Fourth order 2N storage runge kutta schemes

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1、/H--o211NASATechnicalMemorandum109112Fourth-Order2N-StorageRunge-KuttaSchemesMarkH.CarpenterLangleyResearchCenter,Hampton,VirginiaChristopherA.KennedyUniversityofCMifornia,SanDiego,LaJoHa,California(NASA-TM-I09112)FOURTH-ORDERN94-329502N-STORAGERUNGE-KUTTASCHEMES(NASA.LangleyResearchCenter

2、)Unclas26pG3/020009987June1994NationalAeronauticsandSpaceAdministrationLangleyResearchCenterHampton,Virginia23681-0001FOURTH-ORDER2N-STORAGERUNGE-KUTTASCHEMESMarkIt.Carpenter_ChristopherA.KennedytAbstractAfamilyoffive-stagefourth-orderRunge-Kuttaschemesisderived;theseschemesrequireonlytwos

3、toragelocations.Aparticularschemeisidentifiedthathasdesirableefficiencycharacteristicsforhyperbolicandparabolicinitial(boundary)valueproblems.Thisschemeiscompetitivewiththeclassicalfourth-ordermethod(high-storage)andisconsiderablymoreefficientandaccuratethanexistingthird-orderlow-storagesc

4、hemes.Section1:IntroductionManyproblemsofmathematicalphysicsresultinthenumericalapproximationofsystemsofcoupledordinarydifferentialequations(ODE's).Thetechniqueforintegratingtheequationsdependson,amongotherthings,thedesiredaccuracy,efficiency,robustness,andsimplicity.FortheODE'sthatresultf

5、romtiledirectsimulationoftheequationsoffluiddynamics,theoverridingconsiderationisoftentheavailabilityoffastmemory.AnumericalintegrationtechniquethatminimizesmemorystorageisessentialandcanbeformulatedwithaRunge-Kutta(RK)methodology.Williamson[1]showedthatallsecond-orderandsomethird-orderRKs

6、chemescanbecastina2N-storageformat,whereNisthedimensionoftilesystemofODE's.Healsoshowedthat(exceptforcertainexceptionalcasesthatinvolvespecificformsofthefunctiontobeintegrated)tilefour-stagefourth-orderRKschemescannotbeputinto2N-storageformat.Fyfe[2]showedthatallfour-stagefourth-orderRKsch

7、emescanbeimplementedin3N-storagelocations.Inspiteofthereducedaccuracy,formanyapl)licationsthemethodsofchoicearetile2N-storagethird-orderRKscllelIle$.Althoughtheprincipalmotivationofthisworkistoderiveschemesof2Nstorage,boththeaccuracyandelliciencyalsodeterminet