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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
Posted:
July 10, 2012
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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.
- 1.
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Akrivis and Michel
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- 2.
Georgios
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Crouzeix, and Charalambos
Makridakis, Implicit-explicit multistep finite
element methods for nonlinear parabolic problems, Math. Comp. 67 (1998), no. 222, 457–477. MR 1458216
(98g:65088), http://dx.doi.org/10.1090/S0025-5718-98-00930-2
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Georgios
Akrivis, Michel
Crouzeix, and Charalambos
Makridakis, Implicit-explicit multistep methods for quasilinear
parabolic equations, Numer. Math. 82 (1999),
no. 4, 521–541. MR 1701828
(2000e:65075), http://dx.doi.org/10.1007/s002110050429
- 4.
Georgios
Akrivis and Fotini
Karakatsani, Modified implicit-explicit BDF methods for nonlinear
parabolic equations, BIT 43 (2003), no. 3,
467–483. MR 2026710
(2004m:65139), http://dx.doi.org/10.1023/B:BITN.0000007057.13132.b3
- 5.
Georgios
Akrivis and Yiorgos-Sokratis
Smyrlis, Linearly implicit schemes for a class of
dispersive-dissipative systems, Calcolo 48 (2011),
no. 2, 145–172. MR 2796117
(2012f:65083), http://dx.doi.org/10.1007/s10092-010-0033-6
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Crouzeix, Une méthode multipas implicite-explicite pour
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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 -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  - and  -stability,
SIAM J. Numer. Anal. 13 (1976) 8-17. MR 0411174 (53:14913)
- 13.
- W. Liniger,
A criterion for  -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  -stability of linear multistep methods,
BIT 9 (1969) 259-263. MR 0256571 (41:1227)
- 16.
- G. Savaré,
 -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:
http://dx.doi.org/10.1090/S0025-5718-2012-02628-7
PII:
S 0025-5718(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
Posted:
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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