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On the Crank-Nicolson scheme once again

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An Erratum to this article was published on 30 March 2011

Abstract

Let \({\left(\tau_j\right)_{j\in\mathbb{N}}}\) be a sequence of strictly positive real numbers, and let A be the generator of a bounded analytic semigroup in a Banach space X. Put \({A_n=\prod_{j=1}^n\left(I+\frac{1}{2} \tau_jA\right) \left(I-\frac{1}{2} \tau_jA\right)^{-1}}\), and let \({x\in X}\). Define the sequence \({\left(x_n\right)_{n\in\mathbb{N}}\subset X}\) by the Crank–Nicolson scheme: x n  = A n x. In this paper, it is proved that the Crank–Nicolson scheme is stable in the sense that \({\sup_{n\in\mathbb{N}}\left\Vert A_nx\right\Vert<\infty}\). Some convergence results are also given.

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References

  1. Nikolai Bakaev, Alexander Ostermann: Long-term stability of variable stepsize approximations of semigroups. Math. Comp. 71(240), 1545–1567 (2002) (electronic). MR 1933044 (2003j:65055)

    MathSciNet  MATH  Google Scholar 

  2. Michael Cowling, Ian Doust, Alan McIntosh, Atsushi Yagi: Banach space operators with a bounded H ∞ functional calculus. J. Austral. Math. Soc. Ser. A 60(1), 51–89 (1996) MR 1364554 (97d:47023)

    Article  MathSciNet  Google Scholar 

  3. Crank J., Nicolson P.: A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type [MR0019410 (8,409b)]. Adv. Comput. Math. 6(3–4), 207–226 (1996) John Crank 80th birthday special issue. MR MR1440436

    Article  MathSciNet  Google Scholar 

  4. Crouzeix M., Larsson S., Piskarëv S., Thomée V.: The stability of rational approximations of analytic semigroups. BIT 33(1), 74–84 (1993) MR MR1326004 (96f:65069)

    Article  MathSciNet  MATH  Google Scholar 

  5. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR MR1721989 (2000i:47075)

  6. John B. Garnett, Bounded analytic functions, first ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR MR2261424 (2007e:30049)

  7. Davide Guidetti, Bülent Karasözen, Sergei Piskarev: Approximation of abstract differential equations. J. Math. Sci. (N. Y.) 122(2), 3013–3054 (2004) Functional analysis. MR MR2084185 (2005g:34127)

    Article  MathSciNet  MATH  Google Scholar 

  8. Markus Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag, Basel, 2006. MR 2244037 (2007j:47030)

  9. Armen M. Jerbashian, Functions of α-bounded type in the half-plane, Advances in Complex Analysis and its Applications, vol. 4, Springer-Verlag, New York, 2005. MR 2109213 (2006m:30004)

  10. Mihály Kovács, Frank Neubrander: On the inverse Laplace-Stieltjes transform of A-stable rational functions. New Zealand J. Math. 36, 41–56 (2007) MR 2406045 (2009b:44003)

    MathSciNet  MATH  Google Scholar 

  11. Christoph Kriegl and Lutz Weis, Contractivity of the H ∞-calculus and Blaschke products, Operator algebras, Operator theory and Applications, vol. 195, pp. 231–244, Birkhäuser Verlag, Basel, Switzerland, 2010.

  12. Peer C. Kunstmann and Lutz Weis, Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H ∞-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65–311. MR 2108959 (2005m:47088)

  13. Lyubich Yu.: Spectral localization, power boundedness and invariant subspaces under Ritt’s type condition. Studia Math. 134(2), 153–167 (1999) MR MR1688223 (2000a:47011)

    MathSciNet  MATH  Google Scholar 

  14. Palencia C.: A stability result for sectorial operators in Banach spaces. SIAM J. Numer. Anal. 30(5), 1373–1384 (1993) MR MR1239826 (94j:65109)

    Article  MathSciNet  MATH  Google Scholar 

  15. Palencia C.: On the stability of variable stepsize rational approximations of holomorphic semigroups. Math. Comp. 62(205), 93–103 (1994) MR 1201070 (94c:47066)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sergey Piskarev, Hans Zwart: Crank-Nicolson scheme for abstract linear systems. Numer. Funct. Anal. Optim. 28(5–6), 717–736 (2007) MR MR2329298 (2008c:65138)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020, Antwerp, Belgium

    Jan A. van Casteren

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  1. Jan A. van Casteren
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Correspondence to Jan A. van Casteren.

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The author is grateful to Sergey Piskarev for suggesting him the problem which is treated in the present paper, and for his comments. Thanks are also due to an anonymous referee for suggestions to improve the quality of an earlier version of the article. He also thanks the University of Antwerp and the Flemish Fund for Scientific Research for their financial and logistic support.

An erratum to this article can be found at http://dx.doi.org/10.1007/s00028-011-0108-0

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van Casteren, J.A. On the Crank-Nicolson scheme once again. J. Evol. Equ. 11, 457–476 (2011). https://doi.org/10.1007/s00028-010-0084-9

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