional spectral methods. In particular, we consider the Chebyshev τ and sinusoidal- or polynomial-subtracted sine–cosine expansion methods. The Fourier and Chebyshev series are used because of the orthogonality and completeness properties and the existence of fast transforms. The rate of convergence of expansions based on Chebyshev series depends only on the smoothness of the function being expanded, and not on its behavior at the boundaries. The sinusoidal- or polynomial-subtracted sine–cosine expansion Tatsumi-type methods do not, in general, possess the exponential- convergence property. This is due to the fact that the higher derivatives of the polynomial- or sinusoidal- subtracted function are not periodic in a model with time-dependent boundary conditions. The discontinuity in derivatives causes the slow co...