Scale-Dependent Accuracy in Regional Spectral Methods

Abstract

The accuracy of a numerical model is often scale dependent. Large spatial-scale phenomena are expected to be numerically solved with better accuracy, regardless of whether the discretization is spectral, finite difference, or finite element. The purpose of this article is to discuss the scale-dependent accuracy associated with the regional spectral model variables expanded by sine?cosine series. In particular, the scale-dependent accuracy in the Chebyshev-tau, finite difference, and sinusoidal- or polynomial-subtracted sine?cosine expansion methods is considered. With the simplest examples, it is demonstrated that regional spectral models may possess an unusual scale-dependent accuracy. Namely, the numerical accuracy associated with large-spatial-scale phenomena may be worse than the numerical accuracy associated with small-spatial-scale phenomena. This unusual scale-dependent accuracy stems from the higher derivatives of basic-state subtraction functions, which are not periodic. The discontinuity is felt mostly by phenomena with large spatial scale. The derivative discontinuity not only causes the slow convergence of the expanded Fourier series (Gibbs phenomenon) but also results in the unusual scale-dependent numerical accuracy. The unusual scale-dependent accuracy allows large-spatial-scale phenomena in the model perturbation fields to be solved less accurately.