Boundary Effects in Regional Spectral Models

Hung-Chi Kuo Department of Atmospheric Science, National Taiwan University, Taipei, Taiwan, Republic of China

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R. T. Williams Department of Meteorology, Naval Postgraduate School, Monterey, California

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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 convergence of the expanded series (Gibbs phenomenon). When a large disturbance is near the boundary so that derivative discontinuities in the expanded function are large, the Tatsumi-type method causes not only erroneous numerical values in the outgoing boundary, but also spurious oscillations in the incoming boundary region. When the wave is away from the boundary, low resolution in the Tatsumi-type method converges exponentially, just as with the Chebyshev τ method. High-resolution solutions of the Tatsumi-type method do not, however, yield high accuracy due to the discontinuity in higher derivatives of the expanded function.

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 convergence of the expanded series (Gibbs phenomenon). When a large disturbance is near the boundary so that derivative discontinuities in the expanded function are large, the Tatsumi-type method causes not only erroneous numerical values in the outgoing boundary, but also spurious oscillations in the incoming boundary region. When the wave is away from the boundary, low resolution in the Tatsumi-type method converges exponentially, just as with the Chebyshev τ method. High-resolution solutions of the Tatsumi-type method do not, however, yield high accuracy due to the discontinuity in higher derivatives of the expanded function.

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