{
  "id": "1102.3136",
  "version": "v3",
  "published": "2011-02-15T17:40:30.000Z",
  "updated": "2012-02-14T17:42:28.000Z",
  "title": "Representation of the Lagrange reconstructing polynomial by combination of substencils",
  "authors": [
    "G. A. Gerolymos"
  ],
  "comment": "final corrected version; in print J. Comp. Appl. Math",
  "journal": "J. Comp. Appl. Math. 236 (2012) 2763-2794",
  "doi": "10.1016/j.cam.2012.01.008",
  "categories": [
    "math.NA",
    "physics.comp-ph"
  ],
  "abstract": "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).",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-02-14T17:42:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D99",
      "65D05",
      "65D25",
      "65M06",
      "65M08"
    ],
    "keywords": [
      "lagrange reconstructing polynomial",
      "combination",
      "lagrange interpolating polynomial",
      "weight-functions",
      "unique representation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.3136G"
    }
  }
}