Stable high-order finite-difference methods based on non-uniform grid point distributions

Hernández Ramos, Juan Antonio and Hermanns Navarro, Miguel (2008). Stable high-order finite-difference methods based on non-uniform grid point distributions. "International Journal For Numerical Methods In Fluids", v. 56 (n. 3); pp. 233-255. ISSN 0271-2091. https://doi.org/10.1002/fld.1510.

Description

Title: Stable high-order finite-difference methods based on non-uniform grid point distributions
Author/s:
  • Hernández Ramos, Juan Antonio
  • Hermanns Navarro, Miguel
Item Type: Article
Título de Revista/Publicación: International Journal For Numerical Methods In Fluids
Date: January 2008
ISSN: 0271-2091
Volume: 56
Subjects:
Freetext Keywords: high-order scheme; finite difference; piecewise polynomials; stability; Runge phenomenon; pseudospectra
Faculty: E.T.S.I. Aeronáuticos (UPM)
Department: Matemática Aplicada y Estadística [hasta 2014]
Creative Commons Licenses: Recognition - No derivative works - Non commercial

Full text

[img]
Preview
PDF - Requires a PDF viewer, such as GSview, Xpdf or Adobe Acrobat Reader
Download (709kB) | Preview

Abstract

It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundstr¨om theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resulting finite-difference operator. Several heuristic explanations exist for the goodness of the clustering, and attempts have been made to link it to the Runge phenomenon present in polynomial interpolations of high degree. By following the philosophy behind the Chebyshev polynomials, a non-uniform grid for piecewise polynomial interpolations of degree q_N is introduced in this paper, where N + 1 is the total number of grid nodes. It is shown that when q = N, this polynomial interpolation coincides with the Chebyshev interpolation, and the resulting finite-difference schemes are equivalent to Chebyshev collocation methods. Finally, test cases are run showing how stability and correct transient behaviours are achieved for any degree q<N through the use of the proposed non-uniform grids. Discussions are complemented by spectra and pseudospectra of the finite-difference operators.

More information

Item ID: 2439
DC Identifier: http://oa.upm.es/2439/
OAI Identifier: oai:oa.upm.es:2439
DOI: 10.1002/fld.1510
Official URL: http://www3.interscience.wiley.com/journal/117868808/issue
Deposited by: Memoria Investigacion
Deposited on: 16 Apr 2010 08:21
Last Modified: 20 Apr 2016 12:08
  • Logo InvestigaM (UPM)
  • Logo GEOUP4
  • Logo Open Access
  • Open Access
  • Logo Sherpa/Romeo
    Check whether the anglo-saxon journal in which you have published an article allows you to also publish it under open access.
  • Logo Dulcinea
    Check whether the spanish journal in which you have published an article allows you to also publish it under open access.
  • Logo de Recolecta
  • Logo del Observatorio I+D+i UPM
  • Logo de OpenCourseWare UPM