Английская Википедия:Double Fourier sphere method
Шаблон:Technical In mathematics, the double Fourier sphere (DFS) method is a simple technique that transforms a function defined on the surface of the sphere to a function defined on a rectangular domain while preserving periodicity in both the longitude and latitude directions.
Introduction
First, a function <math>f(x, y, z)</math> on the sphere is written as <math>f(\lambda,\theta)</math> using spherical coordinates, i.e.,
- <math>f(\lambda,\theta) = f(\cos\lambda\sin\theta,\sin\lambda\sin\theta, \cos\theta), (\lambda,\theta)\in[-\pi,\pi]\times[0,\pi].</math>
The function <math>f(\lambda, \theta)</math> is <math>2\pi</math>-periodic in <math>\lambda</math>, but not periodic in <math>\theta</math>. The periodicity in the latitude direction has been lost. To recover it, the function is "doubled up” and a related function on <math>[-\pi,\pi]\times[-\pi,\pi]</math> is defined as
- <math>\tilde{f}(\lambda,\theta) = \begin{cases}
g(\lambda + \pi, \theta), & (\lambda, \theta) \in [-\pi, 0] \times [0, \pi],\\
h(\lambda, \theta), &(\lambda, \theta) \in [0, \pi] \times [0, \pi],\\
g(\lambda, -\theta), &(\lambda, \theta) \in [0, \pi] \times [-\pi, 0],\\
h(\lambda + \pi, -\theta), & (\lambda, \theta) \in [-\pi, 0] \times [-\pi, 0],\\
\end{cases}</math>
where <math>g(\lambda, \theta) = f(\lambda- \pi, \theta)</math> and <math>h(\lambda, \theta) = f(\lambda, \theta)</math> for <math>(\lambda, \theta) \in[0, \pi] \times [0, \pi]</math>. The new function <math>\tilde{f}</math> is <math>2\pi</math>-periodic in <math>\lambda</math> and <math>\theta</math>, and is constant along the lines <math>\theta = 0</math> and <math>\theta = \pm\pi</math>, corresponding to the poles.
The function <math>\tilde{f}</math> can be expanded into a double Fourier series
- <math>\tilde{f} \approx \sum_{j=-n}^n \sum_{k=-n}^n a_{jk} e^{ij\theta}e^{ik\lambda}</math>
History
The DFS method was proposed by Merilees[1] and developed further by Steven Orszag.[2] The DFS method has been the subject of relatively few investigations since (a notable exception is Fornberg's work),[3] perhaps due to the dominance of spherical harmonics expansions. Over the last fifteen years it has begun to be used for the computation of gravitational fields near black holes[4] and to novel space-time spectral analysis.[5]
References
- ↑ P. E. Merilees, The pseudospectral approximation applied to the shallow water equations on a sphere, Atmosphere, 11 (1973), pp. 13–20
- ↑ S. A. Orszag, Fourier series on spheres, Mon. Wea. Rev., 102 (1974), pp. 56–75.
- ↑ B. Fornberg, A pseudospectral approach for polar and spherical geometries, SIAM J. Sci. Comp, 16 (1995), pp. 1071–1081
- ↑ R. Bartnik and A. Norton, Numerical methods for the Einstein equations in null quasispherical coordinates, SIAM J. Sci. Comp, 22 (2000), pp. 917–950
- ↑ C. Sun, J. Li, F.-F. Jin, and F. Xie, Contrasting meridional structures of stratospheric and tropospheric planetary wave variability in the northern hemisphere, Tellus A, 66 (2014)
