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A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equations

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Abstract

A modified Crank–Nicolson-type compact alternating direction implicit (ADI) finite difference method is proposed for a class of two-dimensional fractional subdiffusion equations with a time Riemann–Liouville fractional derivative of order \((1-\alpha )\) \((0<\alpha <1)\). This method improves the known compact ADI methods in the sense that it is based on the L1 approximation for the fractional derivative, the truncation errors on all time levels have the same order of \(\mathcal{O}(\tau ^{2\alpha }+h_{x}^{4}+h_{y}^{4})\) and the optimal error estimate \(\mathcal{O}(\tau ^{2\alpha }+h_{x}^{4}+h_{y}^{4})\) can be easily obtained in the standard \(H^{1}\)- and \(L^{2}\)-norms and the weighted \(L^{\infty }\)-norm. The unique solvability, unconditional stability and convergence of the resulting scheme are rigorously proved. A Richardson extrapolation algorithm is presented to increase the temporal accuracy from the order \(2\alpha \) to the order \(\min \{1+\alpha , 4\alpha \}\). Numerical results demonstrate the accuracy of the modified compact ADI method and the high efficiency of the Richardson extrapolation algorithm.

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Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of the paper. This work was supported in part by E-Institutes of Shanghai Municipal Education Commission No. E03004, Science and Technology Commission of Shanghai Municipality (STCSM) No. 13dz2260400, and Shanghai Leading Academic Discipline Project No. B407.

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Authors and Affiliations

  1. Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Department of Mathematics, East China Normal University, Shanghai, 200241, People’s Republic of China

    Tao Wang & Yuan-Ming Wang

  2. Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science, E-Institute of Shanghai Universities, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Yuan-Ming Wang

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  1. Tao Wang
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Correspondence to Yuan-Ming Wang.

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Wang, T., Wang, YM. A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equations. J. Appl. Math. Comput. 52, 439–476 (2016). https://doi.org/10.1007/s12190-015-0949-8

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  • DOI: https://doi.org/10.1007/s12190-015-0949-8

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