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A- andB-stability for Runge-Kutta methods-characterizations and equivalence

  • Numerical Approximation of Transverse Shearing Stress in Bent Plates
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Summary

Using a special representation of Runge-Kutta methods (W-transformation), simple characterizations ofA-stability andB-stability have been obtained in [9, 8, 7]. In this article we will make this representation and their conclusions more transparent by considering the “exact Runge-Kutta method”. Finally we demonstrate by a numerical example that for difficult problemsB-stable methods are superior to methods which are “only”A-stable.

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References

  1. Burrage, K., Butcher, J.C.: Stability criteria for implicit Runge-Kutta methods. SIAM J. Numer. Anal.16, 46–57 (1979)

    Google Scholar 

  2. Butcher, J.C.: A stability property of implicit Runge-Kutta methods. BIT15, 358–361 (1975)

    Google Scholar 

  3. Crouzeix, M.: Sur laB-stabilité des méthodes de Runge-Kutta. Numer. Math.32, 75–82 (1979)

    Google Scholar 

  4. Dahlquist, G.: A special stability problem for linear multistep methods. BIT3, 27–43 (1963)

    Google Scholar 

  5. Dahlquist, G.: Error analysis for a class of methods of stiff non-linear initial value problems. Numerical Analysis, Dundee 1975. Springer Lect. Notes Math.506, 60–74 (1975)

    Google Scholar 

  6. Dekker, K., Verwer, J.G.: Stability of Runge-Kutta methods for stiff nonlinear differential equations. Amsterdam: North-Holland 1984

    Google Scholar 

  7. Hairer, E.: Constructive characterization ofA-stable approximations to exp(z) and its connection with algebraically stable Runge-Kutta methods. Numer. Math.39, 247–258 (1982)

    Google Scholar 

  8. Hairer, E., Türke, H.: The equivalence ofB-stability andA-stability. BIT24, 520–528 (1984)

    Google Scholar 

  9. Hairer, E., Wanner, G.: Algebraically stable and implementable Runge-Kutta methods of high order. SIAM J. Numer. Anal.18, 1098–1108 (1981)

    Google Scholar 

  10. Hairer, E., Wanner, G.: Characterization of non-linearly stable implicit Runge-Kutta methods. Numerical Integration of Differential Equations and Large Linear Systems. Bielefeld 1980. Springer Lect. Notes Math.968, 207–219 (1982)

    Google Scholar 

  11. Varga, R.S.: On high order stable implicit methods for solving parabolic partial differential equations. J. Math. Phys.40, 220–231 (1961)

    Google Scholar 

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  1. Section de mathématiques 2-4, rue du Lièvre, Case postale 240, CH-1211, Genève 24, Switzerland

    E. Hairer

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  1. E. Hairer
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Hairer, E. A- andB-stability for Runge-Kutta methods-characterizations and equivalence. Numer. Math. 48, 383–389 (1986). https://doi.org/10.1007/BF01389646

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  • DOI: https://doi.org/10.1007/BF01389646

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