Gibbs Phenomenon
By Anne Gelb

     A few decades ago spectral methods emerged as an important way to approach a variety of numerical problems.  Particularly appealing are their highly accurate approximations to continuous functions.  While spectral methods yield exponential convergence for continuous functions, the results for discontinuous functions are poor.  Spurious oscillations are exhibited at the jump discontinuities and  the overall convergence rate is reduced to O(1/N).
     This is known as the Gibbs phenomenon.
A large part of my research has been devoted to restoring the exponential convergence qualities to spectral methods for discontinuous functions.  Without this restoration, spectral methods would be limited to solving simple smooth problems, and could not be applied in many physical applications.
     A method for removing the Gibbs phenomenon for one dimensional discontinuous functions was  successfully
developed by Gottlieb and Shu.  They proved that the knowledge of the first N spectral expansion coefficients of a         piecewise analytic function  f(x) is enough  to recover an exponentially convergent approximation to the point values of  f(x) in any subinterval in which the function is analytic.  My research includes the recovery of piecewise analytic functions on spheres, as well as in higher dimensions.
     One recent highlight for me was collaborating with Dr. Antonio Navarra.  We applied the elimination of the Gibbs phenomenon technique to noisy spherical data in spectral climate models.  We have successfully recovered mountain data contaminated by the Gibbs phenomenon, and our current efforts involve creating a robust numerical method that will effectively reduce the Gibbs phenomenon in spectral climate models.  No other numerical approach has successfully eliminated the Gibbs phenomenon in spectral climate models.
     Another aspect of this research involves detecting edges of discontinuous functions from spectral data, an area in which I have been collaborating with Professor Eitan Tadmor.  We developed a general method of detecting the edges of a piecewise smooth function from its (pseudo-)spectral coefficients.  Later we enhanced the method to “pinpoint” the edges of the piecewise smooth function, and refer to it as the enhanced edge detection method. This research, together with the elimination of the Gibbs phenomenon, makes spectral methods a strong competitor in the arena of numerical methods for solving nonlinear partial differential equations.  In particular, we have utilized our enhanced edge detection method in conjunction with the spectral viscosity method in constructing a new procedure for the spectral approximations of nonlinear conservation laws.  This new numerical method, the enhanced spectral viscosity method, displays highly accurate results for both scalar and one-dimensional systems of nonlinear conservation laws, as well as remarkably high resolution at the shock locations.  Currently I am interested in applying the results of the enhanced edge detection method to numerically evaluate Stefan-like problems.

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