دانشگاه تبریز
فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری
پایگاه استنادی علوم جهان اسلام (ISC)

 آمار

تعداد نشریات45
تعداد شماره‌ها1,370
تعداد مقالات16,844
تعداد مشاهده مقاله54,231,594
تعداد دریافت فایل اصل مقاله16,943,340
Marciniak, Andrzej, Szyszka, Barbara, Hoffmann, Tomasz. (1400). An interval version of the Kuntzmann-Butcher method for solving the initial value problem. سامانه مدیریت نشریات علمی, 10(1), 44-60. doi: 10.22034/cmde.2020.39203.1720
Andrzej Marciniak; Barbara Szyszka; Tomasz Hoffmann. "An interval version of the Kuntzmann-Butcher method for solving the initial value problem". سامانه مدیریت نشریات علمی, 10, 1, 1400, 44-60. doi: 10.22034/cmde.2020.39203.1720
Marciniak, Andrzej, Szyszka, Barbara, Hoffmann, Tomasz. (1400). 'An interval version of the Kuntzmann-Butcher method for solving the initial value problem', سامانه مدیریت نشریات علمی, 10(1), pp. 44-60. doi: 10.22034/cmde.2020.39203.1720
Marciniak, Andrzej, Szyszka, Barbara, Hoffmann, Tomasz. An interval version of the Kuntzmann-Butcher method for solving the initial value problem. سامانه مدیریت نشریات علمی, 1400; 10(1): 44-60. doi: 10.22034/cmde.2020.39203.1720

An interval version of the Kuntzmann-Butcher method for solving the initial value problem

