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Finite difference operators from moving least squares interpolation

Published online by Cambridge University Press:  23 October 2007

Hennadiy Netuzhylov ,
Thomas Sonar  and
Warisa Yomsatieankul
Hennadiy Netuzhylov
Affiliation:
TU Braunschweig, Computational Mathematics, Pockelstrasse 14, 38106 Braunschweig, Germany. t.sonar@tu-bs.de
Thomas Sonar
Affiliation:
TU Braunschweig, Computational Mathematics, Pockelstrasse 14, 38106 Braunschweig, Germany. t.sonar@tu-bs.de
Warisa Yomsatieankul
Affiliation:
TU Braunschweig, Computational Mathematics, Pockelstrasse 14, 38106 Braunschweig, Germany. t.sonar@tu-bs.de
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Abstract

In a foregoing paper [Sonar, ESAIM: M2AN39 (2005) 883–908] we analyzed the Interpolating Moving Least Squares (IMLS) method due to Lancaster and Šalkauskas with respect to its approximation powers and derived finite difference expressions for the derivatives. In this sequel we follow a completely different approach to the IMLS method given by Kunle [Dissertation (2001)]. As a typical problem with IMLS method we address the question of getting admissible results at the boundary by introducing “ghost points”. Most interesting in IMLS methods are the finite difference operators which can be computed from different choices of basis functions and weight functions. We present a way of deriving these discrete operators in the spatially one-dimensional case. Multidimensional operators can be constructed by simply extending our approach to higher dimensions. Numerical results ranging from 1-d interpolation to the solution of PDEs are given.

Keywords

Difference operatorsmoving least squares interpolationorder of approximation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 5 , September 2007 , pp. 959 - 974
Copyright
© EDP Sciences, SMAI, 2007

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References

G.H. Golub and C.F. Van Loan, Matrix Computations. Johns Hopkins Univ. Press (1996).
M. Kunle, Entwicklung und Untersuchung von Moving Least Square Verfahren zur numerischen Simulation hydrodynamischer Gleichungen. Dissertation, Fakultät für Physik, Eberhard-Karls-Universität zu Tübingen (2001).
Lancaster, P. and Šalkauskas, K., Surfaces generated by moving least square methods. Math. Comp. 37 (1981) 141–158. CrossRef
P. Lancaster and K. Šalkauskas, Curve and Surface Fitting - An Introduction. Academic Press (1986).
Netuzhylov, H., Meshfree collocation solution of Boundary Value Problems via Interpolating Moving Least Squares. Comm. Num. Meth. Engng. 22 (2006) 893–899. CrossRef
O. Nowak and T. Sonar, Upwind and ENO strategies in Interpolating Moving Least Squares methods (in preparation).
Sonar, T., Difference operators from interpolating moving least squares and their deviation from optimality. ESAIM: M2AN 39 (2005) 883–908. CrossRef