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Long-term stability of variable stepsize approximations of semigroups

Authors: Nikolai Bakaev and Alexander Ostermann
Journal: Math. Comp. 71 (2002), 1545-1567
MSC (2000): Primary 65M12, 65L20
Published electronically: August 3, 2001
MathSciNet review: 1933044
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Abstract: This paper is concerned with the stability of rational one-step approximations of $C_0$ semigroups. Particular emphasis is laid on long-term stability bounds. The analysis is based on a general Banach space framework and allows variable stepsize sequences. Under reasonable assumptions on the stepsize sequence, asymptotic stability bounds for general $C_0$ semigroups are derived. The bounds are typical in the sense that they contain, in general, a factor that grows with the number of steps. Under additional hypotheses on the approximation, more favorable stability bounds are obtained for the subclass of holomorphic semigroups.

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  • 1. N. Bakaev, Stability estimates for a certain general discretization method, Dokl. Akad. Nauk SSSR 309 (1989), 11-15 (in Russian); English transl. in Soviet Math. Dokl. 40 (1990). MR 91e:65104
  • 2. N. Bakaev, Some problems of well-posedness of difference schemes on nonuniform grids, Zhurn. Vychisl. Mat. i Mat. Fiz. 33 (1993), 561-577 (in Russian); English transl. in Comput. Math. Math. Phys. 33 (1993). MR 94g:65085
  • 3. N. Bakaev, On variable stepsize Runge-Kutta approximations of a Cauchy problem for the evolution equation, BIT 38 (1998), 462-485. MR 99i:65069
  • 4. Ph. Brenner and V. Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979), 683-694. MR 80j:47052
  • 5. Ph. Brenner, V. Thomée, and L. Wahlbin, Besov Spaces and Applications to Difference Methods for Initial Value Problems, Lecture Notes in Mathematics 434, Springer-Verlag, Berlin, 1975. MR 57:1106
  • 6. E. Hairer and M. Zennaro, On error growth functions of Runge-Kutta methods, Appl. Numer. Math. 22 (1996), 205-216. MR 97j:65116
  • 7. G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1988. MR 89d:26016
  • 8. E. Hille and R. Phillips, Functional Analysis and Semigroups, AMS, Providence, 1957. MR 19:664d
  • 9. M.-N. LeRoux, Semidiscretizations in time for parabolic problems, Math. Comp. 33 (1979), 919-931. MR 80f:65101
  • 10. Ch. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), 293-313. MR 92h:65145
  • 11. Ch. Lubich and A. Ostermann, Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization, Numer. Math. 81 (1998), 53-84. MR 2000m:37176
  • 12. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. MR 96e:47039
  • 13. C. Palencia, On the stability of variable stepsize rational approximations of holomorphic semigroups, Math. Comp. 62 (1994), 93-103. MR 94c:47066

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Additional Information

Nikolai Bakaev
Affiliation: Department of Mathematics, Air Force Technical University, Planetnaya 3, Moscow 125190, Russia

Alexander Ostermann
Affiliation: Section de mathématiques, Université de Genève, C.P. 240, CH-1211 Genève 24, Switzerland

Received by editor(s): July 10, 2000
Received by editor(s) in revised form: December 26, 2000
Published electronically: August 3, 2001
Additional Notes: The work of the first author was supported by the Swiss National Science Foundation under Grant 20-56577.99.
The second author was on leave from Universität Innsbruck, Institut für Technische Mathematik, Geometrie und Bauinformatik, Technikerstraße 13, A-6020 Innsbruck, Austria
Article copyright: © Copyright 2001 American Mathematical Society

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