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A Fast Algorithm for the Variable-Order Spatial Fractional Advection-Diffusion Equation

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Abstract

We propose a fast algorithm for the variable-order (VO) space-fractional advection-diffusion equations with nonlinear source terms on a finite domain. Due to the impact of the space-dependent the VO, the resulting coefficient matrices arising from the finite difference discretization of the fractional advection-diffusion equation are dense without Toeplitz-like structure. By the properties of the elements of coefficient matrices, we show that the off-diagonal blocks can be approximated by low-rank matrices. Then we present a fast algorithm based on the polynomial interpolation to approximate the coefficient matrices. The approximation can be constructed in \({\mathcal {O}}(kN)\) operations and requires \({\mathcal {O}}(kN)\) storage with N and k being the number of unknowns and the approximants, respectively. Moreover, the matrix-vector multiplication can be implemented in \({\mathcal {O}} (kN\log N)\) complexity, which leads to a fast iterative solver for the resulting linear systems. The stability and convergence of the new scheme are also studied. Numerical tests are carried out to exemplify the accuracy and efficiency of the proposed method.

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Acknowledgements

We thank the anonymous referees for valuable comments and suggestions which lead to a significant improvement of the presentation.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Hong-Kui Pang

  2. Department of Mathematics, University of Macau, Macao, People’s Republic of China

    Hai-Wei Sun

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  1. Hong-Kui Pang
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Correspondence to Hai-Wei Sun.

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The research of Hong-Kui Pang is supported by research Grants 11771189 and 11501562 from the National Natural Science Foundation of China, BK20171162 from the Natural Science Foundation of Jiangsu Province, and the Qing-Lan Project of Jiangsu Province. The research of Hai-Wei Sun is supported by The Science and Technology Development Fund, Macau SAR (File No. 0118/2018/A3), and MYRG2018-00015-FST from University of Macau.

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Pang, HK., Sun, HW. A Fast Algorithm for the Variable-Order Spatial Fractional Advection-Diffusion Equation. J Sci Comput 87, 15 (2021). https://doi.org/10.1007/s10915-021-01427-w

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  • DOI: https://doi.org/10.1007/s10915-021-01427-w

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