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B-convergence of the implicit midpoint rule and the trapezoidal rule

  • Part II Numerical Mathematics
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Abstract

We present upper bounds for the global discretization error of the implicit midpoint rule and the trapezoidal rule for the case of arbitrary variable stepsizes. Specializing our results for the case of constant stepsizes they easily prove second order optimal B-convergence for both methods.

1980 AMS Subject Classification: 65L05, 65L20.

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References

  1. O. Axelsson,Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form, Numer. Math. 28 (1977), 1–14.

    Google Scholar 

  2. O. Axelsson,Error estimates over infinite intervals of some discretizations of evolution equations, BIT 24 (1984), 413–424.

    Google Scholar 

  3. K. Burrage and J. C. Butcher,Non-linear stability of a general class of differential equation methods, BIT 20 (1980), 185–203.

    Google Scholar 

  4. G. Dahlquist,Stability and error bounds in the numerical integration of ordinary differential equations, Trans. Roy. Inst. Techn., no. 130, Stockholm (1959).

  5. G. Dahlquist,A special stability problem for linear multistep methods, BIT 3 (1963), 27–43.

    Google Scholar 

  6. G. Dahlquist and B. Lindberg,On some implicit one-step methods for stiff differential equations, Report TRITA-NA-7302, Dept. Numer, Anal. Comp. Sci., Roy. Inst. Techn., Stockholm (1973).

  7. G. Dahlquist,On stability and error analysis for stiff non-linear problems, part I, Report TRITA-NA-7508, Dept. Numer. Anal. Comp. Sci., Roy. Inst. Techn., Stockholm (1975).

  8. G. Dahlquist,Error analysis for a class of methods for stiff non-linear initial value problems, Numerical Analysis Dundee 1975, Lecture Notes in Mathematics 506, Springer-Verlag, Berlin (1976), 60–72.

    Google Scholar 

  9. G. Dahlquist, W. Liniger and O. Nevanlinna,Stability of two-step methods for variable integration steps, SIAM J. Numer. Anal. 20 (1983), 1071–1085.

    Google Scholar 

  10. K. Dekker and J. G. Verwer,Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland Publ. Co., Amsterdam (1984).

    Google Scholar 

  11. R. Frank, J. Schneid and C. W. Ueberhuber,The concept of B-convergence, SIAM J. Numer. Anal. 18 (1981), 753–780.

    Google Scholar 

  12. R. Frank, J. Schneid and C. W. Ueberhuber,B-convergence of Runge-Kutta methods, Report 48/81, Inst. Numer. Math., Techn. Univ. Wien (1981).

  13. R. Frank, J. Schneid and C. W. Ueberhuber,Stability properties of implicit Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), 497–514.

    Google Scholar 

  14. R. Frank, J. Schneid and C. W. Ueberhuber,Order results for implicit Runge-Kutta methods applied to stiff systems, SIAM J. Numer. Anal. 22 (1985), 515–534.

    Google Scholar 

  15. I. A. M. Goddijn,Stabiliteit van Runge-Kutta methoden, Masters thesis, Univ. Leiden (1981).

  16. A. R. Gourlay,A note on trapezoidal methods for the solution of initial value problems, Math. Comp. 24 (1970), 629–633.

    Google Scholar 

  17. W. H. Hundsdorfer,The numerical solution of nonlinear stiff initial value problems, CWI-Tract 12, Amsterdam (1985).

  18. J. F. B. M. Kraaijevanger and M. N. Spijker,Algebraic stability and error propagation in Runge-Kutta methods, in preparation.

  19. J.D. Lambert,Computational Methods in Ordinary Differential Equations, John Wiley & Sons, London (1973).

    Google Scholar 

  20. W. Liniger and O. Nevanlinna,Contractive methods for stiff differential equations, part II, BIT 19 (1979), 53–72.

    Google Scholar 

  21. O. Nevanlinna,On error bounds for G-stable methods, BIT 16 (1976), 79–84.

    Google Scholar 

  22. A. Prothero and A. Robinson,On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp. 28 (1974), 145–162.

    Google Scholar 

  23. H. J. Stetter,Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, Berlin (1973).

    Google Scholar 

  24. H. J. Stetter,Zur B-Konvergenz der impliziten Trapez- und Mittelpunktregel, unpublished note.

  25. M. van Veldhuizen,Asymptotic expansions of the global error for the implicit midpoint rule (stiff case), Computing 33 (1984), 185–192.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics and Computer Science, University of Leiden, Wassenaarseweg 80, 2333 AL, Leiden, The Netherlands

    J. F. B. M. Kraaijevanger

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  1. J. F. B. M. Kraaijevanger
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Kraaijevanger, J.F.B.M. B-convergence of the implicit midpoint rule and the trapezoidal rule. BIT 25, 652–666 (1985). https://doi.org/10.1007/BF01936143

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  • DOI: https://doi.org/10.1007/BF01936143

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