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.
Highlights
Due to the extended use of reconstruction operators in many fields of application, ranging from hyperbolic conservation laws to computer aided geometric design, it is of great importance to dispose of efficient methods to build them for different situations.In general, and for the sake of simplicity, the considered functions are polynomials
We extend the definition of the Piecewise Polynomial Harmonic (PPH) reconstruction operator to data over non uniform grids and we study some properties of this operator
We prove adaption to the jump discontinuity in the sense that some order of approximation is maintained in the area close to the discontinuity, on the contrary to what happens with linear operators that lose completely the approximation order
Summary
Due to the extended use of reconstruction operators in many fields of application, ranging from hyperbolic conservation laws to computer aided geometric design, it is of great importance to dispose of efficient methods to build them for different situations. Recent approaches to deal with similar problems of functions affected by discontinuities can be found for example in [1,2,3,4,5] These nonlinear methods give rise to interesting applications. The theoretical analysis as much as the practical applications were developed in uniform grids in previous articles (see, for example, [12,13,14,15,16,17,18]) In turn these reconstruction operators are the heart of the definition of associated subdivision and multiresolution schemes [5,19,20,21].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
