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Time Discretisation of Parabolic Problems with the Variable 3-Step BDF

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Abstract

In this paper the stability of the 3-step backward differentiation formula (BDF) on variable grids for the numerical integration of time-dependent parabolic problems is analysed. A stability inequality with a stability constant depending in a controllable way on the mesh is obtained. In particular if the ratios r j of adjacent mesh-sizes of the underlying grid satisfy the bound r j ≤ r¯ < 1.199 then any mixture of the j-step BDF for j ∈ {1, 2, 3} is stable provided the number of changes between increasing and decreasing mesh-sizes is uniformly bounded. From the stability inequality error estimates can be obtained.

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REFERENCES

  1. J. Becker, A second order backward difference method with variable steps for a parabolic problem, BIT, 38 (1998), pp. 644-662.

    Google Scholar 

  2. J. H. Bramble, J. E. Pasciak, P. H. Sammon, and V. Thomée, Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data, Math. Comp., 52 (1989), pp. 339-367.

    Google Scholar 

  3. G. D. Byrne and A. C. Hindmarsh, A polyalgorithm for the numerical solution of ordinary differential equations, ACM Trans. Math. Software, 1 (1975), pp. 71-96.

    Google Scholar 

  4. M. Calvo, T. Grande, and R. D. Grigorieff, On the zero-stability of the variable order variable step-size BDF-formulas, Numer. Math., 57 (1990), pp. 39-50.

    Google Scholar 

  5. E. Emmrich, Stability and error of the variable two-step BDF for parabolic problems, Preprint 703/2000, TU Berlin.

  6. R. D. Grigorieff, Time discretization of semigroups by the variable two-step BDF method, in Numerical Treatment of Differential Equations, K. Strehmel, ed., (NUMDIFF-5, Halle 1989), Teubner, Leipzig, 1991, pp. 204-216.

    Google Scholar 

  7. R. D. Grigorieff, On the variable two-step BDF method for parabolic equations, Preprint 426/1995, TU Berlin.

  8. N. Guglielmi and M. Zennaro, On the zero-stability of the variable stepsize multistep methods: The spectral radius approach, Numer. Math., 88 (2001), pp. 445-458.

    Google Scholar 

  9. M.-N. Le Roux, Semi-discrétisation en temps pour les équations d'évolution parabolique lorsque l'opérateur dépend du temps, RAIRO Anal. Numér., 13 (1997), pp. 119-137.

    Google Scholar 

  10. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 1997.

    Google Scholar 

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Authors and Affiliations

  1. Dpto. de Matemática Aplicada, Facultad de Ciencias, Universidad de Zaragoza, Edificio Matemáticas, 50009, Zaragoza, Spain

    M. Calvo

  2. Technische the Universität Berlin, Straβe des 17. Juni 135, 10623, Berlin, Germany

    R. D. Grigorieff

Authors
  1. M. Calvo
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  2. R. D. Grigorieff
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Calvo, M., Grigorieff, R.D. Time Discretisation of Parabolic Problems with the Variable 3-Step BDF. BIT Numerical Mathematics 42, 689–701 (2002). https://doi.org/10.1023/A:1021992101967

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