Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation

Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation

This paper addresses fundamental challenges in numerical approximation methods, focusing on balancing accuracy with shape-preserving properties. We present novel approaches that combine traditional spline methods with modern numerical techniques, extending existing quasi-interpolation techniques bas...

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Bibliographic Details
Main Authors: Francesc Aràndiga, Sara Remogna
Format: Article
Language:English
Published: MDPI AG 2025-04-01
Series:Mathematics
Subjects:
spline quasi-interpolation
WENO
piecewise smooth functions approximation
shape-preserving
Online Access:https://www.mdpi.com/2227-7390/13/8/1237
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Summary:This paper addresses fundamental challenges in numerical approximation methods, focusing on balancing accuracy with shape-preserving properties. We present novel approaches that combine traditional spline methods with modern numerical techniques, extending existing quasi-interpolation techniques based on B-splines. Our methods maintain computational efficiency while better handling discontinuities, achieving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>1</mn></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>2</mn></msup></semantics></math></inline-formula> reconstructions and preserving essential shape properties. We demonstrate theoretical frameworks showing optimal approximation order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula>, with local reconstruction. Numerical experiments confirm significant improvements in accuracy and smoothness near discontinuities compared to existing methods, particularly in image processing and shock-capturing applications.
ISSN:2227-7390

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