Evaluation of the Accuracy of the Solution to the Heat Conduction Problem with the Interval Method of Crank-Nicolson Type

• Autorzy: Jankowska M. A. , Sypniewska-Kamińska G. , Kamiński H.
• Języki publikacji: PL

Abstrakty

EN

The paper deals with the interval method of Crank-Nicolson type used for some initial-boundary value problem for the onedimensional heat conduction equation. The numerical experiments are directed at a short presentation of advantages of the interval solutions obtained in the floating-point interval arithmetic over the approximate ones. It is also shown how we can deal with errors that occur during computations in terms of interval analysis and interval arithmetic.

Słowa kluczowe

PL

metoda Cranka-Nicholsona; arytmetyka interwałowa; błędy obliczeniowe

EN

interval finite difference method of Crank-Nicholson type; interval arithmetic; computational errors

• Wydawca: Oficyna Wydawnicza Politechniki Białostockiej
• Czasopismo: Acta Mechanica et Automatica
• Rocznik: 2012
• Tom: Vol. 6, no. 1
• Strony: 36--43
• Opis fizyczny: Bibliogr. 22 poz., Wykr.

Twórcy

autor

Jankowska M. A.

autor

Sypniewska-Kamińska G.

autor

Kamiński H.

• Institute of Applied Mechanics, Faculty of Mechanical Engineering and Management, Poznan University of Technology Piotrowo 3, 60-965 Poznań, Poland,

Bibliografia

• 1. Anderson D.A., Tannehill J.C., Pletcher R.H. (1984), Computational Fluid Mechanics and Heat Transfer, Hemisphere Publishing Corporation, New York.
• 2. Jain M.K. (1984), Numerical Solution of Differential Equations - 2nd edition, John Wiley & Sons (Asia) Pte Ltd.
• 3. Jankowska M.A. (2006), Interval Multistep Methods of Adams type and their Implementation in the C++ Language, Ph.D. Thesis, Poznan University of Technology, Poznań.
• 4. Jankowska M.A. (2009a), C++ Library for Floating-Point Conversions. User and Reference Guide, Poznan University of Technology. Last updated 2009, available in Software at http://www.mjank.user.icpnet.pl/.
• 5. Jankowska M.A. (2009b), C++ Library for Floating-Point Interval Arithmetic. User and Reference Guide, Poznan University of Technology. Last updated 2009, available in Software at http://www.mjank.user.icpnet.pl/.
• 6. Jankowska M.A. (2010), Remarks on Algorithms Implemented in Some C++ Libraries for Floating-Point Conversions and Interval Arithmetic, Lecture Notes in Computer Science 6068, 436-445.
• 7. Jankowska M.A., Marciniak A., An Interval Finite Difference Method for Solving the One-Dimensional Heat Conduction Equation, Lecture Notes in Computer Science (accepted).
• 8. Jankowska M.A. (2012), An Interval Finite Difference Method of Crank-Nicolson Type for Solving the One-Dimensional Heat Conduction Equation with Mixed Boundary Conditions, Lecture Notes in Computer Science 7133, 157-167.
• 9. Jankowska M.A. (2011), Some Kind of the Error Term Approximation in Interval Method of Crank-Nicolson Type, Dynamical Systems. Nonlinear Dynamics and Control, Proceedings of DSTA 2011 – 11th Conference on Dynamical Systems – Theory and Applications, Łódź, Poland, 297-304.
• 10. Kamiński H., Stefaniak J., Sypniewska-Kamińska G. (2004), Some remarks on the method of fictitious of direct and inverse problems of heat conduction, Proceedings of the International Symposium on Trends in Continuum Physics, TRECOP 2004, 163-175.
• 11. Manikonda S., Berz M., Makino K. (2005), High-order verified solutions of the 3D Laplace equation, WSEAS Transactions on Computers 4 (11), 1604-1610.
• 12. Marciniak A. (2009), Selected Interval Methods for Solving the Initial Value Problem, Publishing House of Poznan University of Technology, Poznan.
• 13. Marciniak A. (2008), An Interval Difference Method for Solving the Poisson Equation - the First Approach, Pro Dialog 24, 49-61.
• 14. Marciniak A. (2012), An Interval Version of the Crank-Nicolson Method - the First Approach, Lecture Notes in Computer Science 7133, 120-126.
• 15. Marciniak A., Gregulec D., Kaczmarek J. (2000), The basic numerical procedures in Turbo pascal language (in polish), Wydawnictwo Nakom, Poznań.
• 16. Moore R.E. (1966), Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ.
• 17. Moore R.E., Kearfott R.B., Cloud M.J. (2009), Introduction to Interval Analysis, SIAM Philadelphia.
• 18. Nakao M.T. (2001), Numerical verification methods for solutions of ordinary and partial differential equations, Numerical Functional Analysis and Optimization 22 (3-4), 321-356.
• 19. Nagatou K., Hashimoto K., Nakao M.T. (2007), Numerical verification of stationary solutions for Navier-Stokes problems, Journal of Computational and Applied Mathematics 199 (2), 445-451.
• 20. Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.(2007), Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press.
• 21. Sunaga T. (1958), Theory of interval algebra and its application to numerical analysis, in: Research Association of Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuykai. Tokyo, Japan, 1958, Vol. 2, pp. 29-46 (547-564).
• 22. Watanabe Y., Yamamoto N., Nakao M.T. (1999), A Numerical Verification Method of Solutions for the Navier-Stokes Equations, Reliable Computing 5 (3), 347-357.