A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

Abstract

In this paper, we analyze the behavior of a nonlinear reconstruction operator called PPH around discontinuities. The acronym PPH stands for Piecewise Polynomial Harmonic, since it uses piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. This study is carried out in the general case of nonuniform grids, although for some results we restrict to σ quasi-uniform grids. In particular we analyze the numerical order of approximation close to jump discontinuities and the elimination of the Gibbs effects. We show, both theoretically and with numerical examples, that the numerical order is reduced but not completely lost as it is the case in their linear counterparts. Moreover we observe that the reconstruction is free of any Gibbs effects for sufficiently small grid sizes.

Geometric representation of the weighted harmonic mean of $n$ positive values and potential uses

“…We now present two particular properties, which have been already used in [12] in order to work with a nonlinear reconstruction for nonuniform grids adapted to the potential presence of jump discontinuities on the signal. The first property has to do with the adaptation in case of jump discontinuities, while the second property is related to the order of approximation attained by the nonlinear reconstruction operator, see [12] for more details.…”