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Some Numerical Extrapolation Methods for the Fractional Sub-diffusion Equation and Fractional Wave Equation Based on the L1 Formula

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Abstract

With the help of the asymptotic expansion for the classic L1 formula and based on the L1-type compact difference scheme, we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation. Three extrapolation formulas are presented, whose temporal convergence orders in \(L_\infty\)-norm are proved to be 2, \(3-\alpha\), and \(4-2\alpha\), respectively, where \(0<\alpha <1\). Similarly, by the method of order reduction, an extrapolation method is constructed for the fractional wave equation including two extrapolation formulas, which achieve temporal \(4-\gamma\) and \(6-2\gamma\) order in \(L_\infty\)-norm, respectively, where \(1<\gamma <2\). Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation, the fast extrapolation methods are obtained which reduce the computational complexity significantly while keeping the accuracy. Several numerical experiments confirm the theoretical results.

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Acknowledgements

The research is supported by the National Natural Science Foundation of China (grant number 11671081). The authors thank the anonymous referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of the paper. The authors are grateful to Prof. Guang-hua Gao at College of Science, Nanjing University of Posts and Telecommunications, for reminding us to revisit Dimitrov’s work.

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  1. School of Mathematics, Southeast University, Nanjing, 210096, Jiangsu, China

    Ren-jun Qi & Zhi-zhong Sun

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  1. Ren-jun Qi
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Correspondence to Zhi-zhong Sun.

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Qi, Rj., Sun, Zz. Some Numerical Extrapolation Methods for the Fractional Sub-diffusion Equation and Fractional Wave Equation Based on the L1 Formula. Commun. Appl. Math. Comput. 4, 1313–1350 (2022). https://doi.org/10.1007/s42967-021-00177-8

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