Skip to main content
SpringerLink
Log in

Über Die Stabilitätsdefinition Für Differenzengleichungen Die Partielle Differentialgleichungen Approximieren

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

For difference equations with constant coefficients necessary and sufficient algebraic stability conditions are given for the stability definitions used by G. Forsythe and W. Wasow (A) and P. D. Lax and R. D. Richtmyer (B). The application of these conditions for difference equations with variable coefficients is considered and it is shown that the stability condition of definitionA is not sufficient for stability. The same is true with respect to the definitionB if the difference equations are not parabolic and do not approximate first order systems. Therefore another stability definition is proposed and a number of properties are discussed.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literatur

  1. G. E. Forsythe and W. R. Wasow,Finite-Difference Methods For Partial Differential Equations. John Wiley & Sons, New York 1960.

    Google Scholar 

  2. P. D. Lax and R. D. Richtmyer,Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math., 9 (1956), 267–293.

    Google Scholar 

  3. P. D. Lax,Differential equations, difference equations and matrix theory. Comm. Pure Appl. Math., 11 (1958), 175–194.

    Google Scholar 

  4. L. Gårding,Linear hyperbolic partial differential equations with constant coefficients. Acta Math., 85 (1951), 1–62.

    Google Scholar 

  5. V. S. Ryabenkii und A. F. Filippow,Über die Stabilität von Differenzengleichungen. Deutscher Verlag der Wissenschaften, Berlin 1960.

    Google Scholar 

  6. G. Dahlquist,Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand., 4 (1956), 33–53.

    Google Scholar 

  7. F. John,On integration of parabolic equations by difference methods. Comm. Pure Appl. Math., 5 (1952), 155–211.

    Google Scholar 

  8. K. O. Friedrichs,Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math., 7 (1954), 345–392.

    Google Scholar 

  9. H. O. Kreiss, Über die Lösung des Cauchyproblems für lineare partielle Differentialgleichungen mit Hilfe von Differenzengleichungen. Acta Math., 101 (1959), 179–199.

    Google Scholar 

  10. H. O. Kreiss, Über die Differenzapproximation hoher Genauigkeit bei Anfangswertproblemen für partielle Differentialgleichungen. Num. Math., 1 (1959), 186–202.

    Google Scholar 

  11. P. D. Lax, On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients. Comm. Pure Appl. Math., 14 (1961), 497–520.

    Google Scholar 

  12. P. D. Lax and B. Wendroff, On the stability of difference schemes. Manuskript.

  13. I. Petrovskii, Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen im Gebiete der nichtanalytischen Funktionen. Bull. Univ. Etat. Moscou Ser. Int. Sect. A Fasc. 7 (1938), 1–74.

    Google Scholar 

  14. W. G. Strang,Difference methods for mixed boundary-value problems. Duke Math. Journal, 27 (1960), 221–232.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Royal Institute of Technology, Stockholm, Sweden

    Heinz-Otto Kreiss

Authors
  1. Heinz-Otto Kreiss
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kreiss, HO. Über Die Stabilitätsdefinition Für Differenzengleichungen Die Partielle Differentialgleichungen Approximieren. BIT 2, 153–181 (1962). https://doi.org/10.1007/BF01957330

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01957330

Search

Navigation