Representation of the Lagrange reconstructing polynomial by combination of substencils
The Lagrange reconstructing polynomial [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] of a function \(f(x)\) on a given set of equidistant (\(\Delta x=\const\)) points \(\bigl\{x_i+\ell\Delta x;\;\ell\in\{-M_-,...,+M_+\}\bigr\}\) is defined [Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (20...
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| Published in: | arXiv.org |
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| Main Author: | |
| Format: | Paper |
| Language: | English |
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Ithaca
Cornell University Library, arXiv.org
14.02.2012
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| ISSN: | 2331-8422 |
| Online Access: | Get full text |
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| Summary: | The Lagrange reconstructing polynomial [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] of a function \(f(x)\) on a given set of equidistant (\(\Delta x=\const\)) points \(\bigl\{x_i+\ell\Delta x;\;\ell\in\{-M_-,...,+M_+\}\bigr\}\) is defined [Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (2011) 267--305] as the polynomial whose sliding (with \(x\)) averages on \([x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]\) are equal to the Lagrange interpolating polynomial of \(f(x)\) on the same stencil. We first study the fundamental functions of Lagrange reconstruction, show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) \(x_i+n\tfrac{1}{2}\Delta x\) (\(n\in\mathbb{Z}\)), and obtain several identities. Using these identities, by analogy to the recursive Neville-Aitken-like algorithm applied to the Lagrange interpolating polynomial, we show that there exists a unique representation of the Lagrange reconstructing polynomial on \(\{i-M_-,...,i+M_+\}\) as a combination of the Lagrange reconstructing polynomials on the \(K_\mathrm{s}+1\leq M:=M_-+M_+>1\) substencils \(\{i-M_-+k_\mathrm{s},...,i+M_+-K_\mathrm{s}+k_\mathrm{s}\}\) (\(k_\mathrm{s}\in\{0,...,K_\mathrm{s}\}\)), with weights \(\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)\) which are rational functions of \(\xi\) (\(x=x_i+\xi\Delta x\)) [Liu Y.Y., Shu C.W., Zhang M.P.: {\em Acta Math. Appl. Sinica} {\bf 25} (2009) 503--538], and give an analytical recursive expression of the weight-functions. We then use the analytical expression of the weight-functions \(\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)\) to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of \(\xi=\tfrac{1}{2}\), under the condition that all of the substencils contain either point \(i\) or point \(i+1\) (or both). |
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| Bibliography: | SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50 |
| ISSN: | 2331-8422 |
| DOI: | 10.48550/arxiv.1102.3136 |