Transcription of Runge–Kutta methods for ordinary differential equations
1 runge Kuttamethodsforordinarydifferentialequat ionsJohnButcherTheUniversityofAucklandNe w ZealandCOEW orkshoponNumericalAnalysisKyushuUniversi tyMay2005 runge Kuttamethodsforordinarydifferentialequat ions p. 1/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p. 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p.
2 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p. 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p.
3 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p. 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p.
4 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p. 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p.
5 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p. 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p.
6 2/48 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p. 2/48 IntroductiontoRunge KuttamethodsIt willbeconvenienttoconsideronlyautonomous initialvalueproblemsy (x) =f(y(x)),y(x0) =y0,f:RN :yn=yn 1+hf(yn 1),h=xn xn 1canbemademoreaccuratebyusingeitherthemi d-pointorthetrapezoidalrulequadraturefor mula:yn=yn 1+hf(yn 1+12hf(yn 1)).
7 Yn=yn 1+12hf(yn 1) +12hf(yn 1+hf(yn 1)). runge Kuttamethodsforordinarydifferentialequat ions p. 3/48 IntroductiontoRunge KuttamethodsIt willbeconvenienttoconsideronlyautonomous initialvalueproblemsy (x) =f(y(x)),y(x0) =y0,f:RN :yn=yn 1+hf(yn 1),h=xn xn 1canbemademoreaccuratebyusingeitherthemi d-pointorthetrapezoidalrulequadraturefor mula:yn=yn 1+hf(yn 1+12hf(yn 1)).yn=yn 1+12hf(yn 1) +12hf(yn 1+hf(yn 1)). runge Kuttamethodsforordinarydifferentialequat ions p. 3/48 IntroductiontoRunge KuttamethodsIt willbeconvenienttoconsideronlyautonomous initialvalueproblemsy (x) =f(y(x)),y(x0) =y0,f:RN :yn=yn 1+hf(yn 1),h=xn xn 1canbemademoreaccuratebyusingeitherthemi d-pointorthetrapezoidalrulequadraturefor mula:yn=yn 1+hf(yn 1+12hf(yn 1)).
8 Yn=yn 1+12hf(yn 1) +12hf(yn 1+hf(yn 1)). runge Kuttamethodsforordinarydifferentialequat ions p. 3/48 ThesemethodsfromRunge s 1895paperare secondorder becausetheerrorina singlestepbehaveslikeO(h3).A few yearslater, Heungave a fullexplanationoforder3methodsandKuttaga ve a Kuttamethodsforordinarydifferentialequat ions p. 4/48 ThesemethodsfromRunge s 1895paperare secondorder becausetheerrorina singlestepbehaveslikeO(h3).A few yearslater, Heungave a fullexplanationoforder3methodsandKuttaga ve a Kuttamethodsforordinarydifferentialequat ions p. 4/48 ThesemethodsfromRunge s 1895paperare secondorder becausetheerrorina singlestepbehaveslikeO(h3).
9 A few yearslater, Heungave a fullexplanationoforder3methodsandKuttaga ve a Kuttamethodsforordinarydifferentialequat ions p. 4/48 ThesemethodsfromRunge s 1895paperare secondorder becausetheerrorina singlestepbehaveslikeO(h3).A few yearslater, Heungave a fullexplanationoforder3methodsandKuttaga ve a Kuttamethodsforordinarydifferentialequat ions p. 4/48 Withtheemergenceofstiff problemsasanimportantapplicationarea, , Kuttamethodsforordinarydifferentialequat ions p. 5/48 Withtheemergenceofstiff problemsasanimportantapplicationarea, , Kuttamethodsforordinarydifferentialequat ions p.
10 5/48 Withtheemergenceofstiff problemsasanimportantapplicationarea, , Kuttamethodsforordinarydifferentialequat ions p. 5/48 Withtheemergenceofstiff problemsasanimportantapplicationarea, , Kuttamethodsforordinarydifferentialequat ions p. 5/48 Withtheemergenceofstiff problemsasanimportantapplicationarea, , Kuttamethodsforordinarydifferentialequat ions p. 5/48 FormulationofmethodIncarryingouta stepweevaluatesstagevaluesY1,Y2,.. ,YsandsstagederivativesF1,F2,.. ,Fs,usingtheformulaFi=f(Yi).EachYiis foundasa linearcombinationoftheFjaddedontoy0:Yi=y 0+hs j=1aijFjandtheapproximationatx1=x0+his foundfromy1=y0+hs i=1biFiRunge Kuttamethodsforordinarydifferentialequat ions p.
