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Runge–Kutta methods for ordinary differential equations

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.

Methods have been found based on Gaussian quadrature. Later this extended to methods related to Radau and Lobatto quadrature. A-stable methods exist in these classes. Because of the high cost of these methods, attention moved to diagonally and singly implicit methods. Runge–Kutta methods for ordinary differential equations – p. 5/48

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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 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 ContentsIntroductiontoRunge KuttamethodsFormulationofmethodTaylorexp ansionofexactsolutionTaylorexpansionforn umericalapproximationOrderconditionsCons tructionoflow orderexplicitmethodsOrderbarriersAlgebra icinterpretationEffective orderImplicitRunge KuttamethodsSingly-implicitmethodsRunge Kuttamethodsforordinarydifferentialequat ions p.

7 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)).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 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 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 ThesemethodsfromRunge s 1895paperare secondorder becausetheerrorina singlestepbehaveslikeO(h3).A few yearslater, Heungave a fullexplanationoforder3methodsandKuttaga ve a Kuttamethodsforordinarydifferentialequat ions p.

10 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. 5/48 Withtheemergenceofstiff problemsasanimportantapplicationarea, , Kuttamethodsforordinarydifferentialequat ions p.

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