Padé-Gegenbauer suppression of Runge phenomenon in the diagonal limit of Gegenbauer approximations

Trouble with Gegenbauer reconstruction for defeating Gibbs phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations

To defeat Gibbs phenomenon in Fourier and Chebyshev series, Gottlieb et al. [D. Gottlieb, C.-W. Shu, A. Solomonoff, H. Vandeven, On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math. 43 (1992) 81–98] developed a ‘‘Gegenbauer reconstruction’’. The partial sums of the Fourier or other spectral series are ...

Robust reprojection methods for the resolution of the Gibbs phenomenon ✩

The classical Gibbs phenomenon exhibited by global Fourier projections and interpolants can be resolved in smooth regions by reprojecting in a truncated Gegenbauer series, achieving high resolution recovery of the function up to the point of discontinuity. Unfortunately, due to the poor conditioning of the Gegenbauer polynomials, the method suffers both from numerical round-off error and the Ru...

Comparison between Rational Gegenbauer and Modified Generalized Laguerre Functions Collocation Methods for Solving the Case of Heat Transfer Equations Arising in Porous Medium

Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems

Radial basis function(RBF) methods have been actively developed in the last decades. The advantages of RBF methods are that these methods are mesh-free and yield high order accuracy if the function is smooth enough. The RBF approximation for discontinuous problems, however, deteriorates its high order accuracy due to the Gibbs phenomenon. With the Gibbs phenomenon in the RBF approximation, the ...

Comparison between Rational Gegenbauer and Modified Generalized Laguerre Functions Collocation Methods for Solving the Case of Heat Transfer Equations Arising in Porous Medium

—————————————————————————————————— – Abstract In this paper, we provide the collocation method for natural convection heat transfer equations embedded in porous medium which are of great importance in the design of canisters for nuclear waste disposal. This problem is a non-linear, three-point boundary value problem on semi-infinite interval. We use two orthogonal functions namely rational Gege...