G.2079Page1GlobalResearchjournalofNaturalScience&Technology(GRJNST)Volume:04-Issue2(2026),2079ISSNP:2790-7643ISSNE:2790-7651www.grjnst.nethttps://doi.org/10.53762/grjnst.04.02.30ATheoreticalInvestigationofLocalTruncationErrorandConvergenceofanAcceleratedThird-OrderRunge–KuttaMethodforNon-AutonomousOrdinaryDifferentialEquationsReceived:29March2026.Accepted:22April2026.Published:30April2026SubjectSpecialist(BPS-InstituteofMathematicsandComputerScience,Universityo17)inSchoolEducationandLiteracyDepartmenMuhammadDaudKandhro(CorrespondingAuthor)fSindh,Jamshoro(SELD),Govt.ofSindhdaud.kandhro@gmail.comNaveedAhmedTunioHST(BPS-16)inSchoolEducationandLiteracyDepartment(SELD),Govt.ofSindhnaveedtunio@gmail.comQureshiInstituteofMathematicsandComputerScience,UniversityofSindh,Jamshoroli@usindh.edu.pkZohaibAlizohaib.aGRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30Copyright©2026GRJNST.ThisarticleispublishedunderanOpenAccessmodel.ItismadeavailabletothepublicunderthetermsoftheCreativeCommonsAttribution4.0International(CCBY4.0)license,whichpermitsunrestricteduseanddistributionG.2079Page2Abstract:Inthisarticle,wetheoreticallyexaminedwell-organizedthirdordernumericaltechniqueforIVPofODE’sincludingpartialderivativewhichhasenhanceditscompetencyregardingtruncationerror.Acceleratednumericalmethodforlocaltruncationerrorandconvergenceistheoreticallyinvestigatedtoassesshowaccurateandreliableproposedmethodis.ExpansionofTaylor’sseriesisdonethatprovidesrightpatterntoexpandandevaluatefunctionevaluation.WiththehelpofTaylor’sseries,Localtruncationerrorisanexplicitlyderivedtoclarifytheorderofaccuracy.Linearstandardtestisdiscussedforcalculatingstability.Stabilityisinvestigatedforknowingbehaviorofmethod.StabilityregionisdrawnbyMATLAB2023asoftware.Stabilityregionisvisualizedtochecknumericalmethodpossessaboundedsolutionwhenappliedfrequentlyandrepeatedly.Consistencyisprovedwhichtellsuserrorgoestozeroasmuchaswedecreasestepsizewhichguaranteesthedesiredresult.Convergencecriteriatheoreticallydiscussedherebyprovingconsistencyandstability.Theoreticallyfindingsshowingthatthenumericalmethodisstable,attainshighaccuracyandgivesreliableperformance.Therefore,methodisefficientandapplicableforextensiveclassofIVParisingintheareaofODE’s.Keywords:NumericalMethod,LocalTruncationError,Stability,Consistency,Convergence1.INTRODUCTIONOrdinarydifferentialequations(ODEs)takeverycrucialroleintheformationofmodelingarisinginscience,engineering,physics,biology,andeconomics.Manyreal-worldproblemsaregovernedbyinitialvalueproblemswhoseanalyticalsolutionsareeitherdifficultorimpossibletoobtaininclosedform.Asaresult,numericalmethodshavebecomeindispensabletoolsforapproximatingsolutionsofsuchequations.Whendesigningnumericalschemes,computationalefficiencyisnottheonlyconsiderationforobtainingreliableresults;it'salsoimportanttounderstandthetheoreticalaspectsoftheschemes,suchastheiraccuracyandstability.Oneofthekeyelementsintheanalysisofnumericalschemesiserroranalysis.TheanalysisoferrorhelpsdeterminetheimpactofdiscretizationonthenumericalsolutionGRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30G.2079Page3andenablesthecomparisonbetweendifferentnumericalschemes.Inthesestudies,itiscommontorefertolocalandglobalerrors.Thelocaltruncationerror(LTE)isdefinedastheerrorpresentedinasinglestepofthenumericalmethod,assumingthattheearlierstepsareexact.Itservesasaprimarymeasurefordeterminingtheorderofaccuracy.Specifically,iftheLTEbehaveslikethenthemethodhasorderp.Thus,thederivationandanalysisoftheLTEareessentialforconstructingofhigh-accuracynumericalmethods.Closelylinkedtoerroranalysisistheconceptofconsistency,whichguaranteesthattheconsequenceapproximatetechniquepreciselyexemplifiestheunderlyingdifferentialequationasstepsizenearertozero.Consistencyguaranteesthatthediscreteschemeisafaithfulapproximationofthecontinuousproblem,forminganecessaryconditionforconvergence.However,consistencyaloneisnotsufficienttoensurereliablenumericalresults.Anotherimportantpropertyisstability,relatingtothebehaviorofnumericalerrorsintheiterativeprocess.Stabilityisapropertythatguaranteesthaterrors(duetotruncation,round-offorsmallchangesininitialconditions)donotblowupduringtheiterativeprocess.Thisiscrucialforstiffproblemsandwhenintegratingoverlongtimes.Averypopularmethodforinvestigatingstabilityinvolvesapplicationofastandardlineartestequation.Usingsuchanapproach,oneestablishestheso-calledstabilityfunction,whichdescribesthegrowthinerrorineachstep.Thestabilityregionisintroducedasapartofthisprocess.Theregioninthecomplexplaneforwhichthenumericalsolutionisboundedisknownasthestabilityregion.Theextentandshapeofthestabilityregiongiveanideaofthestepsizesthatcanbetakenandthestabilityofthemethod.Itispreferableforamethodtohavealargerstabilityregion,particularlyforstifforhighlyoscillatoryproblems,becauseitmeansthatlargerstepsizescanbetaken.Themainrequirementofmethodisthatitisconvergent,whichmeansthatthenumericalsolutionconvergestotheexactsolutionasthestepsizeapproacheszero.Convergenceisacrucialpropertyforanumericalmethod.Afundamentalresultinnumericalanalysisisthat,undercertainconditions,consistencyandstabilityleadtoGRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30G.2079Page4convergence.Thisconnectionunderscorestheneedtoestablishbothofthesepropertiesofanumericalmethod.Inlightoftheseconsiderations,thedevelopmentofeffectiveandreliablenumericalschemesrequiresacomprehensiveframeworkthatintegrateserroranalysis,consistency,stability,andconvergence.Inthispaper,wefocusonthesystematicinvestigationofthesepropertiesforamethoddesignedtosolveODE’s.TheanalysiscontainsthederivationoftheLTEtoestablishtheorderofaccuracy,verificationofconsistency,detailedstabilityanalysisthroughtheconstructionofthestabilityfunction,andcharacterizationofthecorrespondingstabilityregion.Finally,convergenceisestablishedbycombiningtheresultsofconsistencyandstability.2.MATERIALSANDMETHODSManynumericalmethodshavedesignedtoattaintheestimatedconsequencesofIVP’sfordifferentialequationinnatureordinary.Wecanconsidergeneralfunctionform:Intervalwehavegivenasispartedintoauniformparts,,thestepsizeisequidistantnodetoattendapproximatevaluesGenerally,functionoftwospacevariablealsofamiliarasaderivativeofthedependentvariableregardingtoindependenthasbeencalculatedthroughintegrating.Thetosolvethefunctioninadiscreteseries.Manyequationsnameddifferentialdon’treachtothesolutionsuchasparticularandanalytical.Todealwithsuchdifficultynewinnovationstakeplaceinnumericalanalysis.Bestadvantageofnumericalschemeisthatthesehavebetterperformancethananalytical.Researchersdaybydayaretryingtoinnovatemanymethodstoobtainbetterconsequences.Butstillhugenumberofworkisneededtounderstandproperbehavesofmethod.Researchersandscholarsareoutonthefieldofinnovationtohonetheirskills.Manyofthemhavealsodoneagreatjobtoconstructandmodifiednewmethods.Kandhro[4]hasdevelopedacceleratedmethodwhoseIterativeIntegratorwhichisGRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30G.2079Page5Thisisnewlydevelopedanacceleratedexplicitschemehavingthreefunctionevaluationspertimestep.Nowerroranalysis.3.ERRORANALYSISItismainpurposetosolveordinarydifferentialequationnumericallytoattainresultswhichareascloseaspossibletotheexactsolution.Twosourcesoferrorwhichaffecttheaccuracyofnumericalmethodnamedasround-off(RO)andtruncation(TR).ROround-offerrortakestheplacewhencomputerscanonlystocknumberswithlimitedprecisionandTRarisesbecausemathematicalproceduresareapproximated(e.g.,bydesertinghigher-orderterms).Anaccuracycantotallyrelyonblunderwhatsizeofstepsizeistaken.WeconsiderTaylor’sseriesfortwovariableuptothefourthpowerofstepsize,wehaveThelocaltruncationerrorisanerrorwhichspawnedinauniquestepoftheproposedimprovedschemethatisdocumentedaswhereisthesolutionobtainedbyTaylors’sSeriesandWhereapproximatesolution.Taylorseriesisutilizedtoexpandthesearoundtermscollectin(2)uptoisusedasanandsimilar.NowweexpandproposedacceleratedexplicitMethodpresentineqn,wegetGRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30G.2079Page6Subtract(5)from(3).Theproposedschemehasalocaltruncationerrorthatis:Thelocaltruncationerror(LTE)oftencallswithname"discretizationerrorpertimestep",whichhelpstoevaluatetheorderofaccuracy.FromLTEitisobserved,LTEhas4rthorderthentheproposedschemein[4]hasonelessthantheLTEinorder.Therefore,theproposedschemein[4]hasthirdorderofaccuracy.4.CONSISTENCYANALYSISDefinition5.1Anincrementfunctionbeconsistent,whenofnumericalmethodisentitledtoConsistencyofthenumericalmethodstellslocaltruncationerror(LTE)tendstozeroasthestepsizedecreasesor.Therefore,newlyproposedschemehasincrementfunctionasByutilizingbothsidesGRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.300limhG.2079Page7Hence,thenewlyproposedschemewithatleastthirdorderaccuracyisproventobeconsistent.5.LINEARSTABILITYANALYSISDahlquist’stestproblemistobeconsideredforverifyingthestabilityoftheMethodin[4]whereasisacomplexconstanti.e,where.AfterexecutingMethod(2)onthistest,weattainpolynomialfunctionacknowledgedasstabilityfunctionwithlinearregionwhichdisplayedviaunfilledareainfig1.Substituteallvaluesin(5),polynomialformofstabilityfunctionisderivedaswhereThisisStabilityFunctionforacceleratedproposedmethodin[4]whichreflectsthethird-orderaccuracyofthemethod.Whenerrorsareintroducedthenlinearstabilityischeckedtoknowhownumericalmethodbehaves.Inparticular,itexpressesthatwhetherthoseerrorsdecay,remainbounded,orgrowuncontrollablyduringthecomputation.Stabilitycontrolserrorpropagation.GRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30G.2079Page86.REGIONOFABSOLUTESTABILITYTheregionoftheacceleratedproposedmethod[4]iscomputedandvisualizedusingMATLAB.Agridofcomplexvaluesisgenerated,andthestabilityfunctionisevaluatedoverthisdomain.Thesetofpointssatisfyingisidentifiedandplottedtoillustratethestabilityregioninthecomplexplane.Thisstabilityregionisafundamentalconceptusedtodeterminewhetheranumericalmethodproducesboundedsolutionswhenappliedrepeatedly.Figure1illustratesthetwo-dimensionalregionofabsolutestabilityfortheproposednumericalmethodintheisdemonstratedonhorizontallineofaxis,whilecomplexplane.Therealparttheimaginarypartisdemonstratedonverticalaxisofline,where.Thewhite(unshaded)regionsindicatethesetofvaluesforwhichthestabilityconditionissatisfied.Theseregionsrepresentthedomainwhensolutionapproximatelyremainsstableandkeepbounded.Theorange(shaded)regioncorrespondstovaluesofGRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30G.2079Page9forwhichindicatinginstability,wherenumericalerrorsmaygrowexponentially.7.CONVERGENCEOFTHEMETHODAmethodistobenumericallyconvergentwhenbothconsistencyandstabilityjustified.Thenecessaryconditionforconvergenceofanumericalmethodisconsistency,whichguaranteesthatthelocaldiscretizationerrorvanishesasstep-sizeverynearlyapproacheszero.However,convergenceisachievedonlywhenconsistencyiscombinedwithstability.Therefore,abovesaidwordstheacceleratedproposedmethodisconverges.8.CONCLUSIONInthiswork,theconstructionanddetailedmathematicalanalysisofanumericalmethodforsolvinginitialvalueproblemsofordinarydifferentialequations.wetheoreticallyexaminedwell-organizedthirdordernumericaltechniqueforIVPofODE’sincludingpartialderivativewhichhasenhanceditscompetencyregardingtruncationerror.Themainfocusisnotheretoderiveschemebuthereisjustifyitstheoreticallybehavewhichisanalyzedbysystematicinvestigationoferrorandconvergence.TheLTEhasderivedperfectlywhichendorsingaccuracybyidentifyitsorder.Stabilityisinvestigatedforknowingbehaviorofmethod.Stabilityregionisenvisionedtochecknumericalmethodpossessaboundedsolutionwhenappliedfrequentlyandrepeatedly.Convergencecriteriatheoreticallydiscussedherebyprovingconsistencyandstability.Theoreticallyfindingsshowingthatthenumericalmethodisstable,attainshighaccuracyandgivesreliableperformance.Therefore,methodisefficientandapplicableforextensiveclassofIVParisingintheareaofODE’s.REFERENCES[1]AshiriboSenaponWusu,MosesAdebowaleAkanbiandSolomonAdebolaOkunuga‘’AThree-StageMultiderivativeExplicitRunge-KuttaMethod’’,AmericanJournalofComputationalMathematics,2013,3,121-126[2]Butcher,JohnCharles.Numericalmethodsforordinarydifferentialequations.JohnWiley&Sons,2016.GRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30G.2079Page10[3]Owolanke,A.O.,Uwaheren,O.AndObarhua,F.O.(2017)AnEightOrderTwo-StepTaylorSeriesAlgorithmforTheNumericalSolutionsofInitialValueProblemsofSecondOrderOrdinaryDifferentialEquations.OpenAccessLibraryJournal,4:E3486.[4]DaudKandhro1*,ImranSoomro3,DevelopmentofAcceleratedThirdOrderRungeKuttaMethodforNon-AutonomousOrdinaryDifferentialEquation[5]AThirdRungeKuttaMethodBasedonaLinearCombinationofArithmeticMean,HarmonicMeanandGeometricMean[5]MukaddesOktenTuracı1,2·Turgut¨Ozis¸1‘’DerivationofThree-DerivativeRunge-KuttaMethods’’SpringerScienceBusinessMediaNewYork2016[6]ErwinKreyszig,HerbertKreyszig,EdwardJ.Norminton"AdvancedEngineeringMathematics."TenthEdition2011[7]Wazwaz,Abdul-Majid."AmodifiedthirdorderRunge-Kuttamethod."AppliedMathematicsLetters3.3(1990):123-125.[8]AbdulMajidWazwak‘’AModifiedThirdOrderRungeKuttaMethod”AlQudsuniversitycollegeofscienceandtechnology(1990).[9]RaoV.Dukkipati"NumericalMethods"FirstEdition2010[10]Burden,RichardL.,andJ.DouglasFaires."Numericalanalysis.2001."Brooks/Cole,USA(2001).[11]PeterV.O’Neil"AdvancedEngineeringMathematics"7thedition[12]Akanbi,M.A.,(2010),On3-stagegeometricexplicitRunge–Kuttamethodforsingularautonomousinitialvalueproblemsinordinarydifferentialequations.Springer-Verlag.GRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30G.2079Page11[13]Butcher,J.C.,(1964),OnRunge-KuttaprocessesofhighOrder,J.Austral.Math.Soc.,4[14]Fatunla,S.O.,(1991),NumericalMethodsforIVPsinODEs.AcademicPress,NewYork,[15]DrNajmuddinAhmed,StudyofNumericalAccuracyofRunge-KuttaSecond,ThirdandFourthOrderMethod[16]Hairer,E.;Narsett,S.P.andWanner,G.(1993):SolvingOrdinaryDifferentialEquations.Ι.Nonstiffproblems,Vol.8ofSpringerSeriesinComputationalMathematicsSpringer–Verlag(Berlin),SecondEd.[17]Hundsdofer,W.andVerwer,J.G.(2003):NumericalSolutionofTimedependentAdvection–Diffusion–ReactionEquations,Vol.33ofSpringerSeriesinComputationalMathematics.Springer(Berlin)[18]Kaw,Autar;Kalu,Egwu(2008):NumericalMethodswithApplications(1stEd.),www.autarkaw.com.[19]Runge–Kutta–FehlbergTypeProcedureonTwoNodesforNumericalIntegrationofSystemsDifferentialEquations.Dumitras,DariaElenaAutomat.Comput.Appl.Math.,Vol.2,pp139–143,Math.Sci.Net.[20]TestresultsonInitialValueMethodsforNon–StiffOrdinaryDifferentialEquationsW.H.Enright;T.E.HullSIAMJournalonNumericalAnalysis,Vol.13,No.6.(Dec.,1976),pp944–961,Jstor.GRJNST,Volume:04-Issue2(2026)/ISSNP:2790-7643ArticleID:2079https://doi.org/10.53762/grjnst.04.02.30G.2079Page12[21]Verner,J.H.(1991):SomeRunge–KuttaFormulaPairs.SIAM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