Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
SIAM Journal on Numerical Analysis
Analysis and Application of Fourier--Gegenbauer Method to Stiff Differential Equations
SIAM Journal on Numerical Analysis
The resolution of the Gibbs phenomenon for spherical harmonics
Mathematics of Computation
Spectrally Accurate Solution of Nonperiodic Differential Equations by the Fourier--Gegenbauer Method
SIAM Journal on Numerical Analysis
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Spectral Simulation of Supersonic Reactive Flows
SIAM Journal on Numerical Analysis
Spectral methods for hyperbolic problems
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions
Journal of Scientific Computing
Applied Numerical Mathematics
Chebyshev super spectral viscosity method for a fluidized bed model
Journal of Computational Physics
Determination of Optimal Parameters for the Chebyshev-Gegenbauer Reconstruction Method
SIAM Journal on Scientific Computing
Towards the resolution of the Gibbs phenomena
Journal of Computational and Applied Mathematics
Parameter Optimization and Reduction of Round Off Error for the Gegenbauer Reconstruction Method
Journal of Scientific Computing
Journal of Computational and Applied Mathematics
Determining Analyticity for Parameter Optimization of the Gegenbauer Reconstruction Method
SIAM Journal on Scientific Computing
Recovering High-Order Accuracy in WENO Computations of Steady-State Hyperbolic Systems
Journal of Scientific Computing
IEEE Transactions on Image Processing
Journal of Computational Physics
Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data
Journal of Scientific Computing
Construction of Lanczos type filters for the Fourier series approximation
Applied Numerical Mathematics
An SVD analysis of equispaced polynomial interpolation
Applied Numerical Mathematics
Algorithm 899: The Matlab postprocessing toolkit
ACM Transactions on Mathematical Software (TOMS)
Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems
Journal of Scientific Computing
Discontinuous functions represented by exact, closed, continuous parametric equations
AMERICAN-MATH'10 Proceedings of the 2010 American conference on Applied mathematics
Gibbs phenomenon removal by adding Heaviside functions
Advances in Computational Mathematics
Hi-index | 31.46 |
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 reexpanded as a series of Gegenbauer polynomials C"n^m(x), recovering spectral accuracy even in the presence of shock waves or other discontinuities. To achieve a rate of convergence which is exponential in N, however, Gegenbauer reconstruction, requires increasing the order m of the polynomials linearly with the truncation N of the series: m=@bN for some constant @b0. When the order m is fixed, it is well-known that the Gegenbauer series converges as N-~ everywhere on x@?[-1,1] if f(x), the function being expanded, is analytic on the interval. But what happens in the diagonal limit in which m, N tend to infinity simultaneously? We show that singularities of f(x) off the real axis can destroy convergence of this diagonal approximation process in the sense that the error diverges for subintervals of x@?[-1,1]. Gegenbauer reconstruction must therefore be constrained to use a sufficiently small ratio of order m to truncation N. This ''off-axis singularity'' constraint is likely to impair the effectiveness of the reconstruction in some applications.