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Abstrakty
In a number of our previous papers we have presented interval versions of Runge-Kutta methods (explicit and implicit) in which the step size was constant. Such an approach has required to choose manually the step size in order to ensure an interval enclosure to the solution with the smallest width. In this paper we propose an algorithm for choosing automatically the step size which guarantees the best (i.e., the tiniest) interval enclosure. This step size is determined with machine accuracy.
Rocznik
Tom
Strony
17--29
Opis fizyczny
Bibliogr. 29 poz., rys.
Twórcy
autor
- Institute of Computing Science Poznan University of Technology Piotrowo 2, 60-965 Poznan, Poland
- Department of Computer Science State University of Applied Sciences in Kalisz Poznanska 201-205, 62-800 Kalisz, Poland
autor
- Institute of Mathematics Poznan University of Technology Piotrowo 3A, 60-965 Poznan, Poland
Bibliografia
- [1] M. Berz, G. Hoffstätter, Computation and Application of Taylor Polynomials with Interval Remainder Bounds, Reliable Computing 4(1), 83–97 (1998).
- [2] M. Berz, K. Makino, Performance of Taylor Model Methods for Validated Integration of ODEs, [In:] J. Dongarra, K. Madsen, J. Wasniewski (eds.) Applied Parallel Computing. State of the Art in Scientific Computing, Lecture Notes in Computer Science 3732, 65–73 (2005).
- [3] J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester (1987).
- [4] G.F. Corliss, R. Rihm, Validating an A Priori Enclosure Using High-Order Taylor Series, [In:] Scientific Computing, Computer Arithmetic, and Validated Numerics, 228–238, Akademie Verlag (1996).
- [5] W.H. Enright, J.D. Pryce, Two Fortran Packages for Assessing Initial Value Methods, ACM Transactions on Mathematical Software 13(1), 1–27 (1987).
- [6] K. Gajda, M. Jankowska, A. Marciniak, B. Szyszka, A Survey of Interval Runge-Kutta and Multistep Methods for Solving the Initial Value Problem, [In:] R. Wyrzykowski, J. Dongarra, K. Karczewski, J. Wasniewski (eds.) Parallel Processing and Applied Mathematics, Lecture Notes in Computer Science 4967, 1361–1371. Springer-Verlag, Berlin (2007).
- [7] K. Gajda, A. Marciniak, B. Szyszka, Three-and Four-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 6, 41–59 (2000).
- [8] E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations I – Nonstiff Problems, Springer–Verlag, Berlin (1987).
- [9] R. Hammer, M. Hocks, U. Kulisch, D. Ratz, Numerical Toolbox for Verified Computing I. Basic Numerical Problems, Theory, Algorithms, and Pascal-XSC Programs, SpringerVerlag, Berlin (1993).
- [10] E.R. Hansen, Topics in Interval Analysis, Oxford University Press, London (1969).
- [11] K.R. Jackson, N.S. Nedialkov, Some Recent Advances in Validated Methods for IVPs for ODEs, Applied Numerical Mathematics 42, 269–284 (2002).
- [12] M. Jankowska, A. Marciniak, Implicit Interval Methods for Solving the Initial Value Problem, Computational Methods in Science and Technology 8(1), 17–30 (2002).
- [13] M. Jankowska, A. Marciniak, On Explicit Interval Methods of Adams-Bashforth Type, Computational Methods in Science and Technology 8(2), 46–57 (2002).
- [14] M. Jankowska, A. Marciniak, On Two Families of Implicit Interval Methods of Adams-Moulton Type, Computational Methods in Science and Technology 12(2), 109–113 (2006).
- [15] S.A. Kalmykov, J.I. Shokin, Z.H. Juldashev, Solving Ordinary Differential Equations by Interval Methods [in Russian], Doklady AN SSSR 230(6) (1976).
- [16] A. Marciniak, Implicit Interval Methods for Solving the Initial Value Problem, Numerical Algorithms 37, 241–251 (2004).
- [17] A. Marciniak, Selected Interval Methods for Solving the Initial Value Problem, Publishing House of Poznan University of Technology, Poznan (2009). http://www.cs.put.poznan.pl/amarciniak/IMforIVP-book/IMforIVP.pdf
- [18] A. Marciniak, Interval Arithmetic Unit (2016). http://www.cs .put.poznan.pl/amarciniak/IAUnits/IntervalArithmetic32and64.pas
- [19] A. Marciniak, Delphi Pascal Programs for Step Size Control in Interval Runge-Kutta Methods (2017). http://www.cs.put.poznan.pl/amarciniak/VSSIRKM-Examples
- [20] A. Marciniak, M.A. Jankowska, Interval Versions for Special Kinds of Explicit Linear Multistep Methods (in review, available from the authors).
- [21] A. Marciniak, M.A. Jankowska, Interval Versions of Milne’s Multistep Methods, Numerical Algorithms 79(1), 87–105 (2018).
- [22] A. Marciniak, M.A. Jankowska, T. Hoffmann, On Interval Predictor-Corrector Methods, Numerical Algorithms 77(3), 777–808 (2017).
- [23] A. Marciniak, B. Szyszka, One-and Two-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 5, 53–65 (1999).
- [24] A. Marciniak, B. Szyszka, T. Hoffmann, An Interval Version of Kuntzmann-Butcher Method for Solving the Initial Value Problem (in review, available from the authors).
- [25] R.E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs (1966).
- [26] R.E. Moore, Methods and Applications of Interval Analysis, SIAM Society for Industrial & Applied Mathematics, Philadelphia (1979).
- [27] N.S. Nedialkov, VNODE-LP – a Validated Solver for Initial Value Problems in Ordinary Differential Equations, Tech. Rep. CAS 06-06-NN, Department of Computing and Software, McMaster University, Hamilton (2006).
- [28] N.S. Nedialkov, K.R. Jackson, G.F. Corliss, Validated Solutions of Initial Value Problems for Ordinary Differential Equations, Applied Mathematics and Computation 105(1), 21–68 (1999).
- [29] Y.I. Shokin, Interval Analysis [in Russian], Nauka, Novosibirsk (1981).
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
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