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Numerical Algorithm With High Spatial Accuracy for the Fractional Diffusion-Wave Equation With Neumann Boundary Conditions

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Abstract

A fourth-order compact algorithm is discussed for solving the time fractional diffusion-wave equation with Neumann boundary conditions. The \(L1\) discretization is applied for the time-fractional derivative and the compact difference approach for the spatial discretization. The unconditional stability and the global convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis. The convergence order is \(\mathcal{O }(\tau ^{3-\alpha }+h^4)\) in the maximum norm, where \(\tau \) is the temporal grid size and \(h\) is the spatial grid size, respectively. In addition, a Crank–Nicolson scheme is presented and the corresponding error estimates are also established. Meanwhile, a compact ADI difference scheme for solving two-dimensional case is derived and the global convergence order of \(\mathcal{O }(\tau ^{3-\alpha }+h_1^4+h_2^4)\) is given. Then extension to the case with Robin boundary conditions is also discussed. Finally, several numerical experiments are included to support the theoretical results, and some comparisons with the Crank–Nicolson scheme are presented to show the effectiveness of the compact scheme.

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Acknowledgments

We would like to thank the anonymous referees for many constructive comments and suggestions which led to an improved presentation of this paper. The research is supported by National Natural Science Foundation of China (No. 11271068) and the Research and Innovation Project for College Graduates of Jiangsu Province (No. CXZZ11_0134) and Foundation for Key Teacher of Shangqiu Normal University (No.2012GGJS14).

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Jincheng Ren & Zhi-zhong Sun

  2. Department of Mathematics, Shangqiu Normal University, Henan, 476000, People’s Republic of China

    Jincheng Ren

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  1. Jincheng Ren
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Correspondence to Zhi-zhong Sun.

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Ren, J., Sun, Zz. Numerical Algorithm With High Spatial Accuracy for the Fractional Diffusion-Wave Equation With Neumann Boundary Conditions. J Sci Comput 56, 381–408 (2013). https://doi.org/10.1007/s10915-012-9681-9

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  • DOI: https://doi.org/10.1007/s10915-012-9681-9

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