Strong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Order

Handle URI:
http://hdl.handle.net/10754/333578
Title:
Strong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Order
Authors:
Hadjimichael, Yiannis ( 0000-0003-3517-8557 ) ; Macdonald, Colin B.; Ketcheson, David I. ( 0000-0002-1212-126X ) ; Verner, James H.
Abstract:
We apply the concept of effective order to strong stability preserving (SSP) explicit Runge--Kutta methods. Relative to classical Runge--Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods---like classical order five methods---require the use of nonpositive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge--Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice.
KAUST Department:
Numerical Mathematics Group; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Strong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Order 2013, 51 (4):2149 SIAM Journal on Numerical Analysis
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Numerical Analysis
Issue Date:
23-Jul-2013
DOI:
10.1137/120884201
ARXIV:
arXiv:1207.2902
Type:
Article
ISSN:
0036-1429; 1095-7170
Additional Links:
http://epubs.siam.org/doi/abs/10.1137/120884201; http://arxiv.org/abs/1207.2902
Appears in Collections:
Articles; Numerical Mathematics Group; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorHadjimichael, Yiannisen
dc.contributor.authorMacdonald, Colin B.en
dc.contributor.authorKetcheson, David I.en
dc.contributor.authorVerner, James H.en
dc.date.accessioned2014-11-03T16:17:40Z-
dc.date.available2014-11-03T16:17:40Z-
dc.date.issued2013-07-23en
dc.identifier.citationStrong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Order 2013, 51 (4):2149 SIAM Journal on Numerical Analysisen
dc.identifier.issn0036-1429en
dc.identifier.issn1095-7170en
dc.identifier.doi10.1137/120884201en
dc.identifier.urihttp://hdl.handle.net/10754/333578en
dc.description.abstractWe apply the concept of effective order to strong stability preserving (SSP) explicit Runge--Kutta methods. Relative to classical Runge--Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods---like classical order five methods---require the use of nonpositive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge--Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice.en
dc.language.isoenen
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/abs/10.1137/120884201en
dc.relation.urlhttp://arxiv.org/abs/1207.2902en
dc.rightsArchived with thanks to SIAM Journal on Numerical Analysisen
dc.subjectstrong stability preserving (SSP)en
dc.subjecteffective orderen
dc.subjectmonotonicityen
dc.subjectRunge–Kutta methodsen
dc.subjecttime integrationen
dc.subjecthigh-order accuracyen
dc.titleStrong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Orderen
dc.typeArticleen
dc.contributor.departmentNumerical Mathematics Groupen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalSIAM Journal on Numerical Analysisen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionMathematical Institute, University of Oxford, Oxford OX1 3LB, UKen
dc.contributor.institutionDepartment of Mathematics, Simon Fraser University, Burnaby V5A 1S6, BC, Canadaen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
dc.identifier.arxividarXiv:1207.2902en
kaust.authorHadjimichael, Yiannisen
kaust.authorKetcheson, David I.en
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