01643704

j

01643704 Gelb, Anne A hybrid approach to spectral reconstruction of piecewise smooth functions. J. Sci. Comput. 15, No.3, 293-322 (2000). 2000 Springer, Dordrecht EN Gibbs phenomenon spectral reconstruction piecewise smooth function Gegenbauer polynomial Gegenbauer reconstruction method edge detection numerical examples exponential fitting method Zbl 0781.42022

doi:10.1023/A:1011126400782

The Gibbs phenomenon is an essential drawback in spectral reconstruction of a piecewise smooth function $f$. In this paper, a new approach to the Gegenbauer reconstruction method [see {\it D. Gottlieb}, {\it C.-W. Shu}, {\it A. Solomonoff} and {\it H. Vandeven}, J. Comput. Appl. Math. 43, No. 1-2, 81-98 (1992; Zbl 0781.42022)] is presented. The new method is less computationally intensive. The author combines the exponential fitting method in smooth regions away from the discontinuities of $f$ with the Gegenbauer reconstruction method in regions close to the discontinuities of $f$. Further, a new way of computing the Gegenbauer coefficients from Jacobian polynomial expansion is proposed. Several numerical applications with univariate and bivariate functions are presented. Manfred Tasche (Rostock)