Volume 28, Issue 4
The Collocation Basis of Compact Finite Differences for Moment-Preserving Interpolations: Review, Extension and Applications

Julián T. Becerra-Sagredo, Rolf JeltschCarlos Málaga

Commun. Comput. Phys., 28 (2020), pp. 1245-1273.

Published online: 2020-08

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  • Abstract

The diagnostic of the performance of numerical methods for physical models, like those in computational fluid mechanics and other fields of continuum mechanics, rely on the preservation of statistical moments of extensive quantities. Dynamic and adaptive meshing often use interpolations to represent fields over a new set of elements and require to be conservative and moment-preserving. Denoising algorithms should not affect moment distributions of data. And numerical deltas are described using the number of moments preserved. Therefore, all these methodologies benefit from the use of moment-preserving interpolations. In this article, we review the presentation of the piecewise polynomial basis functions that provide moment-preserving interpolations, better described as the collocation basis of compact finite differences, or Z-splines. We present different applications of these basis functions that show the improvement of numerical algorithms for fluid mechanics, discrete delta functions and denoising. We also provide theorems of the extension of the properties of the basis, previously known as the Strang and Fix theory, to the case of arbitrary knot partitions.

  • Keywords

Conservation of moments, moment-preserving interpolation, conservative interpolation, high-order interpolation, regularized delta function, numerical advection, denoising.

  • AMS Subject Headings

41A05, 65D05, 76M23, 97N50

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COPYRIGHT: © Global Science Press

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@Article{CiCP-28-1245, author = {T. Becerra-Sagredo , Julián and Jeltsch , Rolf and Málaga , Carlos}, title = {The Collocation Basis of Compact Finite Differences for Moment-Preserving Interpolations: Review, Extension and Applications}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {4}, pages = {1245--1273}, abstract = {

The diagnostic of the performance of numerical methods for physical models, like those in computational fluid mechanics and other fields of continuum mechanics, rely on the preservation of statistical moments of extensive quantities. Dynamic and adaptive meshing often use interpolations to represent fields over a new set of elements and require to be conservative and moment-preserving. Denoising algorithms should not affect moment distributions of data. And numerical deltas are described using the number of moments preserved. Therefore, all these methodologies benefit from the use of moment-preserving interpolations. In this article, we review the presentation of the piecewise polynomial basis functions that provide moment-preserving interpolations, better described as the collocation basis of compact finite differences, or Z-splines. We present different applications of these basis functions that show the improvement of numerical algorithms for fluid mechanics, discrete delta functions and denoising. We also provide theorems of the extension of the properties of the basis, previously known as the Strang and Fix theory, to the case of arbitrary knot partitions.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0170}, url = {http://global-sci.org/intro/article_detail/cicp/18100.html} }
TY - JOUR T1 - The Collocation Basis of Compact Finite Differences for Moment-Preserving Interpolations: Review, Extension and Applications AU - T. Becerra-Sagredo , Julián AU - Jeltsch , Rolf AU - Málaga , Carlos JO - Communications in Computational Physics VL - 4 SP - 1245 EP - 1273 PY - 2020 DA - 2020/08 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0170 UR - https://global-sci.org/intro/article_detail/cicp/18100.html KW - Conservation of moments, moment-preserving interpolation, conservative interpolation, high-order interpolation, regularized delta function, numerical advection, denoising. AB -

The diagnostic of the performance of numerical methods for physical models, like those in computational fluid mechanics and other fields of continuum mechanics, rely on the preservation of statistical moments of extensive quantities. Dynamic and adaptive meshing often use interpolations to represent fields over a new set of elements and require to be conservative and moment-preserving. Denoising algorithms should not affect moment distributions of data. And numerical deltas are described using the number of moments preserved. Therefore, all these methodologies benefit from the use of moment-preserving interpolations. In this article, we review the presentation of the piecewise polynomial basis functions that provide moment-preserving interpolations, better described as the collocation basis of compact finite differences, or Z-splines. We present different applications of these basis functions that show the improvement of numerical algorithms for fluid mechanics, discrete delta functions and denoising. We also provide theorems of the extension of the properties of the basis, previously known as the Strang and Fix theory, to the case of arbitrary knot partitions.

Julián T. Becerra-Sagredo, Rolf Jeltsch & Carlos Málaga. (2020). The Collocation Basis of Compact Finite Differences for Moment-Preserving Interpolations: Review, Extension and Applications. Communications in Computational Physics. 28 (4). 1245-1273. doi:10.4208/cicp.OA-2019-0170
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