Discretization of Dirac delta functions in level set methods
type: Item
instance of: scholarly article
title: Discretization of Dirac delta functions in level set methods
zbMATH Open document ID: 1074.65025
author: Björn Engquist
author: Anna-Karin Tornberg
author: Richard Tsai
published in: Journal of Computational Physics
publication date: 2005-06-13T00:00:00Z
review text: The authors analyze the accuracy of regularizations of Dirac delta functions in the context of the level set method. Following a recent work by \textit{A. K. Tornberg} and \textit{B. Engquist} [J. Comput Phys. 200, No. 2, 462--488 (2004; Zbl 1115.76392)] showing that the most common technique for the regularization of the delta function in level set methods is inconsistent and may lead to \(O(1)\) errors, the authors present two techniques to construct consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both techniques are based solely on the distance to the singularity and thus are independent of the grid. The first of them is based on a tensor product of regularized one-dimensional delta functions. In the second technique a variable support of the regularization domain is used. The regularization is constructed. from a one-dimensional regularization that is extended to multi-dimensions. Being the Dirac delta function \(\delta\) previously replaced by a more regular function \(\delta_\varepsilon\), the multidimensional regularized delta function is then defined as \(\delta_\varepsilon(\Gamma, x)= \delta_\varepsilon(d(\Gamma, x))\) where \(d(\Gamma, x)\) stands for the distance function. Convergence analysis, numerical results and some applications to a class of partial differential equations are also given.
reviewed by: Nácere Hayek
Mathematics Subject Classification ID: 65D20
Mathematics Subject Classification ID: 46F10
zbMATH DE Number: 2175309
zbMATH Keywords: discretization of Dirac delta functions
zbMATH Keywords: level set methods
zbMATH Keywords: consistent approximations
zbMATH Keywords: regularized delta functions
zbMATH Keywords: regularization
zbMATH Keywords: convergence analysis
zbMATH Keywords: numerical results
MaRDI profile type: Publication
full work available at URL: j.jcp.2004.09.018
OpenAlex ID: W2139971200
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DOI: 10.1016/J.JCP.2004.09.018
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description: scientific article; zbMATH DE number 2175309