Skip to main content
SpringerLink
Log in

On the Sign-Stability of Finite Difference Solutions of Semilinear Parabolic Problems

  • Conference paper
Numerical Analysis and Its Applications (NAA 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5434))

Included in the following conference series:

Abstract

The sign-stability property is one of the important qualitative properties of the one-dimensional heat conduction equation, or more generally, of one-dimensional parabolic problems. This property means that the number of the spatial sign-changes of the solution function cannot increase in time. In this paper, sufficient conditions will be given that guarantee the fulfillment of a numerical analogue of the sign-stability for the finite difference solution of a semilinear parabolic problem. The results are demonstrated on a numerical test problem.

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979)

    MATH  Google Scholar 

  2. Faragó, I.: Nonnegativity of the Difference Schemes. Pure Math. Appl. 6, 38–56 (1996)

    MathSciNet  Google Scholar 

  3. Faragó, I., Horváth, R.: On the Connections Between the Qualitative Properties of the Numerical Solutions of Linear Parabolic Problems. SIAM J. Sci. Comput. 28, 2316–2336 (2006)

    Article  Google Scholar 

  4. Faragó, I., Horváth, R., Korotov, S.: Discrete Maximum Principle for Linear Parabolic Problems Solved on Hybrid Meshes. Appl. Num. Math. 53(2-4), 249–264 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gantmacher, F.R., Krein, M.G.: Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme. Akademie-Verlag, Berlin (1960)

    Google Scholar 

  6. Glashoff, K., Kreth, H.: Vorzeichenstabile Differenzenverfaren für parabolische Anfangsrandwertaufgaben. Numerische Mathematik 35, 343–354 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Horváth, R.: Maximum Norm Contractivity in the Numerical Solution of the One-Dimensional Heat Equation. Appl. Num. Math. 31, 451–462 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Horváth, R.: On the Sign-Stability of the Numerical Solution of the Heat Equation. Pure Math. Appl. 11, 281–291 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Horváth, R.: On the Sign-Stability of the Finite Difference Solutions of One-Dimensional Parabolic Problems. In: Boyanov, T., Dimova, S., Georgiev, K., Nikolov, G. (eds.) NMA 2006. LNCS, vol. 4310, pp. 458–465. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  10. Horváth, R.: On the Sign-Stability of Numerical Solutions of One-Dimensional Parabolic Problems. Appl. Math. Model. 32(8), 1570–1578 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pólya, G.: Qualitatives über Wärmeausgleich. Ztschr. f. angew. Math. und Mech. 13, 125–128 (1933)

    Article  MATH  Google Scholar 

  12. Schoenberg, I.J.: Über variationsvermindernde lineare Transformationen. Math. Z. 32 (1930)

    Google Scholar 

  13. Sturm, Ch.: Journal de Math. pures et appliquées. 1, 373–444 (1836)

    Google Scholar 

  14. Tabata, M.: A Finite Difference Approach to the Number of Peaks of Solutions for Semilinear Parabolic Problems. J. Math. Soc. Japan 32, 171–192 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Budapest University of Technology and Economics, Egry J. u. 1, 1111, Budapest, Hungary

    Róbert Horváth

Authors
  1. Róbert Horváth
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. Bl. 25-A, 1113, Sofia, Bulgaria

    Svetozar Margenov

  2. FNSE, Department of Numerical Methods and Statistics, University of Rousse, 8 Studentska str., 7017, Rousse, Bulgaria

    Lubin G. Vulkov

  3. Department of Informatics and Mathematical Modelling, Technical University of Denmark, 2800, Kongens, Lyngby, Denmark

    Jerzy Waśniewski

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Horváth, R. (2009). On the Sign-Stability of Finite Difference Solutions of Semilinear Parabolic Problems. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2008. Lecture Notes in Computer Science, vol 5434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00464-3_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-00464-3_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00463-6

  • Online ISBN: 978-3-642-00464-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation