On global error control in nested implicit Runge-Kutta methods of the gauss type
Automatic global error control based on a combined control of step size and order presented by Kulikov and Khrustaleva in 2008 is investigated. Special attention is given to the efficiency of computation, because the implicit extrapolation based on multistage implicit Runge-Kutta schemes may be expe...
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| Vydáno v: | Numerical analysis and applications Ročník 4; číslo 3; s. 199 - 209 |
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| Jazyk: | angličtina |
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Dordrecht
SP MAIK Nauka/Interperiodica
01.07.2011
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| ISSN: | 1995-4239, 1995-4247 |
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| Abstract | Automatic global error control based on a combined control of step size and order presented by Kulikov and Khrustaleva in 2008 is investigated. Special attention is given to the efficiency of computation, because the implicit extrapolation based on multistage implicit Runge-Kutta schemes may be expensive. Specifically, we discuss a technique of global error estimation and control in order to compute a numerical solution satisfying the user-supplied accuracy conditions (in exact arithmetic) automatically. The theoretical results of this paper are confirmed by numerical experiments on test problems. |
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| AbstractList | Automatic global error control based on a combined control of step size and order presented by Kulikov and Khrustaleva in 2008 is investigated. Special attention is given to the efficiency of computation, because the implicit extrapolation based on multistage implicit Runge-Kutta schemes may be expensive. Specifically, we discuss a technique of global error estimation and control in order to compute a numerical solution satisfying the user-supplied accuracy conditions (in exact arithmetic) automatically. The theoretical results of this paper are confirmed by numerical experiments on test problems. |
| Author | Kulikov, G. Yu Khrustaleva, E. Yu Kuznetsov, E. B. |
| Author_xml | – sequence: 1 givenname: G. Yu surname: Kulikov fullname: Kulikov, G. Yu email: gkulikov@math.ist.utl.pt, kulgyu@yahoo.com organization: CEMAT, Instituto Superior Técnico, TU Lisbon, Av. Rovisco Pais – sequence: 2 givenname: E. B. surname: Kuznetsov fullname: Kuznetsov, E. B. organization: Moscow Aviation Institute – sequence: 3 givenname: E. Yu surname: Khrustaleva fullname: Khrustaleva, E. Yu organization: Ulyanovsk State University |
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| Cites_doi | 10.1016/S0168-9274(99)00126-9 10.1007/11758501_104 10.1023/B:BITN.0000046811.70614.38 10.1093/imanum/2.2.211 10.1002/0470868279 10.1007/BF01933583 10.1515/rnam.2007.029 10.1007/BF01947741 10.1007/BF01963532 10.1093/imamat/19.4.455 10.1137/0714069 10.1007/BF01932774 10.1137/090764840 10.1093/imamat/20.4.425 10.1007/BF01932722 10.1016/j.apnum.2008.03.019 10.1137/0714068 10.1515/RJNAMM.2009.009 10.1007/BF01932265 10.1137/0727044 10.1007/978-3-540-72584-8_18 10.1090/S0025-5718-1974-0331793-2 10.1007/BF01932291 |
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| Keywords | implicit Runge-Kutta formulas efficient implementation nested implicit schemes of the Gauss type global error estimation and control |
