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Fractional centered difference schemes and banded preconditioners for nonlinear Riesz space variable-order fractional diffusion equations

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Abstract

In this paper, high-order finite difference methods are proposed to solve the initial-boundary value problem for one- and two-dimensional Riesz space variable-order fractional diffusion equations. We first introduce fractional centered difference (FCD) and weighted and shifted fractional centered difference (WSFCD) schemes for Riesz space variable-order fractional derivatives. Then the Crank-Nicolson (CN) scheme and the linearly implicit conservative (LIC) difference scheme are applied to discretize the time derivative in linear and nonlinear problems, respectively. Thus, we get CN-FCD and CN-WSFCD schemes, and LIC-FCD and LIC-WSFCD schemes, respectively. Theoretical results about the stability and convergence for the above-mentioned schemes are presented and proved. Banded preconditioners are introduced to speed up GMRES methods for solving the discretization linear systems. The spectral property of the preconditioned matrix is analyzed. Numerical results show that the proposed schemes and preconditioners are very efficient.

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Acknowledgements

We thank the referees for providing detailed and valuable comments and suggestions, which are very helpful for improving our paper.

Funding

This research was supported by National Natural Science Foundation of China (11771265), basic research and applied basic research projects in Guangdong Province (Projects of Guangdong-Hong Kong-Macao Center for Applied Mathematics) (2020B1515310018) and key research projects of general universities in Guangdong Province (2019KZDXM034).

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  1. Department of Mathematics, Shantou University, Shantou, 515063, Guangdong, People’s Republic of China

    Qiu-Ya Wang, Zi-Hang She, Cheng-Xue Lao & Fu-Rong Lin

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  1. Qiu-Ya Wang
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Correspondence to Fu-Rong Lin.

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Wang, QY., She, ZH., Lao, CX. et al. Fractional centered difference schemes and banded preconditioners for nonlinear Riesz space variable-order fractional diffusion equations. Numer Algor 95, 859–895 (2024). https://doi.org/10.1007/s11075-023-01592-z

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