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Stability and error estimates for the variable step-size BDF2 method for linear and semilinear parabolic equations

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Abstract

In this paper, stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the \(l^{\infty }(0,T;H)\)-stability of the BDF2 method for linear and semilinear parabolic equations is identical with the upper bound for the zero-stability. The \(l^{\infty }(0,T;V)\)-stability of the variable step-size BDF2 method is also established under more relaxed condition on the ratios of consecutive step-sizes. Based on these stability results, error estimates in several different norms are derived. To utilize the BDF method, the trapezoidal method and the backward Euler scheme are employed to compute the starting value. For the latter choice, order reduction phenomenon of the constant step-size BDF2 method is observed theoretically and numerically in several norms. Numerical results also illustrate the effectiveness of the proposed method for linear and semilinear parabolic equations.

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Acknowledgements

The authors would like to thank the anonymous referees for comments and suggestions that led to improvements in the presentation of this paper.

Funding

This work was supported by a grant from the National Natural Science Foundation of China (Grant No. 11771060), Science and Technology Innovation Plan Of Shanghai, China (No. 20JC1414200), and sponsored by Natural Science Foundation of Shanghai, China (No. 20ZR1441200) .

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, 200234, Shanghai, China

    Wansheng Wang, Mengli Mao & Zheng Wang

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  1. Wansheng Wang
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  2. Mengli Mao
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  3. Zheng Wang
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Correspondence to Wansheng Wang.

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Communicated by: Long Chen

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Wang, W., Mao, M. & Wang, Z. Stability and error estimates for the variable step-size BDF2 method for linear and semilinear parabolic equations. Adv Comput Math 47, 8 (2021). https://doi.org/10.1007/s10444-020-09839-2

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