Integrating Moving Least Squares with non-linear WENO method: A novel Partition of Unity approach in 1D
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866929685823225856 |
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| author | Garcés, Inmaculada Ramón, José M. Ruiz-Álvarez, Juan Yáñez, Dionisio F. |
| author_facet | Garcés, Inmaculada Ramón, José M. Ruiz-Álvarez, Juan Yáñez, Dionisio F. |
| contents | The approximation of data is a fundamental challenge encountered in various fields, including computer-aided geometric design, the numerical solution of partial differential equations, or the design of curves and surfaces. Numerous methods have been developed to address this issue, providing good results when the data is continuous. Among these, the Moving Least Squares (MLS) method has proven to be an effective strategy for fitting data, finding applications in both statistics and applied mathematics. However, the presence of isolated discontinuities in the data can lead to undesirable artifacts, such as the Gibbs phenomenon, which adversely affects the quality of the approximation.
In this paper, we propose a novel approach that integrates the Moving Least Squares method with the well-established non-linear Weighted Essentially Non-Oscillatory (WENO) method. This combination aims to construct a non-linear operator that enhances the accuracy of approximations near discontinuities while maintaining the order of accuracy in smooth regions. We investigate the properties of this operator in one dimension, demonstrating its effectiveness through a series of numerical experiments that validate our theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14536 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrating Moving Least Squares with non-linear WENO method: A novel Partition of Unity approach in 1D Garcés, Inmaculada Ramón, José M. Ruiz-Álvarez, Juan Yáñez, Dionisio F. Numerical Analysis 41A05, 41A10, 65D05, 65M06, 65N06 The approximation of data is a fundamental challenge encountered in various fields, including computer-aided geometric design, the numerical solution of partial differential equations, or the design of curves and surfaces. Numerous methods have been developed to address this issue, providing good results when the data is continuous. Among these, the Moving Least Squares (MLS) method has proven to be an effective strategy for fitting data, finding applications in both statistics and applied mathematics. However, the presence of isolated discontinuities in the data can lead to undesirable artifacts, such as the Gibbs phenomenon, which adversely affects the quality of the approximation. In this paper, we propose a novel approach that integrates the Moving Least Squares method with the well-established non-linear Weighted Essentially Non-Oscillatory (WENO) method. This combination aims to construct a non-linear operator that enhances the accuracy of approximations near discontinuities while maintaining the order of accuracy in smooth regions. We investigate the properties of this operator in one dimension, demonstrating its effectiveness through a series of numerical experiments that validate our theoretical findings. |
| title | Integrating Moving Least Squares with non-linear WENO method: A novel Partition of Unity approach in 1D |
| topic | Numerical Analysis 41A05, 41A10, 65D05, 65M06, 65N06 |
| url | https://arxiv.org/abs/2501.14536 |