{
  "id": "1102.1826",
  "version": "v3",
  "published": "2011-02-09T10:44:19.000Z",
  "updated": "2012-05-07T09:58:52.000Z",
  "title": "A general recurrence relation for the weight-functions in Mühlbach-Neville-Aitken representions with application to WENO interpolation and differentiation",
  "authors": [
    "G. A. Gerolymos"
  ],
  "comment": "extended revised version",
  "journal": "Appl. Math. Comp. 219 (2012) 4133--4142",
  "doi": "10.1016/j.amc.2012.09.044",
  "categories": [
    "math.NA",
    "physics.comp-ph"
  ],
  "abstract": "In several applications, such as \\tsc{weno} interpolation and reconstruction [Shu C.W.: SIAM Rev. 51 (2009) 82--126], we are interested in the analytical expression of the weight-functions which allow the representation of the approximating function on a given stencil (Chebyshev-system) as the weighted combination of the corresponding approximating functions on substencils (Chebyshev-subsystems). We show that the weight-functions in such representations [M\\\"uhlbach G.: Num. Math. 31 (1978) 97--110] can be generated by a general recurrence relation based on the existence of a 1-level subdivision rule. As an example of application we apply this recurrence to the computation of the weight-functions for Lagrange interpolation [Carlini E., Ferretti R., Russo G.: SIAM J. Sci. Comp. 27 (2005) 1071--1091] for a general subdivision of the stencil ${x_{i-M_-},...,x_{i+M_+}}$ of $M+1:=M_-+M_++1$ distinct ordered points into $K_\\mathrm{s}+1\\leq M:=M_-+M_+>1$ (Neville) substencils ${x_{i-M_-+k_\\mathrm{s}},...,x_{i+M_+-K_\\mathrm{s}+k_\\mathrm{s}}}$ ($k_\\mathrm{s}\\in{0,...,K_\\mathrm{s}}$) all containing the same number of $M-K_\\mathrm{s}+1$ points but each shifted by 1 cell with respect to its neighbour, and give a general proof for the conditions of positivity of the weight-functions (implying convexity of the combination), extending previous results obtained for particular stencils and subdvisions [Liu Y.Y., Shu C.W., Zhang M.P.: Acta Math. Appl. Sinica 25 (2009) 503--538]. Finally, we apply the recurrence relation to the representation by combination of substencils of derivatives of arbitrary order of the Lagrange interpolating polynomial.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-07T09:58:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D99",
      "65D05",
      "65D25"
    ],
    "keywords": [
      "general recurrence relation",
      "weno interpolation",
      "weight-functions",
      "mühlbach-neville-aitken representions",
      "application"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.1826G"
    }
  }
}