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A fast second-order scheme for nonlinear Riesz space-fractional diffusion equations

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Abstract

In this paper, the fractional centered difference formula is employed to discretize the Riesz derivative, while the Crank-Nicolson scheme is approximated the time derivative, and the explicit linearized technique is used to deal with nonlinear term, a second-order finite difference method is obtained for the nonlinear Riesz space-fractional diffusion equations, and the resulting system is symmetric positive definite ill-conditioned Toeplitz matrix and then the fast sine transform can be used to reduce the computational cost of the matrix-vector multiplication. The preconditioned conjugate gradient method with a preconditioner based on sine transform is proposed to solve the linear system. Theoretically, the spectrum of the preconditioned matrix falling in an open interval (1/2,3/2) is proved, which can guarantee the linear convergence rate of the proposed methods. By the similar technique, the two-dimension case is also studied. Finally, numerical experiments are carried out to demonstrate that the proposed preconditioner works well.

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Acknowledgements

The authors thank professors Hai-Wei Sun and Xin Huang for discussion on the τ preconditioner.

Funding

This work was supported in part by the Science and Technology Research Project of Jiangxi Provincial Education Department of China (no. GJJ200919) and initial fund for Doctors (EA202007384), the National Natural Science Foundation of China (no. 11961048), and the Natural Science Foundation of Jiangxi Province of China (no. 20181ACB20001).

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Authors and Affiliations

  1. College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, 330063, People’s Republic of China

    Chun-Hua Zhang & Jian-Wei Yu

  2. Department of Mathematics, School of Sciences, Nanchang University, Nanchang, 330031, People’s Republic of China

    Xiang Wang

  3. Numerical Simulation and High-Performance Computing Laboratory, School of Sciences, Nanchang University, Nanchang, 330031, People’s Republic of China

    Xiang Wang

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  1. Chun-Hua Zhang
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Correspondence to Xiang Wang.

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Zhang, CH., Yu, JW. & Wang, X. A fast second-order scheme for nonlinear Riesz space-fractional diffusion equations. Numer Algor 92, 1813–1836 (2023). https://doi.org/10.1007/s11075-022-01367-y

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