Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-08T07:46:19.727Z Has data issue: false hasContentIssue false
  • English
  • Français

Difference operators from interpolating moving least squares and theirdeviation from optimality

Published online by Cambridge University Press:  15 September 2005

Thomas Sonar
Thomas Sonar*
Affiliation:
Institut Computational Mathematics, TU Braunschweig, Pockelsstraße 14, 38106 Braunschweig, Germany. [email protected]
Get access

Abstract

We consider the classical Interpolating Moving Least Squares (IMLS)interpolant as defined by Lancaster and Šalkauskas [Math. Comp.37 (1981) 141–158] andcompute the first and second derivative of this interpolant at the nodes of agiven grid with the help of a basic lemma on Shepard interpolants. We comparethe difference formulae with those defining optimal finite difference methods anddiscuss their deviation from optimality.

Keywords

Difference operatorsmoving least squares interpolationorder of approximation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 5 , September 2005 , pp. 883 - 908
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. and Krysl, P., Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Engrg. 139 (1996) 3–47. CrossRef
J.P. Boyd, Chebyshev and Fourier Spectral Methods. Springer Verlag (1989).
Fornberg, B., Generation of Finite Difference Formulas on Arbitrarily Spaced Grids. Math. Comp. 51 (1988) 699–706. CrossRef
B. Fornberg, A Practical Guide to Pseudospectral Methods. Cambridge University Press (1996).
Fürst, J. and Sonar, Th., On meshless collocation approximations of conservation laws: preliminary investigations on positive schemes and dissipation models. ZAMM Z. Angew. Math. Mech. 81 (2001) 403–415. 3.0.CO;2-T>CrossRef
M. Kunle, Entwicklung und Untersuchung von Moving Least Square Verfahren zur numerischen Simulation hydrodynamischer Gleichungen. Doktorarbeit, Fakultät für Physik, Eberhard-Karls-Universität zu Tübingen (2001).
Lancaster, P. and Šalkauskas, K., Surfaces generated by moving least squares methods. Math. Comp. 37 (1981) 141–158. CrossRef
P. Lancaster and K. Šalkauskas, Curve and Surface Fitting: An Introduction. Academic Press (1986).
Liszka, T. and Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput. Structures 11 (1980) 83–95. CrossRef
H. Netuzylov, Th. Sonar and W. Yomsatieankul, Finite difference operators from moving least squares interpolation. Manuscript, Institut Computational Mathematics, TU Braunschweig (2004).
Perrone, N. and Kao, R., A general finite difference method for arbitrary meshes. Comput. Structures 5 (1975) 45–58. CrossRef
Schönauer, W., Generation of difference and error formulae of arbitrary consistency order on an unstructured grid. ZAMM Z. Angew. Math. Mech. 78 (1998) S1061–S1062. CrossRef
L. Theilemann, Ein gitterfreies differenzenverfahren. Doktorarbeit, Institut für Aerodynamik und Gasdynamik, Universität Stuttgart (1983).
W. Yomsatieankul, Th. Sonar and H. Netuzhylov, Spatial difference operators from moving least squares interpolation. Manuscript, Institut Computational Mathematics, TU Braunschweig (2004).