Skip to main content
SpringerLink
Log in
Book cover

International Conference on Differential & Difference Equations and Applications

ICDDEA 2017: Differential and Difference Equations with Applications pp 259–270Cite as

Interval Difference Methods for Solving the Poisson Equation

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 230))

Abstract

In the paper we resemble interval difference method of second order designed by us earlier and present new, fourth order interval difference methods for solving the Poisson equation with Dirichlet boundary conditions. Interval solutions obtained contain all possible numerical errors. Numerical solutions presented confirm the fact that the exact solutions are within the resulting intervals.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    In our opinion, it is rather impossible to obtain a theoretical proof of this fact.

References

  1. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)

    MATH  Google Scholar 

  2. Burden, R.L., Faires, J.D.: Numerical Analysis, 3rd edn. Weber & Schmidt Publishers, Prindle (1985)

    MATH  Google Scholar 

  3. Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: Numerical Toolbox for Verified Computing I. Basic Numerical Problems, Theory, Algorithms, and Pascal-XSC Programs. Springer, Berlin (1993)

    MATH  Google Scholar 

  4. Hansen, E.R.: Topics in Interval Analysis. Oxford University Press, London (1969)

    MATH  Google Scholar 

  5. Harfash, A.J., Jalob, H.A.: Sixth and fourth order compact finite difference schemes for two and three dimension poisson equation with two methods to derive these schemes. Basrah J. Sci. 2, 1–20 (2006)

    Google Scholar 

  6. Hoffmann, T., Marciniak, A.: Solving the poisson equation by an interval difference method of the second order. Comput. Methods Sci. Technol. 19(1), 13–21 (2013)

    Article  Google Scholar 

  7. Hoffmann, T., Marciniak, A.: Solving the generalized poisson equation in proper and directed interval arithmetic. Comput. Methods Sci. Technol. 22(4), 225–232 (2016)

    Article  Google Scholar 

  8. Hoffmann, T., Marciniak, A., Szyszka, B.: Interval versions of central difference method for solving the poisson equation in proper and directed interval arithmetic. Found. Comput. Decis. Sci. 38(3), 193–206 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Kincaid, D.R., Cheney, E.W.: Numerical analysis: mathematics of scientific computing, 3rd edn. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  10. Marciniak, A.: An interval difference method for solving the poisson equation the first approach. Pro Dialog 24, 49–61 (2008)

    Google Scholar 

  11. Marciniak, A.: On Realization of Floating-Point Directed Interval Arithmetic (2012). http://www.cs.put.poznan.pl/amarciniak/KONF-referaty/DirectedArithmetic.pdf

  12. Marciniak, A.: Interval Arithmetic Unit (2016). http://www.cs.put.poznan.pl/amarciniak/IAUnits/IntervalArithmetic32and64.pas

  13. Marciniak, A.: Delphi Pascal Programs for Interval Difference Methods for Solving the Poisson Equation (2017). http://www.cs.put.poznan.pl/amarciniak/IDM-PoissonEqn-Examples/

  14. Meis, T., Marcowitz, U.: Numerical Solution of Partial Differential Equations. Springer, Berlin (1981)

    Book  Google Scholar 

  15. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  16. Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)

    Book  Google Scholar 

  17. Nakao, M.T.: On verified computations of solutions for nonlinear parabolic problems. IEICE Nonlinear Theory Appl. 5(3), 320–338 (2014)

    Article  Google Scholar 

  18. Popova, E.D.: Extended interval arithmetic in IEEE floating-point environment. Interval Comput. 4, 100–129 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Zhang, J.: Multigrid method and fourth-order compact scheme for 2D poisson equation with unequal mesh- size discretization. J. Comput. Phys. 179, 170–179 (2002)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computing Science, Poznan University of Technology, Piotrowo 2, 60-965, Poznan, Poland

    Andrzej Marciniak

  2. Department of Computer Science, Higher Vocational State School in Kalisz, Poznanska 201-205, 62-800, Kalisz, Poland

    Andrzej Marciniak

  3. Poznan Supercomputing and Networking Center, Jana Pawla II 10, 61-139, Poznan, Poland

    Tomasz Hoffmann

Authors
  1. Andrzej Marciniak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomasz Hoffmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrzej Marciniak .

Editor information

Editors and Affiliations

  1. Departamento de Ciências Exactas e Engenharia, Academia Militar, Amadora, Portugal

    Sandra Pinelas

  2. Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain

    Tomás Caraballo

  3. Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei, China

    Peter Kloeden

  4. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee, USA

    John R. Graef

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Marciniak, A., Hoffmann, T. (2018). Interval Difference Methods for Solving the Poisson Equation. In: Pinelas, S., Caraballo, T., Kloeden, P., Graef, J. (eds) Differential and Difference Equations with Applications. ICDDEA 2017. Springer Proceedings in Mathematics & Statistics, vol 230. Springer, Cham. https://doi.org/10.1007/978-3-319-75647-9_21

Download citation

Publish with us

Policies and ethics

Search

Navigation