A note on the Gibbs phenomenon with multiquadric radial basis functions

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.