Mitigating Gibbs Phenomenon: A Localized Padé-Chebyshev Approach and Its Conservation Law Applications

Abstract

Approximating non-smooth functions presents a significant challenge due to the emergence of unwanted oscillations near discontinuities, commonly known as Gibbs' phenomena. Traditional methods like finite Fourier or Chebyshev representations only achieve convergence on the order of $O(1)$. A promising avenue in addressing this issue lies in nonlinear and essentially non-oscillatory approximation techniques, such as rational or Padé approximation. A recent and notable endeavor to mitigate Gibbs' oscillations is through singular Padé-Chebyshev approximation. However, a drawback of this approach is the requirement to specify the discontinuity location within the algorithm, which is often unknown in practical applications. To tackle this obstacle, we propose a localized Pad\'e-Chebyshev approximation method. Fortunately, our efforts yield success; the proposed localized variant effectively captures jump locations in non-smooth functions while maintaining an essentially non-oscillatory character. Furthermore, we employ Padé-Chebyshev approximation within a finite volume framework to address scalar hyperbolic conservation laws. Remarkably, the resulting rational numerical scheme demonstrates stability regardless of wave propagation direction. Consequently, we introduce a central rational numerical scheme for scalar hyperbolic conservation laws, offering robust and accurate computation of solutions.