Family of spectral filters for discontinuous problems
Journal of Scientific Computing
Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation
Journal of Approximation Theory
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Solving partial differential equations by collocation using radial basis functions
Applied Mathematics and Computation
An efficient numerical scheme for Burgers' equation
Applied Mathematics and Computation
Overlapping domain decomposition method by radial basis functions
Applied Numerical Mathematics
Radial Basis Functions
Towards the resolution of the Gibbs phenomena
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Inverse Polynomial Reconstruction of Two Dimensional Fourier Images
Journal of Scientific Computing
Applied Numerical Mathematics
Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems
Journal of Scientific Computing
Iterative adaptive RBF methods for detection of edges in two-dimensional functions
Applied Numerical Mathematics
Skew-Radial Basis Function Expansions for Empirical Modeling
SIAM Journal on Scientific Computing
Compact RBF meshless methods for photonic crystal modelling
Journal of Computational Physics
Spectral collocation and radial basis function methods for one-dimensional interface problems
Applied Numerical Mathematics
Journal of Scientific Computing
Gibbs phenomenon removal by adding Heaviside functions
Advances in Computational Mathematics
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Any global or high order approximation method suffers from the Gibbs phenomenon if the approximant has a jump discontinuity in the given domain. In this note, we present a numerical study of the Gibbs phenomenon with multiquadric radial basis functions for discontinuous functions. Unlike the Gibbs phenomenon found in other global methods such as the Fourier or polynomial methods, the Gibbs phenomenon with multiquadric radial basis functions is characterized by two parameters: the shape parameter and the magnitude of the jump discontinuity. The numerical study shows that the accuracy can be enhanced by adaptively applying piecewise linear basis functions in the vicinity of the discontinuity. By exploiting nonsmooth basis functions locally, the Gibbs oscillations are considerably attenuated and, consequently, the accuracy in the smooth region is also enhanced. We also discuss on the expansion coefficients defined in the physical domain as an indicator of nonsmoothness of the given function.