Computational Methods for Differential Equations
مقاله 4، دوره 10، شماره 1، فروردین 2022، صفحه 44-60    اصل مقاله (169.19 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2020.39203.1720
نویسندگان
Andrzej Marciniak1، 2؛ Barbara Szyszka* 3؛ Tomasz Hoffmann4
1Institute of Computing Science, Poznan University of Technology, Piotrowo 2, 60-965 Poznan, Poland.
2Department of Computer Science, Higher Vocational State School in Kalisz Poznanska 201-205, 62-800 Kalisz, Poland.
3Institute of Mathematics, Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland.
4Poznan Supercomputing and Networking Center, Jana Pawła II 10, 61-139 Poznan, Poland.
چکیده
The Kutzmann-Butcher method is the unique implicit four-stage Runge-Kutta method of order 8. In many problems in ordinary differential equations this method realized in floating-point arithmetic gives quite good approximations to the exact solutions, but the results obtained do not contain any information on rounding errors, representation errors and the error of the method. Thus, we describe an interval version of this method, which realized in floating-point interval arithmetic gives approximations (enclosures in the form of an interval) containing all these errors. The described method can also include data uncertainties in the intervals obtained.
کلیدواژه‌ها
Initial value problem؛ Runge-Kutta methods؛ Kuntzmann-Butcher method؛ Interval Runge-Kutta methods؛ Floating-point interval arithmetic
مراجع
  • [1]          G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, 1983, Doi: 10.1016/C2009-0-21898-8.
  • [2]          H. Bauch, On the iterative inclusion of solutions in initial-value problems for ordinary differential equations, Computing, 22 (1979), 339–354, Doi: 10.1007/BF02265314.
  • [3]          M. Berz and G. Hoffst¨atter, Computation and application of Taylor polynomials with interval remainder bounds, Reliable Computing, 4 (1998), 83–97, Doi: 10.1023/A:1009958918582.
  • [4]          M. Berz and K. Makino,  Performance of Taylor  model methods for validated integration of ODEs,  in:  J. Don-  garra, and K. Madsen, and J. Wasniewski, J. (eds) Applied Parallel Computing. State of the Art in Scientific Computing, (PARA 2004), Lecture Notes in Computer Science, Springer, Berlin, Heidelberg 3732 (2006), 65–73, Doi: 10.1007/11558958-8.
  • [5]          J. C. Butcher, Implicit Runge-Kutta processes,  Mathematics of Computation, American Mathematical Society, 18 (1964), 50–64, Doi: 10.1090/S0025-5718-1964-0159424-9.
  • [6]          , J. C. Butcher, The numerical analysis of Ordinary Differential Equations: Runge-Kutta and general linear methods, Wiley-Interscience, New York, 1987.
  • [7]          G. F. Corliss and R. Rihm, Validating an a priori enclosure using high-order Taylor series, in Scientific Comput-  ing, Computer Arithmetic, and Validated Numerics 90 (SCAN-95), Mathematical Research, Akademie Verlag, 1996, 228–238.
  • [8]          W. H. Enright, and T. E. Hull, and B. Lindberg, Comparing numerical methods for stiff systems of O.D.E:s, BIT Numerical Mathematics, Springer 15 (1975), 10–48, Doi: 10.1007/BF01932994.
  • [9]          W. H. Enright, and J. D. Pryce, Two FORTRAN packages for assessing initial value methods, ACM Transactions    on Mathematical Software (TOMS), ACM, 13(1) (1987), 1–27, Doi: 10.1145/23002.27645.
  • [10]        K. Gajda, M. Jankowska, A. Marciniak, and B. Szyszka, A survey of interval  Runge-Kutta  and  multistep  methods for solving the Initial Value Problem, in R. Wyrzykowski, and J. Dongarra, and K. Karczewski, and J. Wasniewski, (eds) Parallel Processing and Applied Mathematics, (PPAM 2007), Lecture Notes in Computer Science, Springer, Berlin, Heidelberg 4967 (2008), 1361–1371, Doi: 10.1007/978-3-540-68111-3-144.
  • [11]        K. Gajda, A. Marciniak, and B. Szyszka, Three-and four-stage implicit interval methods of Runge-Kutta type, Computational Methods in Science and Technology, 6 (2000), 41–59.
  • [12]        E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I – nonstiff problems , Springer- Verlag Berlin Heidelberg, 8 1993, Doi: 10.1007/978-3-540-78862-1.
  • [13]        E. Hairer and G. Wanner, Solving Ordinary Differential Equations II – stiff and differential - algerbaic problems, Springer-Verlag Berlin Heidelberg, 14 1996, Doi: 10.1007/978-3-642-05221-7.
  • [14]        R. Hammer, M. Hocks, U.  Kulisch,  and  D.  Ratz,  Numerical  toolbox  for  verified  computing  I.  Basic  numeri-  cal problems, theory, algorithms, and Pascal-XSC programs, Springer-Verlag Berlin Heidelberg, 21 1993, Doi: 10.1007/978-3-642-78423-1.
  • [15]        E. R. Hansen, Topics in interval analysis, Oxford University Press, London, 1969.
  • [16]        K. R. Jackson and N. S. Nedialkov, Some recent advances in validated methods for IVPs for ODEs, Applied Numerical Analysis and Computational Mathematics, 42 (2002), 269–284.
  • [17]        M. Jankowska and A. Marciniak, Implicit interval multistep methods for solving the initial value problem, Com- putational Methods in Science and Technology, 8(1) (2002), 17–30.
  • [18]        M. Jankowska and A. Marciniak, On explicit interval methods of Adams-Bashforth type,  Computational Methods   in Science and Technology, 8(2) (2002), 46–57.
  • [19]        M. Jankowska and A. Marciniak, On two families of implicit interval methods of Adams-Moulton type, Compu- tational Methods in Science and Technology, 12(2) (2006), 109–114.
  • [20]        S. A. Kalmykov, Y. I. Shokin, and Z. H. Juldashev, On the solution of ordinary differential equations by interval methods [in Russian], Doklady Akad. Nauk SSSR 230, 6 (1976), 1267–1270.
  • [21]        R. J. Lohner, Computation of guaranteed enclosures for the solutions of ordinary  initial  and  boundary  value  problems, in J. R. Cash, and I. Gladwell, (eds), Computational Ordinary Differential Equations, Clarendon Press, Oxford, 1992, 425–435.
  • [22]        A. Marciniak, Implicit interval methods for solving the initial value problem, Numerical Algorithms, Springer, 37 (2004), 241–251.
  • [23]        A. Marciniak, Multistep interval methods of Nystr¨om and Milne-Simpson types, Computational Methods in Science and Technology, 13(1) (2007), 23–40.
  • [24]        A. Marciniak, On multistep interval methods for solving the initial value problem, Journal of Computational and Applied Mathematics, 199(2) (2007), 229–237.
  • [25]        A. Marciniak, Selected interval methods for solving the initial value problem, http://www.cs.put.poznan.pl/amarciniak/IMforIVP-book/IMforIVP.pdf, Publishing House of Poznan Uni- versity of Technology, Poznan, 2009.
  • [26]        A. Marciniak, Delphi Pascal programs for an interval Kutzmann-Butcher method, http://www.cs.put.poznan.pl/amarciniak/IKBM-Examples/, 2016.
  • [27]        A. Marciniak, Interval arithmetic module, http://www.cs.put.poznan.pl/amarciniak/IAUnits/IntervalArithmetic32and64.pas, 2016.
  • [28]        A. Marciniak, M. Jankowska, and T. Hoffmann, On interval predictor-corrector methods, Numerical Algorithms, 75(3) (2017), 777–808.
  • [29]        A. Marciniak and B. Szyszka, One-and two-stage implicit interval methods of Runge-Kutta type, Computational Methods in Science and Technology, 5 (1999), 53–65.
  • [30]        R. E. Moore, The automatic analysis and control of error in digital computation  based  on  the  use  of  interval  numbers, in L. B. Rall (ed) Error in Digital Computation, John Wiley & Sons, New York, 1 1965, 61–130.
  • [31]        R. E. Moore, Interval analysis, Prentice-Hall, Englewood Cliffs, 1966.
  • [32]        R. E. Moore, Methods and applications of interval analysis, SIAM Studies in applied and numerical mathematics, Soc. for Industrial & Applied Math, Philadelphia, 1979, Doi: 10.1137/1.9781611970906.
  • [33]        N. S. Nedialkov, Interval tools for ODEs and DAEs, Technical Report CAS 06-09-NN, Department of Computing and Software, McMaster University, Hamilton, 2006.
  • [34]        N. S. Nedialkov, VNODE-LP - a validated solver for initial value problems in Ordinary Differential Equations, Technical Report CAS 06-06-NN, Department of Computing and Software, McMaster University, Hamilton, 2006.
  • [35]        N. S. Nedialkov, K. R. Jackson, and G. F. Corliss, Validated solutions of initial value problems for Ordinary Differential Equations, Applied Mathematics and Computation, Elsevier, 105(1) (1999), 21–68.
  • [36]        K. Nickel, Using interval methods for the numerical solution of ODE’s, ZAMM-Journal of applied Mathematics and Mechanics/Zeitschrift fu¨r Angewandte Mathematik und Mechanik, 66 (1986), 513–523.
  • [37]        Y. I. Shokin, Interval analysis [in Russian], Nauka, Novosibirsk, 1981.
آمار
تعداد مشاهده مقاله: 691
تعداد دریافت فایل اصل مقاله: 712


سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب