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Interpolatory quadrature rules for oscillatory integrals

Veerle Ledoux and Marnix Van Daele UGent (2012) JOURNAL OF SCIENTIFIC COMPUTING. 53(3). p.586-607
abstract
In this paper we revisit some quadrature methods for highly oscillatory integrals of the form . Exponentially Fitted (EF) rules depend on frequency dependent nodes which start off at the Gauss-Legendre nodes when the frequency is zero and end up at the endpoints of the integral when the frequency tends to infinity. This makes the rules well suited for small as well as for large frequencies. However, the computation of the EF nodes is expensive due to iteration and ill-conditioning. This issue can be resolved by making the connection with Filon-type rules. By introducing some S-shaped functions, we show how Gauss-type rules with frequency dependent nodes can be constructed, which have an optimal asymptotic rate of decay of the error with increasing frequency and which are effective also for small or moderate frequencies. These frequency-dependent nodes can also be included into Filon-Clenshaw-Curtis rules to form a class of methods which is particularly well suited to be implemented in an automatic software package.
Please use this url to cite or link to this publication:
author
organization
year
type
journalArticle (original)
publication status
published
subject
keyword
CLENSHAW-CURTIS, Numerical quadrature · High oscillation, GAUSS QUADRATURE, EQUATIONS, SERIES
journal title
JOURNAL OF SCIENTIFIC COMPUTING
J. Sci. Comput.
volume
53
issue
3
pages
586 - 607
Web of Science type
Article
Web of Science id
000311400300006
JCR category
MATHEMATICS, APPLIED
JCR impact factor
1.71 (2012)
JCR rank
22/247 (2012)
JCR quartile
1 (2012)
ISSN
0885-7474
DOI
10.1007/s10915-012-9589-4
language
English
UGent publication?
yes
classification
A1
copyright statement
I have transferred the copyright for this publication to the publisher
id
3079175
handle
http://hdl.handle.net/1854/LU-3079175
date created
2012-12-26 09:54:00
date last changed
2016-12-19 15:43:38
@article{3079175,
  abstract     = {In this paper we revisit some quadrature methods for highly oscillatory integrals of the form . Exponentially Fitted (EF) rules depend on frequency dependent nodes which start off at the Gauss-Legendre nodes when the frequency is zero and end up at the endpoints of the integral when the frequency tends to infinity. This makes the rules well suited for small as well as for large frequencies. However, the computation of the EF nodes is expensive due to iteration and ill-conditioning. This issue can be resolved by making the connection with Filon-type rules. By introducing some S-shaped functions, we show how Gauss-type rules with frequency dependent nodes can be constructed, which have an optimal asymptotic rate of decay of the error with increasing frequency and which are effective also for small or moderate frequencies. These frequency-dependent nodes can also be included into Filon-Clenshaw-Curtis rules to form a class of methods which is particularly well suited to be implemented in an automatic software package.},
  author       = {Ledoux, Veerle and Van Daele, Marnix},
  issn         = {0885-7474},
  journal      = {JOURNAL OF SCIENTIFIC COMPUTING},
  keyword      = {CLENSHAW-CURTIS,Numerical quadrature {\textperiodcentered} High oscillation,GAUSS QUADRATURE,EQUATIONS,SERIES},
  language     = {eng},
  number       = {3},
  pages        = {586--607},
  title        = {Interpolatory quadrature rules for oscillatory integrals},
  url          = {http://dx.doi.org/10.1007/s10915-012-9589-4},
  volume       = {53},
  year         = {2012},
}

Chicago
Ledoux, Veerle, and Marnix Van Daele. 2012. “Interpolatory Quadrature Rules for Oscillatory Integrals.” Journal of Scientific Computing 53 (3): 586–607.
APA
Ledoux, Veerle, & Van Daele, M. (2012). Interpolatory quadrature rules for oscillatory integrals. JOURNAL OF SCIENTIFIC COMPUTING, 53(3), 586–607.
Vancouver
1.
Ledoux V, Van Daele M. Interpolatory quadrature rules for oscillatory integrals. JOURNAL OF SCIENTIFIC COMPUTING. 2012;53(3):586–607.
MLA
Ledoux, Veerle, and Marnix Van Daele. “Interpolatory Quadrature Rules for Oscillatory Integrals.” JOURNAL OF SCIENTIFIC COMPUTING 53.3 (2012): 586–607. Print.