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Nezhostkie zadachi1990MoscowMir(Solving Ordinary Differential Equations: Nonstiff Problems) KulikovG.Yu.Automatic Error Control in the Gauss-Type Nested Implicit Runge-Kutta Formula of Order 6Russ. J. Numer. Anal. Math. Model.2009242123144252273010.1515/RJNAMM.2009.0091168.65368 AltR.Methodes A-Stables pour l’Integration de Systemes Differentiels Mal Conditionnes1971ParisUniversite Paris Dahlquist, G., Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, Trans. Royal Inst. Technol., 1959, no. 130, pp. 1–87. AlexanderR.Diagonally Implicit Runge-Kutta Methods for Stiff ODEsSIAM J. Numer. Anal.1977141006102445889010.1137/07140680374.65038 ButcherJ.C.Numerical Methods for Ordinary Differential Equations2003ChichesterWiley10.1002/04708682791040.65057 HairerE.WannerG.Reshenie obyknovennykh differentsialnykh uravnenii. Zhostkie i differentsialno-algebraicheskie zadachi1999MoscowMir(Solving Ordinary Differential Equations: Stiff and Differential-Algebraic Problems) ButcherJ.C.ChenD.J.A New Type of Singly Implicit Runge-Kutta MethodsAppl. Numer. Math.200034179188177042210.1016/S0168-9274(99)00126-90954.65058 van BokhovenW.M.Efficient Higher Order Implicit One-Step Methods for Integration of Stiff Differential EquationsBIT198020344356997410.1007/BF019335830448.65047 CrouzeixM.Sur l’Approximation des Equations Differentielles Operationnelles Lineaires par de Methodes de Runge-Kutta1975ParisUniversite Paris KværnøA.Singly Diagonally Implicit Runge-Kutta Methods with an Explicit First StageBIT200444489502210601210.1023/B:BITN.0000046811.70614.38 ProtheroA.RobinsonA.On the Stability and Accuracy of One-Step Methods for Solving Stiff Systems of Ordinary Differential EquationsMath. Comp.19742414516233179310.1090/S0025-5718-1974-0331793-2 KulikovG.Yu.WeinerR.Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error ControlSIAM J. Sci. Comp.201032416951723267129310.1137/0907648401215.65125 KurdiM.Stable High Order Methods for Time Discretization of Stiff Differential Equations1974CaliforniaUniversity of California CashJ.R.On a Class of Implicit Runge-Kutta ProceduresJ. Inst. Math. Appl.19771945547043659710.1093/imamat/19.4.4550364.65051 NørsettS.P.Runge-Kutta Methods with Multiple Real Eigenvalue OnlyBIT19761638839310.1007/BF01932722 Kulikov, G.Yu. and Shindin, S.K., On a Family of Cheap Symmetric One-Step Methods of Order Four, Alexandrov, V.N. et al., Eds., Computational Science-ICCS 2006, 6th Int. Conf., Reading, UK, May 28–31, 2006, part 1, 2006, vol. 3991, pp. 781–785. KulikovG.Yu.KhrustalevaE.Yu.On Automatic Control of Mesh Size and Order in Implicit Single-Step Extrapolation MethodsSib. Zh. Vych. Mat. Mat. Fiz.2008489158016062500112 BurrageK.A Special Family of Runge-Kutta Methods for Solving Stiff Differential EquationsBIT197818224148345810.1007/BF019477410384.65034 CashJ.R.SinghalA.Mono-Implicit Runge-Kutta Formulae for the Numerical Integration of Stiff Differential SystemsIMA J. Numer. Anal.1982221122766859310.1093/imanum/2.2.2110488.65031 Nørsett, S.P., Semi-Explicit Runge-Kutta Methods, Report, Trondheim: University of Trondheim, 1974, no. 6/74. ButcherJ.C.CashJ.R.Towards Efficient Runge-Kutta Methods for Stiff SystemsSIAM J. Numer. Anal.199027753761104126110.1137/07270440702.65072 KulikovG.Yu.ShindinS.K.Adaptive Nested Implicit Runge-Kutta Formulas of Gauss TypeAppl. Numer. Math.200959707722249228610.1016/j.apnum.2008.03.0191161.65055 Kulikov, G.Yu. and Shindin, S.K., Numerical Tests with Gauss-Type Nested Implicit Runge-Kutta Formulas, Y Shi et al., Eds., Computational Science-ICCS 2007, 7th Int. Conf., Beijing, China, May 27–30, 2007, part 1, 2007, vol. 4487, pp. 136–143. BurrageK.ButcherJ.C.ChipmanF.H.An Implementation of