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Fast IIF–WENO Method on Non-uniform Meshes for Nonlinear Space-Fractional Convection–Diffusion–Reaction Equations

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Abstract

In this article, our goal is to establish fast and efficient numerical methods for nonlinear space-fractional convection–diffusion–reaction (CDR) equations in the 1, 2, and 3 dimensions. For the spatial discretization of the CDR equations, the weighted essentially non-oscillatory (WENO) scheme is used to approximate the convection term, and the fractional centered difference formula is applied to deal with the diffusion term. As a result, a nonlinear system of ordinary differential equations (ODEs) is derived. Since implicit integration factor (IIF) methods are a class of time-stepping schemes with good robustness and stability, the second order IIF–WENO scheme on non-uniform meshes in Jiang and Zhang (J Comput Phys 253:368–388, 2013) is applied to solve the nonlinear ODEs system. In order to obtain an efficient implementation of the IIF–WENO scheme, we propose an adaptive restarting Krylov subspace method to compute the action of matrix exponentials arising in IIF–WENO. Numerical examples are presented to confirm the validity of the IIF–WENO scheme, and to verify that the proposed fast solution algorithm is extremely attractive in terms of computational complexity and memory storage.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (61772003 and 11801463), the Applied Basic Research Project of Sichuan Province (2020YJ0007) and the Fundamental Research Funds for the Central Universities (No. zdjs2021002). The first author is also supported by the China Scholarship Council.

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Authors and Affiliations

  1. School of Computer Science and Engineering, Southwest Minzu University, Chengdu, 610225, Sichuan, People’s Republic of China

    Huan-Yan Jian

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, People’s Republic of China

    Huan-Yan Jian, Ting-Zhu Huang & Yong-Liang Zhao

  3. Department of Mathematics, University of Innsbruck, Technikerstr. 13, 6020, Innsbruck, Austria

    Alexander Ostermann

  4. Institute of Mathematics, School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, Sichuan, People’s Republic of China

    Xian-Ming Gu

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  1. Huan-Yan Jian
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Jian, HY., Huang, TZ., Ostermann, A. et al. Fast IIF–WENO Method on Non-uniform Meshes for Nonlinear Space-Fractional Convection–Diffusion–Reaction Equations. J Sci Comput 89, 13 (2021). https://doi.org/10.1007/s10915-021-01622-9

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