no. 5
Finite difference operators from moving least squares interpolation
ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 5, pp. 959-974.

In a foregoing paper [Sonar, ESAIM: M2AN 39 (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.

DOI : 10.1051/m2an:2007042
Classification : 65M06, 65M60, 65F05
Mots clés : difference operators, moving least squares interpolation, order of approximation 
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     title = {Finite difference operators from moving least squares interpolation},
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Netuzhylov, Hennadiy; Sonar, Thomas; Yomsatieankul, Warisa. Finite difference operators from moving least squares interpolation. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 5, pp. 959-974. doi : 10.1051/m2an:2007042. http://www.numdam.org/articles/10.1051/m2an:2007042/

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[2] 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).

[3] P. Lancaster and K. Šalkauskas, Surfaces generated by moving least square methods. Math. Comp. 37 (1981) 141-158. | Zbl

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[5] H. Netuzhylov, Meshfree collocation solution of Boundary Value Problems via Interpolating Moving Least Squares. Comm. Num. Meth. Engng. 22 (2006) 893-899. | Zbl

[6] O. Nowak and T. Sonar, Upwind and ENO strategies in Interpolating Moving Least Squares methods (in preparation).

[7] T. Sonar, Difference operators from interpolating moving least squares and their deviation from optimality. ESAIM: M2AN 39 (2005) 883-908. | Numdam | Zbl

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