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# Computing Tutorials on Fourier, Harmonic and Related Topics

June 2003

Four tutorials on Fourier, harmonic and related topics will be presented by Paul Swarztrauber on June 17, 19, 24, and 26, 2003 at 10:00am in the Walter Orr Roberts room in the Fleischman Building at NCAR.

 June 17: Computing on the Rectangle Much of the theory and analysis for computations on the sphere can be best understood in terms of comparable computations on the rectangle where Fourier theory and analysis apply. Topics Trigonometric representation Nonperiodic functions Aliasing Interpolation error Two-thirds rule Using EZFFT Staggered grids Real in terms of complex Multiprocessor FFTs Fractional FFT Accessing FFTPACK Spectral accuracy The discrete basis Trig interpolation Alias control Subroutine EZFFT FFT for any N Complex transform The FFT Symmetric FFTs FFTPACK June 19: Computing on the Sphere: Part I Here we discuss the basic tools that are used for the spectral representation of scalar functions (such as temperature, pressure, divergence) on the sphere. Topics Sphere vs rectangle Assoc. Legendre fns. Computing the ALFs ALFPACK Scalar harmonic analysis Aliases and Aliasing Selecting a finite basis Least squares representation Double Fourier series Integration formulas Gauss points and weights Generalized harmonic analysis Harmonic projectors June 24: Computing on the Sphere: Part II Vectors on the sphere are discontinuous at the poles and therefore scalar spectral analysis of vectors is quite different than the analysis of scalars on a rectangle. Topics Discontinuous vectors Unbounded derivatives Bounded differential expressions Vector Harmonics Vector Harmonic Analysis Computing Vorticity, divergence, and gradients Robert's variables U and V June 26: Computing on the Sphere: Part III Here we describe the spectral transform method for modeling geophysical fluids. Actually we discuss two popular methods plus the vector harmonic transform method and note that others exist. The methods are presented by application to the shallow water equations. Topics Vector harmonic method and attributes Shallow water equations with bounded terms Model results Ritchie's U, V model (ECMWF) Vorticity and divergence (NCAR model) SPHEREPACK

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