Skip to main content
Log in

Dahlquist’s classical papers on stability theory

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

This text, which is based on the author’s talk in honour of G. Dahlquist at the SciCade05 meeting in Nagoya, describes the two classical papers from 1956 and 1963 of Dahlquist and their enormous impact on the research of decades to come; it also allows the author to present a personal testimony of his never ending admiration for the scientific and personal qualities of this great man.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. Abdulle, Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comp., 23 (2002), pp. 2042–2055.

    Article  MathSciNet  Google Scholar 

  2. G. Birkhoff and R. S. Varga, Discretization errors for well-set Cauchy problems, 1, J. Math. Phys., 44 (1965), pp. 1–23.

    MathSciNet  MATH  Google Scholar 

  3. J. Butcher, Thirty years of G-stability, this issue, DOI: 10.1007/s10543-006-0078-8.

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

    MathSciNet  MATH  Google Scholar 

  5. G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Trans. Royal Inst. Technol., Stockholm, Sweden, Nr.130, p. 87, 1959.

  6. J. W. Daniel and R. E. Moore, Computation and theory in ordinary differential equations, W. H. Freeman and Company, San Francisco, p. 172, 1970.

  7. B. L. Ehle, High order A-stable methods for the numerical solution of systems of D.E.’s, BIT, 8 (1968), pp. 276–278.

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Genin, An algebraic approach to A-stable linear multistep-multiderivative integration formulas, BIT, 14 (1974), pp. 382–406.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Hairer, Symmetric linear multistep methods, this issue, DOI: 10.1007/s10543-006-0066-z.

  10. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, 2nd edn., Springer, Berlin, Heidelberg, 1993.

  11. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, 2nd. edn., Springer, Berlin, Heidelberg, 1991, 1996.

  12. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons Inc., New York, London, Sydney, 1962.

  13. W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advectiondiffusion-Reaction Equations, Springer, Berlin, Heidelberg, 2003.

  14. R. Jeltsch, Note on A-stability of multistep multiderivative methods, BIT, 16 (1976), pp. 74–78.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Jeltsch and O. Nevanlinna, Largest disk of stability of explicit Runge–Kutta methods, BIT, 18 (1978), pp. 500–502.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Jeltsch and O. Nevanlinna, Stability of explicit time discretizations for solving initial value problems, Numer. Math., 37 (1981), pp. 61–91 (Corrigendum: Numer. Math., 39 (1982), p. 155).

  17. S. P. Nørsett, C-polynomials for rational approximations to the exponential function, Numer. Math., 25 (1975), pp. 39–56.

    Article  MathSciNet  Google Scholar 

  18. M. Reimer, Zur Theorie der linearen Differenzenformeln, Math. Z., 95 (1967), pp. 373–402.

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Wanner, E. Hairer, and S. P. Nørsett, Order stars and stability theorems, BIT, 18 (1978), pp. 475–489.

    Article  MathSciNet  MATH  Google Scholar 

  20. G. Wanner, E. Hairer, and S. P. Nørsett, When I-stability implies A-stability, BIT, 18 (1978), p. 503.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to G. Wanner.

Additional information

In memory of Germund Dahlquist (1925–2005).

AMS subject classification (2000)

65F05, 65F07

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wanner, G. Dahlquist’s classical papers on stability theory . Bit Numer Math 46, 671–683 (2006). https://doi.org/10.1007/s10543-006-0072-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-006-0072-1

Key words

Navigation