Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
A note on the accuracy of spectral method applied to nonlinear conservation laws
Journal of Scientific Computing
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Regularized radial basis functional networks: theory and applications
Regularized radial basis functional networks: theory and applications
Reconstruction and representation of 3D objects with radial basis functions
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
Radial Basis Functions
Parameter Optimization and Reduction of Round Off Error for the Gegenbauer Reconstruction Method
Journal of Scientific Computing
3D Scattered Data Approximation with Adaptive Compactly Supported Radial Basis Functions
SMI '04 Proceedings of the Shape Modeling International 2004
Polynomials and Potential Theory for Gaussian Radial Basis Function Interpolation
SIAM Journal on Numerical Analysis
Recovering High-Order Accuracy in WENO Computations of Steady-State Hyperbolic Systems
Journal of Scientific Computing
A note on the Gibbs phenomenon with multiquadric radial basis functions
Applied Numerical Mathematics
Computers & Mathematics with Applications
Applied Numerical Mathematics
Adaptive radial basis function methods for time dependent partial differential equations
Applied Numerical Mathematics
Iterative adaptive RBF methods for detection of edges in two-dimensional functions
Applied Numerical Mathematics
Spectral collocation and radial basis function methods for one-dimensional interface problems
Applied Numerical Mathematics
Optimal Gegenbauer quadrature over arbitrary integration nodes
Journal of Computational and Applied Mathematics
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Radial basis function (RBF) methods have been actively developed in the last decades. RBF methods are global methods which do not require the use of specialized points and that yield high order accuracy if the function is smooth enough. Like other global approximations, the accuracy of RBF approximations of discontinuous problems deteriorates due to the Gibbs phenomenon, even as more points are added. In this paper we show that it is possible to remove the Gibbs phenomenon from RBF approximations of discontinuous functions as well as from RBF solutions of some hyperbolic partial differential equations. Although the theory for the resolution of the Gibbs phenomenon by reprojection in Gegenbauer polynomials relies on the orthogonality of the basis functions, and the RBF basis is not orthogonal, we observe that the Gegenbauer polynomials recover high order convergence from the RBF approximations of discontinuous problems in a variety of numerical examples including the linear and nonlinear hyperbolic partial differential equations. Our numerical examples using multi-quadric RBFs suggest that the Gegenbauer polynomials are Gibbs complementary to the RBF multi-quadric basis.