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.