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A compact finite difference method for solving a class of time fractional convection-subdiffusion equations

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Abstract

A high-order compact finite difference method is proposed for solving a class of time fractional convection-subdiffusion equations. The convection coefficient in the equation may be spatially variable, and the time fractional derivative is in the Caputo’s sense with the order \(\alpha \) (\(0<\alpha <1\)). After a transformation of the original equation, the spatial derivative is discretized by a fourth-order compact finite difference method and the time fractional derivative is approximated by a \((2-\alpha )\)-order implicit scheme. The local truncation error and the solvability of the method are discussed in detail. A rigorous theoretical analysis of the stability and convergence is carried out using the discrete energy method, and the optimal error estimates in the discrete \(H^{1}\), \(L^{2}\) and \(L^{\infty }\) norms are obtained. Applications using several model problems give numerical results that demonstrate the effectiveness and the accuracy of this new method.

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Acknowledgments

The author would like to thank the referees for their valuable comments and suggestions which improved the presentation of the paper.

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Authors and Affiliations

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

    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. Yuan-Ming Wang
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Correspondence to Yuan-Ming Wang.

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Communicated by Jan Hesthaven.

This work was supported in part by E-Institutes of Shanghai Municipal Education Commission No. E03004 and Shanghai Leading Academic Discipline Project No. B407.

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Wang, YM. A compact finite difference method for solving a class of time fractional convection-subdiffusion equations. Bit Numer Math 55, 1187–1217 (2015). https://doi.org/10.1007/s10543-014-0532-y

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