Propagation of internal errors in explicit Runge–Kutta methods and internal stability of SSP and extrapolation methods

Handle URI:
http://hdl.handle.net/10754/333680
Title:
Propagation of internal errors in explicit Runge–Kutta methods and internal stability of SSP and extrapolation 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
Issue Date:
11-Apr-2014
Type:
Technical Report
Sponsors:
This publication is based on work supported by Award No. FIC/2010/05 2000000231, made by KAUST.
Additional Links:
http://arxiv.org/abs/1309.1317
Appears in Collections:
Technical Reports; 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.accessioned2014-11-04T12:21:09Z-
dc.date.available2014-11-04T12:21:09Z-
dc.date.issued2014-04-11en
dc.identifier.urihttp://hdl.handle.net/10754/333680en
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.sponsorshipThis publication is based on work supported by Award No. FIC/2010/05 2000000231, made by KAUST.en
dc.language.isoenen
dc.relation.urlhttp://arxiv.org/abs/1309.1317en
dc.titlePropagation of internal errors in explicit Runge–Kutta methods and internal stability of SSP and extrapolation methodsen
dc.typeTechnical Reporten
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentNumerical Mathematics Groupen
dc.eprint.versionPre-printen
dc.contributor.institutionComputational Aerosciences Branch, NASA Langley Research Center, Hampton, VA 23681, USA.en
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
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