Internal Error Propagation in Explicit Runge--Kutta Methods

Handle URI:
http://hdl.handle.net/10754/333640
Title:
Internal Error Propagation in Explicit Runge--Kutta Methods
Authors:
Ketcheson, David I. ( 0000-0002-1212-126X ) ; Loczi, Lajos ( 0000-0002-7999-5658 ) ; Parsani, Matteo ( 0000-0001-7300-1280 )
Abstract:
In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Numerical Mathematics Group
Citation:
Internal Error Propagation in Explicit Runge--Kutta Methods 2014, 52 (5):2227 SIAM Journal on Numerical Analysis
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Numerical Analysis
Issue Date:
11-Sep-2014
DOI:
10.1137/130936245
ARXIV:
arXiv:1309.1317
Type:
Article
ISSN:
0036-1429; 1095-7170
Sponsors:
The authors were supported by award FIC/2010/05 2000000231 made by KAUST.
Additional Links:
http://epubs.siam.org/doi/abs/10.1137/130936245; http://arxiv.org/abs/1309.1317
Appears in Collections:
Articles; Numerical Mathematics Group; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorKetcheson, David I.en
dc.contributor.authorLoczi, Lajosen
dc.contributor.authorParsani, Matteoen
dc.date.accessioned2014-11-04T07:18:16Z-
dc.date.available2014-11-04T07:18:16Z-
dc.date.issued2014-09-11en
dc.identifier.citationInternal Error Propagation in Explicit Runge--Kutta Methods 2014, 52 (5):2227 SIAM Journal on Numerical Analysisen
dc.identifier.issn0036-1429en
dc.identifier.issn1095-7170en
dc.identifier.doi10.1137/130936245en
dc.identifier.urihttp://hdl.handle.net/10754/333640en
dc.description.abstractIn practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.en
dc.description.sponsorshipThe authors were supported by award FIC/2010/05 2000000231 made by KAUST.en
dc.language.isoenen
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/abs/10.1137/130936245en
dc.relation.urlhttp://arxiv.org/abs/1309.1317en
dc.rightsArchived with thanks to SIAM Journal on Numerical Analysisen
dc.subjectRunge–Kutta methodsen
dc.subjectinternal stabilityen
dc.subjectroundoff erroren
dc.subjectstrong stability preservationen
dc.subjectextrapolationen
dc.subjectordinary differential equationsen
dc.titleInternal Error Propagation in Explicit Runge--Kutta Methodsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentNumerical Mathematics Groupen
dc.identifier.journalSIAM Journal on Numerical Analysisen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionComputational Aerosciences Branch, NASA Langley Research Center, Hampton, VA 23681en
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
dc.identifier.arxividarXiv:1309.1317en
kaust.authorKetcheson, David I.en
kaust.authorLoczi, Lajosen
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.