Finite difference operators from moving least squares interpolation

Hennadiy Netuzhylov; Thomas Sonar; Warisa Yomsatieankul

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 5, page 959-974
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Netuzhylov, Hennadiy, Sonar, Thomas, and Yomsatieankul, Warisa. "Finite difference operators from moving least squares interpolation." ESAIM: Mathematical Modelling and Numerical Analysis 41.5 (2007): 959-974. <http://eudml.org/doc/250081>.

@article{Netuzhylov2007,
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. },
author = {Netuzhylov, Hennadiy, Sonar, Thomas, Yomsatieankul, Warisa},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Difference operators; moving least squares interpolation; order of approximation.},
language = {eng},
month = {10},
number = {5},
pages = {959-974},
publisher = {EDP Sciences},
title = {Finite difference operators from moving least squares interpolation},
url = {http://eudml.org/doc/250081},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Netuzhylov, Hennadiy
AU - Sonar, Thomas
AU - Yomsatieankul, Warisa
TI - Finite difference operators from moving least squares interpolation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/10//
PB - EDP Sciences
VL - 41
IS - 5
SP - 959
EP - 974
AB - 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.
LA - eng
KW - Difference operators; moving least squares interpolation; order of approximation.
UR - http://eudml.org/doc/250081
ER -

References

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  1. G.H. Golub and C.F. Van Loan, Matrix Computations. Johns Hopkins Univ. Press (1996).  
  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.  
  4. P. Lancaster and K. Šalkauskas, Curve and Surface Fitting - An Introduction. Academic Press (1986).  
  5. H. Netuzhylov, Meshfree collocation solution of Boundary Value Problems via Interpolating Moving Least Squares. Comm. Num. Meth. Engng.22 (2006) 893–899.  
  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: M2AN39 (2005) 883–908.  

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