×

zbMATH — the first resource for mathematics

A convergence rate theorem for finite difference approximations to delta functions. (English) Zbl 1155.65016
A new rate of convergence for approximations to certain integrals over codimension one manifolds in \(\mathbb R^n\), is proved. The type of manifold is defined by the zero level set of a smooth mapping \(u:\mathbb R^n\to\mathbb R\). The approximation method used, is based on two finite difference methods for the discretizing of the delta function, originally presented by the author in a previous research work [J. Comput. Phys. 220, No. 2, 915–931 (2007; Zbl 1115.65028)], were empirical convergence rates had indicated the first and second order accuracy. In this work these convergence rates are proved for the two proposed algorithms, under fairly general hypotheses.

MSC:
65D15 Algorithms for approximation of functions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J.T. Beale, A proof that a discrete delta function is second order accurate, Preprint at <http://www.math.duke.edu/beale/papers/ddel.pdf>. · Zbl 1136.65017
[2] V.F. Candela, A. Marquina, On the numerical approximation of the length of (implicit) level curves, Preprint 2006. · Zbl 1203.65040
[3] Chang, Y.C.; Hou, T.Y.; Merriman, B.; Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. comput. phys., 124, 449-464, (1996) · Zbl 0847.76048
[4] Engquist, B.; Tornberg, A.K.; Tsai, R., Discretization of Dirac delta functions in level set methods, J. comput. phys., 207, 28-51, (2005) · Zbl 1074.65025
[5] Liu, H.; Osher, S.; Tsai, R., Multi-valued solution and level set methods in computational high frequency wave propagation, Commun. comput. phys., 1, 5, 765-804, (2006) · Zbl 1120.65110
[6] Mayo, A., The fast solution of poisson’s and the biharmonic equations on irregular regions, SIAM J. numer. anal., 21, 285-299, (1984) · Zbl 1131.65303
[7] Min, C.; Gibou, F., Geometric integration over irregular domains with application to level-set methods, J. comput. phys., 226, 1432-1443, (2007) · Zbl 1125.65021
[8] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2003), Springer-Verlag New York · Zbl 1026.76001
[9] Osher, S.; Sethian, J., Fronts propagating with curvature dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132
[10] Peng, D.; Merriman, B.; Osher, S.; Zhao, H.; Kang, M., A PDE-based fast local level set method, J. comput. phys., 155, 410-438, (1999) · Zbl 0964.76069
[11] Runborg, O., Mathematical models and numerical methods for high frequency waves, Commun. comput. phys., 2, 5, 827-880, (2007) · Zbl 1164.78300
[12] Sethian, J.A., Level set methods and fast marching methods, (1999), Cambridge University Press Cambridge · Zbl 0929.65066
[13] Smereka, P., The numerical approximation of a delta function with application to level set methods, J. comput. phys., 211, 77-90, (2006) · Zbl 1086.65503
[14] Tornberg, A.K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 462-488, (2004) · Zbl 1115.76392
[15] Tornberg, A.K.; Engquist, B., Regularization techniques for numerical approximation of PDEs with singularities, J. sci. comput., 19, 527-552, (2003) · Zbl 1035.65085
[16] Towers, J.D., Two methods for discretizing a delta function supported on a level set, J. comput. phys., 220, 915-931, (2007) · Zbl 1115.65028
[17] J.D. Towers, Discretizing delta functions via finite difference and gradient normalization, Preprint at <http://www.miracosta.edu/home/jtowers/>. · Zbl 1167.65007
[18] Wen, X., High order numerical methods to a type of delta function integrals, J. comput. phys., 226, 1952-1967, (2007) · Zbl 1125.65024
[19] X. Wen, High order numerical quadratures to one dimensional delta function integrals, Preprint at <http://lsec.cc.ac.cn/wenxin/research.html>. · Zbl 1170.65008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.