Singly Implicit Runge-Kutta MethodsBIT19802032634059521310.1007/BF019327740456.65040 DahlquistG.A Special Stability Problem for Linear Multistep MethodsBIT19633274317047710.1007/BF019635320123.11703 CashJ.R.On a Note of the Computational Aspects of a Class of Implicit Runge-Kutta ProceduresJ. Inst. Math. Appl.19772042544145540810.1093/imamat/20.4.4250386.65033 G.Yu. Kulikov (4092_CR4) 2009; 59 G.Yu. Kulikov (4092_CR2) 2007; 22 R. Alexander (4092_CR16) 1977; 14 W.M. Bokhoven van (4092_CR31) 1980; 20 J.C. Butcher (4092_CR9) 2003 M. Kurdi (4092_CR19) 1974 E. Hairer (4092_CR8) 1999 4092_CR21 K. Burrage (4092_CR24) 1980; 20 G. Dahlquist (4092_CR13) 1963; 3 4092_CR1 A. Prothero (4092_CR22) 1974; 24 4092_CR3 J.C. Butcher (4092_CR26) 1990; 27 J.R. Cash (4092_CR29) 1977; 19 A. Kværnø (4092_CR20) 2004; 44 J.R. Cash (4092_CR30) 1977; 20 G.Yu. Kulikov (4092_CR10) 2008; 48 R. Alt (4092_CR17) 1971 J.C. Butcher (4092_CR27) 2000; 34 S.P. Nørsett (4092_CR28) 1977; 17 M. Crouzeix (4092_CR18) 1975 S.P. Nørsett (4092_CR25) 1976; 16 E. Hairer (4092_CR7) 1990 4092_CR12 K. Burrage (4092_CR23) 1978; 18 J.R. Cash (4092_CR32) 1982; 2 G.Yu. Kulikov (4092_CR5) 2009; 24 K. Dekker (4092_CR6) 1988 J.C. Butcher (4092_CR14) 1976; 16 T.A. Bickart (4092_CR15) 1977; 14 G.Yu. Kulikov (4092_CR11) 2010; 32 |
| References_xml | – reference: KurdiM.Stable High Order Methods for Time Discretization of Stiff Differential Equations1974CaliforniaUniversity of California – reference: ProtheroA.RobinsonA.On the Stability and Accuracy of One-Step Methods for Solving Stiff Systems of Ordinary Differential EquationsMath. Comp.19742414516233179310.1090/S0025-5718-1974-0331793-2 – reference: NørsettS.P.Runge-Kutta Methods with Multiple Real Eigenvalue OnlyBIT19761638839310.1007/BF01932722 – reference: BickartT.A.An Efficient Solution Process for Implicit Runge-Kutta MethodsSIAM J. Numer. Anal.1977141022102745889310.1137/07140690368.65037 – reference: KulikovG.Yu.Automatic Error Control in the Gauss-Type Nested Implicit Runge-Kutta Formula of Order 6Russ. J. Numer. Anal. Math. Model.2009242123144252273010.1515/RJNAMM.2009.0091168.65368 – reference: HairerE.NersettS.WannerG.Reshenie obyknovennykh differentsialnykh uravnenii. Nezhostkie zadachi1990MoscowMir(Solving Ordinary Differential Equations: Nonstiff Problems) – reference: Nørsett, S.P., Semi-Explicit Runge-Kutta Methods, Report, Trondheim: University of Trondheim, 1974, no. 6/74. – reference: BurrageK.A Special Family of Runge-Kutta Methods for Solving Stiff Differential EquationsBIT197818224148345810.1007/BF019477410384.65034 – reference: Kulikov, G.Yu. and Shindin, S.K., Numerical Tests with Gauss-Type Nested Implicit Runge-Kutta Formulas, Y Shi et al., Eds., Computational Science-ICCS 2007, 7th Int. Conf., Beijing, China, May 27–30, 2007, part 1, 2007, vol. 4487, pp. 136–143. – reference: van BokhovenW.M.Efficient Higher Order Implicit One-Step Methods for Integration of Stiff Differential EquationsBIT198020344356997410.1007/BF019335830448.65047 – reference: BurrageK.ButcherJ.C.ChipmanF.H.An Implementation of Singly Implicit Runge-Kutta MethodsBIT19802032634059521310.1007/BF019327740456.65040 – reference: AltR.Methodes A-Stables pour l’Integration de Systemes Differentiels Mal Conditionnes1971ParisUniversite Paris – reference: KulikovG.Yu.MerkulovA.I.ShindinS.K.Asymptotic Error Estimate for General Newton-Type Methods and Its Application to Differential EquationsRuss. J. Numer. Anal. Math. Model.2007226567590237503210.1515/rnam.2007.0291145.65033 – reference: Dahlquist, G., Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, Trans. Royal Inst. Technol., 1959, no. 130, pp. 1–87. – reference: HairerE.WannerG.Reshenie obyknovennykh differentsialnykh uravnenii. Zhostkie i differentsialno-algebraicheskie zadachi1999MoscowMir(Solving Ordinary Differential Equations: Stiff and Differential-Algebraic Problems) – reference: CrouzeixM.Sur l’Approximation des Equations Differentielles Operationnelles Lineaires par de Methodes de Runge-Kutta1975ParisUniversite Paris – reference: ButcherJ.C.CashJ.R.Towards Efficient Runge-Kutta Methods for Stiff SystemsSIAM J. Numer. Anal.199027753761104126110.1137/07270440702.65072 – reference: ButcherJ.C.On the Implementation of Implicit Runge-Kutta MethodsBIT19761623724048874610.1007/BF019322650336.65037 – reference: KulikovG.Yu.ShindinS.K.Adaptive Nested Implicit Runge-Kutta Formulas of Gauss TypeAppl. Numer. Math.200959707722249228610.1016/j.apnum.2008.03.0191161.65055 – reference: ButcherJ.C.ChenD.J.A New Type of Singly Implicit Runge-Kutta MethodsAppl. Numer. Math.200034179188177042210.1016/S0168-9274(99)00126-90954.65058 – reference: NørsettS.P.WolfbrandtA.Attainable Order of Rational Approximations to the Exponential Function with Only Real PolesBIT19771720020810.1007/BF01932291 – reference: DahlquistG.A Special Stability Problem for Linear Multistep MethodsBIT19633274317047710.1007/BF019635320123.11703 – reference: CashJ.R.On a Note of the Computational Aspects of a Class of Implicit Runge-Kutta ProceduresJ. Inst. Math. Appl.19772042544145540810.1093/imamat/20.4.4250386.65033 – reference: KulikovG.Yu.KhrustalevaE.Yu.On Automatic Control of Mesh Size and Order in Implicit Single-Step Extrapolation MethodsSib. Zh. Vych. Mat. Mat. Fiz.2008489158016062500112 – reference: KværnøA.Singly Diagonally Implicit Runge-Kutta Methods with an Explicit First StageBIT200444489502210601210.1023/B:BITN.0000046811.70614.38 – reference: CashJ.R.SinghalA.Mono-Implicit Runge-Kutta Formulae for the Numerical Integration of Stiff Differential SystemsIMA J. Numer. Anal.1982221122766859310.1093/imanum/2.2.2110488.65031 – reference: Kulikov, G.Yu. and Shindin, S.K., On a Family of Cheap Symmetric One-Step Methods of Order Four, Alexandrov, V.N. et al., Eds., Computational Science-ICCS 2006, 6th Int. Conf., Reading, UK, May 28–31, 2006, part 1, 2006, vol. 3991, pp. 781–785. – reference: CashJ.R.On a Class of Implicit Runge-Kutta ProceduresJ. Inst. Math. Appl.19771945547043659710.1093/imamat/19.4.4550364.65051 – reference: DekkerK.VerverJ.Ustoichivost’ metodov Runge-Kutty dlya zhyostkikh nelineinykh differentsialnykh uravnenii1988MoscowMir(Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations) – reference: KulikovG.Yu.WeinerR.Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error ControlSIAM J. Sci. Comp.201032416951723267129310.1137/0907648401215.65125 – reference: ButcherJ.C.Numerical Methods for Ordinary Differential Equations2003ChichesterWiley10.1002/04708682791040.65057 – reference: AlexanderR.Diagonally Implicit Runge-Kutta Methods for Stiff ODEsSIAM J. Numer. Anal.1977141006102445889010.1137/07140680374.65038 – volume: 34 start-page: 179 year: 2000 ident: 4092_CR27 publication-title: Appl. Numer. Math. doi: 10.1016/S0168-9274(99)00126-9 – volume-title: Stable High Order Methods for Time Discretization of Stiff Differential Equations year: 1974 ident: 4092_CR19 – ident: 4092_CR12 – ident: 4092_CR1 doi: 10.1007/11758501_104 – volume: 44 start-page: 489 year: 2004 ident: 4092_CR20 publication-title: BIT doi: 10.1023/B:BITN.0000046811.70614.38 – volume: 2 start-page: 211 year: 1982 ident: 4092_CR32 publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/2.2.211 – volume-title: Numerical Methods for Ordinary Differential Equations year: 2003 ident: 4092_CR9 doi: 10.1002/0470868279 – volume: 20 start-page: 34 year: 1980 ident: 4092_CR31 publication-title: BIT doi: 10.1007/BF01933583 – volume: 22 start-page: 567 issue: 6 year: 2007 ident: 4092_CR2 publication-title: Russ. J. Numer. Anal. Math. Model. doi: 10.1515/rnam.2007.029 – ident: 4092_CR21 – volume: 18 start-page: 22 year: 1978 ident: 4092_CR23 publication-title: BIT doi: 10.1007/BF01947741 – volume: 3 start-page: 27 year: 1963 ident: 4092_CR13 publication-title: BIT doi: 10.1007/BF01963532 – volume: 19 start-page: 455 year: 1977 ident: 4092_CR29 publication-title: J. Inst. Math. Appl. doi: 10.1093/imamat/19.4.455 – volume-title: Methodes A-Stables pour l’Integration de Systemes Differentiels Mal Conditionnes year: 1971 ident: 4092_CR17 – volume: 14 start-page: 1022 year: 1977 ident: 4092_CR15 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0714069 – volume: 20 start-page: 326 year: 1980 ident: 4092_CR24 publication-title: BIT doi: 10.1007/BF01932774 – volume: 32 start-page: 1695 issue: 4 year: 2010 ident: 4092_CR11 publication-title: SIAM J. Sci. Comp. doi: 10.1137/090764840 – volume: 20 start-page: 425 year: 1977 ident: 4092_CR30 publication-title: J. Inst. Math. Appl. doi: 10.1093/imamat/20.4.425 – volume: 16 start-page: 388 year: 1976 ident: 4092_CR25 publication-title: BIT doi: 10.1007/BF01932722 – volume-title: Reshenie obyknovennykh differentsialnykh uravnenii. Zhostkie i differentsialno-algebraicheskie zadachi year: 1999 ident: 4092_CR8 – volume-title: Reshenie obyknovennykh differentsialnykh uravnenii. Nezhostkie zadachi year: 1990 ident: 4092_CR7 – volume: 59 start-page: 707 year: 2009 ident: 4092_CR4 publication-title: Appl. Numer. Math. doi: 10.1016/j.apnum.2008.03.019 – volume: 14 start-page: 1006 year: 1977 ident: 4092_CR16 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0714068 – volume: 24 start-page: 123 issue: 2 year: 2009 ident: 4092_CR5 publication-title: Russ. J. Numer. Anal. Math. Model. doi: 10.1515/RJNAMM.2009.009 – volume: 16 start-page: 237 year: 1976 ident: 4092_CR14 publication-title: BIT doi: 10.1007/BF01932265 – volume: 27 start-page: 753 year: 1990 ident: 4092_CR26 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0727044 – ident: 4092_CR3 doi: 10.1007/978-3-540-72584-8_18 – volume-title: Ustoichivost’ metodov Runge-Kutty dlya zhyostkikh nelineinykh differentsialnykh uravnenii year: 1988 ident: 4092_CR6 – volume: 24 start-page: 145 year: 1974 ident: 4092_CR22 publication-title: Math. Comp. doi: 10.1090/S0025-5718-1974-0331793-2 – volume: 48 start-page: 1580 issue: 9 year: 2008 ident: 4092_CR10 publication-title: Sib. Zh. Vych. Mat. Mat. Fiz. – volume-title: Sur l’Approximation des Equations Differentielles Operationnelles Lineaires par de Methodes de Runge-Kutta year: 1975 ident: 4092_CR18 – volume: 17 start-page: 200 year: 1977 ident: 4092_CR28 publication-title: BIT doi: 10.1007/BF01932291 |
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| Snippet | Automatic global error control based on a combined control of step size and order presented by Kulikov and Khrustaleva in 2008 is investigated. Special... |
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| SubjectTerms | Mathematics Mathematics and Statistics Numerical Analysis |
| Title | On global error control in nested implicit Runge-Kutta methods of the gauss type |
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| Volume | 4 |
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