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Implicit-explicit multistep methods for nonlinear parabolic equations


Author: Georgios Akrivis
Journal: Math. Comp. 82 (2013), 45-68
MSC (2010): Primary 65M12, 65M60; Secondary 65L06
DOI: https://doi.org/10.1090/S0025-5718-2012-02628-7
Published electronically: July 10, 2012
MathSciNet review: 2983015
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Abstract: Implicit-explicit multistep methods for nonlinear parabolic equations were recently analyzed in [2, 3, 1]. In these papers the linear operator of the equation is assumed to be time-independent, self-adjoint and positive definite; then, the linear part is discretized implicitly and the remaining part explicitly. Here we slightly relax the hypotheses on the linear operator by allowing part of it to be time-dependent or nonself-adjoint. We establish optimal order a priori error estimates.


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  • 1. G. Akrivis and M. Crouzeix,
    Linearly implicit methods for nonlinear parabolic equations,
    Math. Comp. 73 (2003) 613-635. MR 2031397 (2005a:65097)
  • 2. G. Akrivis, M. Crouzeix and Ch. Makridakis,
    Implicit-explicit multistep finite element methods for nonlinear parabolic problems,
    Math. Comp. 67 (1998) 457-477. MR 1458216 (98g:65088)
  • 3. G. Akrivis, M. Crouzeix and Ch. Makridakis,
    Implicit-explicit multistep methods for quasilinear parabolic equations,
    Numer. Math. 82 (1999) 521-541. MR 1701828 (2000e:65075)
  • 4. G. Akrivis and F. Karakatsani,
    Modified implicit-explicit BDF methods for nonlinear parabolic equations,
    BIT Numer. Math. 43 (2003) 467-483. MR 2026710 (2004m:65139)
  • 5. G. Akrivis and Y.-S. Smyrlis,
    Linearly implicit schemes for a class of dispersive-dissipative systems,
    Calcolo 48 (2011) 145-172. MR 2796117
  • 6. M. Crouzeix,
    Une méthode multipas implicite-explicite pour l'approximation des équations d'évolution paraboliques,
    Numer. Math. 35 (1980) 257-276. MR 592157 (82b:65084)
  • 7. R. D. Grigorieff,
    Numerik gewöhnlicher Differentialgleichungen, Bd. 2, Mehrschrittverfahren,
    Teubner Studienbücher, Stuttgart, 1977. MR 0657222 (58:31842)
  • 8. R. D. Grigorieff and J. Schroll,
    Über A$ (\alpha )$-stabile Verfahren hoher Konsistenzordnung,
    Computing 20 (1978) 343-350. MR 619908 (83b:65086)
  • 9. E. Hairer and G. Wanner,
    Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems,
    2nd revised ed., Springer-Verlag, Berlin Heidelberg, Springer Series in Computational Mathematics v. 14, 2010. MR 2657217 (2011f:65001)
  • 10. P. Henrici,
    Discrete Variable Methods in Ordinary Differential Equations,
    J. Wiley & Sons, New York, London, 1962. MR 0135729 (24:B1772)
  • 11. W. Hundsdorfer and S. J. Ruuth,
    IMEX extensions of linear multistep methods with general monotonicity and boundedness properties,
    J. Comp. Phys. 225 (2007) 2016-2042. MR 2349693 (2009f:65185)
  • 12. R. Jeltsch,
    Stiff stability and its relation to $ A_0$- and $ A(0)$-stability,
    SIAM J. Numer. Anal. 13 (1976) 8-17. MR 0411174 (53:14913)
  • 13. W. Liniger,
    A criterion for $ A$-stability of linear multistep integration formulae,
    Computing 3 (1968) 280-285. MR 0239763 (39:1120)
  • 14. C. Lubich,
    On the convergence of multistep methods for nonlinear stiff differential equations,
    Numer. Math. 58 (1991) 839-853. MR 1098868 (92d:65127)
  • 15. S. Nørsett,
    A criterion for $ A(\alpha )$-stability of linear multistep methods,
    BIT 9 (1969) 259-263. MR 0256571 (41:1227)
  • 16. G. Savaré,
    $ A(\varTheta )$-stable approximations of abstract Cauchy problems,
    Numer. Math. 65 (1993) 319-335. MR 1227025 (94h:65062)
  • 17. V. Thomée,
    Galerkin Finite Element Methods for Parabolic Problems. 2nd ed.,
    Springer-Verlag, Berlin, 2006. MR 2249024 (2007b:65003)

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Additional Information

Georgios Akrivis
Affiliation: Computer Science Department, University of Ioannina, 45110 Ioannina, Greece
Email: akrivis@cs.uoi.gr

DOI: https://doi.org/10.1090/S0025-5718-2012-02628-7
Keywords: Nonlinear parabolic equations, implicit–explicit multistep methods, BDF methods, strong $A(0)$-stability, $A(𝜗)$-stability, $G$-stability
Received by editor(s): April 15, 2011
Received by editor(s) in revised form: September 12, 2011
Published electronically: July 10, 2012
Additional Notes: This work was supported in part by University of Cyprus grant no. 8037P-3/311-21028.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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