On the Convexity Preservation of a Quasi <i>C</i><sup>3</sup> Nonlinear Interpolatory Reconstruction Operator on <i>σ</i> Quasi-Uniform Grids

On the Convexity Preservation of a Quasi <i>C</i><sup>3</sup> Nonlinear Interpolatory Reconstruction Operator on <i>σ</i> Quasi-Uniform Grids

This paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity p...

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Main Authors: Pedro Ortiz, Juan Carlos Trillo
Format: Article
Language:English
Published: MDPI AG 2021-02-01
Series:Mathematics
Subjects:
interpolation
reconstruction
nonlinearity
nonuniform
σ quasi-uniform
convexity
Online Access:https://www.mdpi.com/2227-7390/9/4/310
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spelling doaj-75afb0b390f44d3da1629e881671b5052021-02-05T00:04:24ZengMDPI AGMathematics2227-73902021-02-01931031010.3390/math9040310On the Convexity Preservation of a Quasi <i>C</i><sup>3</sup> Nonlinear Interpolatory Reconstruction Operator on <i>σ</i> Quasi-Uniform GridsPedro Ortiz0Juan Carlos Trillo1Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202 Cartagena, SpainDepartamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202 Cartagena, SpainThis paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity preservation. In particular, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> quasi-uniform grids with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>≤</mo><mn>4</mn></mrow></semantics></math></inline-formula>, we get a quasi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>3</mn></msup></semantics></math></inline-formula> reconstruction that maintains the convexity properties of the initial data. We give some numerical experiments regarding the approximation order and the convexity preservation.https://www.mdpi.com/2227-7390/9/4/310interpolationreconstructionnonlinearitynonuniformσ quasi-uniformconvexity
collection DOAJ
language English
format Article
sources DOAJ
author Pedro Ortiz
Juan Carlos Trillo
spellingShingle Pedro Ortiz
Juan Carlos Trillo
On the Convexity Preservation of a Quasi <i>C</i><sup>3</sup> Nonlinear Interpolatory Reconstruction Operator on <i>σ</i> Quasi-Uniform Grids
Mathematics
interpolation
reconstruction
nonlinearity
nonuniform
σ quasi-uniform
convexity
author_facet Pedro Ortiz
Juan Carlos Trillo
author_sort Pedro Ortiz
title On the Convexity Preservation of a Quasi <i>C</i><sup>3</sup> Nonlinear Interpolatory Reconstruction Operator on <i>σ</i> Quasi-Uniform Grids
title_short On the Convexity Preservation of a Quasi <i>C</i><sup>3</sup> Nonlinear Interpolatory Reconstruction Operator on <i>σ</i> Quasi-Uniform Grids
title_full On the Convexity Preservation of a Quasi <i>C</i><sup>3</sup> Nonlinear Interpolatory Reconstruction Operator on <i>σ</i> Quasi-Uniform Grids
title_fullStr On the Convexity Preservation of a Quasi <i>C</i><sup>3</sup> Nonlinear Interpolatory Reconstruction Operator on <i>σ</i> Quasi-Uniform Grids
title_full_unstemmed On the Convexity Preservation of a Quasi <i>C</i><sup>3</sup> Nonlinear Interpolatory Reconstruction Operator on <i>σ</i> Quasi-Uniform Grids
title_sort on the convexity preservation of a quasi <i>c</i><sup>3</sup> nonlinear interpolatory reconstruction operator on <i>σ</i> quasi-uniform grids
publisher MDPI AG
series Mathematics
issn 2227-7390
publishDate 2021-02-01
description This paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity preservation. In particular, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> quasi-uniform grids with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>≤</mo><mn>4</mn></mrow></semantics></math></inline-formula>, we get a quasi <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>3</mn></msup></semantics></math></inline-formula> reconstruction that maintains the convexity properties of the initial data. We give some numerical experiments regarding the approximation order and the convexity preservation.
topic interpolation
reconstruction
nonlinearity
nonuniform
σ quasi-uniform
convexity
url https://www.mdpi.com/2227-7390/9/4/310
work_keys_str_mv AT pedroortiz ontheconvexitypreservationofaquasiicisup3supnonlinearinterpolatoryreconstructionoperatoronisiquasiuniformgrids
AT juancarlostrillo ontheconvexitypreservationofaquasiicisup3supnonlinearinterpolatoryreconstructionoperatoronisiquasiuniformgrids
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