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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.

MSC:
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
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References:
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