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Linearly implicit methods for nonlinear parabolic equations. (English) Zbl 1045.65079
Authors’ 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”.
It is interesting to note that, when applied to the nonlinear parabolic problem studied by the authors, S. L. Keeling’s schemes [SIAM J. Numer. Anal. 27, No. 2, 394–418 (1990; Zbl 0697.65083)] become a particular case of the schemes studied in this paper.

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] G. Akrivis, M. Crouzeix and Ch. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems, Math. Comp. 67 (1998) 457-477. · Zbl 0896.65066
[2] Georgios Akrivis, Michel Crouzeix, and Charalambos Makridakis, Implicit-explicit multistep methods for quasilinear parabolic equations, Numer. Math. 82 (1999), no. 4, 521 – 541. · Zbl 0936.65118
[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). · Zbl 0477.65040
[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. · Zbl 0697.65083
[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. · Zbl 0963.65102
[7] C. Lubich, On the convergence of multistep methods for nonlinear stiff differential equations, Numer. Math. 58 (1991), no. 8, 839 – 853. · Zbl 0729.65055
[8] C. Lubich and A. Ostermann, Linearly implicit time discretization of non-linear parabolic equations, IMA J. Numer. Anal. 15 (1995) 555-583. · Zbl 0834.65092
[9] Giuseppe Savaré, \?(\Theta )-stable approximations of abstract Cauchy problems, Numer. Math. 65 (1993), no. 3, 319 – 335. · Zbl 0791.65077
[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. · Zbl 0451.65055
[11] Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. · Zbl 1105.65102
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