Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Linearly implicit methods for nonlinear parabolic equations

Authors: Georgios Akrivis and Michel Crouzeix
Journal: Math. Comp. 73 (2004), 613-635
MSC (2000): Primary 65M60, 65M12; Secondary 65L06
Published electronically: June 19, 2003
MathSciNet review: 2031397
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Abstract: We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit-explicit multistep schemes as well as the combination of implicit Runge-Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes are locally stable. We establish optimal order error estimates.

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  • 1. G. Akrivis, M. Crouzeix and Ch. Makridakis,
    Implicit-explicit multistep finite element methods for nonlinear parabolic problems,
    Math. Comp. 67 (1998) 457-477. MR 98q:65088
  • 2. 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, 10.1007/s002110050429
  • 3. Philip Brenner, Michel Crouzeix, and Vidar Thomée, Single-step methods for inhomogeneous linear differential equations in Banach space, RAIRO Anal. Numér. 16 (1982), no. 1, 5–26 (English, with French summary). MR 648742
  • 4. M. Crouzeix,
    Sur l'approximation des équations différentielles opérationnelles linéaires par des méthodes de Runge-Kutta.
    Thèse, Université de Paris VI, 1975.
  • 5. Stephen L. Keeling, Galerkin/Runge-Kutta discretizations for semilinear parabolic equations, SIAM J. Numer. Anal. 27 (1990), no. 2, 394–418. MR 1043612, 10.1137/0727024
  • 6. Jens Lang, Adaptive multilevel solution of nonlinear parabolic PDE systems, Lecture Notes in Computational Science and Engineering, vol. 16, Springer-Verlag, Berlin, 2001. Theory, algorithm, and applications. MR 1801795
  • 7. C. Lubich, On the convergence of multistep methods for nonlinear stiff differential equations, Numer. Math. 58 (1991), no. 8, 839–853. MR 1098868, 10.1007/BF01385657
  • 8. C. Lubich and A. Ostermann,
    Linearly implicit time discretization of non-linear parabolic equations,
    IMA J. Numer. Anal. 15 (1995) 555-583. MR 96q:65085
  • 9. Giuseppe Savaré, 𝐴(Θ)-stable approximations of abstract Cauchy problems, Numer. Math. 65 (1993), no. 3, 319–335. MR 1227025, 10.1007/BF01385755
  • 10. T. Steihaug and A. Wolbrandt,
    An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations,
    Math. Comp. 33 (1979) 521-534. MR 80q:65087
  • 11. Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR 744045

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Additional Information

Georgios Akrivis
Affiliation: Computer Science Department, University of Ioannina, 451 10 Ioannina, Greece

Michel Crouzeix
Affiliation: IRMAR, Université de Rennes I, Campus de Beaulieu, F-35042 Rennes, France

Keywords: Nonlinear parabolic equations, linearly implicit methods, strong $A(0)$-stability, implicit-explicit multistep schemes, polynomial order
Received by editor(s): May 2, 2001
Received by editor(s) in revised form: October 2, 2002
Published electronically: June 19, 2003
Additional Notes: The work of the first author was supported in part by the Greek Secretariat for Research and Technology through the PENED Program, no 99ED 275
Article copyright: © Copyright 2003 American Mathematical Society