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Selected Presentations

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2006

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2003

Tutorials on Fourier, harmonic and related topics, presented by Paul Swarztrauber, June 2003:

  • Fourier Analysis and Related Topics. 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 Discussed: trigonometric representation, spectral accuracy, nonperiodic functions, the discrete basis, aliasing, trig interpolation, interpolation error, alias control, two-thirds rule, subroutine EZFFT, using EZFFT, FFT for any N, staggered grids, complex transform, real in terms of complex, the FFT, multiprocessor FFTs, symmetric FFTs, fractional FFT, FFTPACK, accessing FFTPACK

  • 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 Discussed: sphere vs rectangle, least squares representation, assoc. Legendre fns., double Fourier series, computing the ALFs, integration formulas, ALFPACK, Gauss points and weights, scalar harmonic analysis, generalized harmonic analysis, aliases and Aliasing, harmonic projectors, selecting a finite basis

  • 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 Discussed: discontinuous vectors, vector harmonic analysis, unbounded derivatives, computing vorticity, divergence and gradients, bounded differential expressions, Robert's variables U and V, vector harmonics

  • 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 Dicussed: vector harmonic method and attributes, Ritchie's U, V model (ECMWF), shallow water equations with bounded terms, vorticity and divergence (NCAR model), model results SPHEREPACK