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The fast method and convergence analysis of the fractional magnetohydrodynamic coupled flow and heat transfer model for the generalized second-grade fluid

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Abstract

In this paper, we first establish a new fractional magnetohydrodynamic (MHD) coupled flow and heat transfer model for a generalized second-grade fluid. This coupled model consists of a fractional momentum equation and a heat conduction equation with a generalized form of Fourier law. The second-order fractional backward difference formula is applied to the temporal discretization and the Legendre spectral method is used for the spatial discretization. The fully discrete scheme is proved to be stable and convergent with an accuracy of O(τ2 + N−r), where τ is the time step-size and N is the polynomial degree. To reduce the memory requirements and computational cost, a fast method is developed, which is based on a globally uniform approximation of the trapezoidal rule for integrals on the real line. The strict convergence of the numerical scheme with this fast method is proved. We present the results of several numerical experiments to verify the effectiveness of the proposed method. Finally, we simulate the unsteady fractional MHD flow and heat transfer of the generalized second-grade fluid through a porous medium. The effects of the relevant parameters on the velocity and temperature are presented and analyzed in detail.

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Acknowledgements

This work was supported by the Project of the National Key R&D Program (Grant No. 2021YFA1000202), National Natural Science Foundation of China (Grant Nos. 12120101001, 12001326 and 12171283), Natural Science Foundation of Shandong Province (Grant Nos. ZR2021ZD03, ZR2020QA032 and ZR2019ZD42), China Postdoctoral Science Foundation (Grant Nos. BX20190191 and 2020M672038) and the Startup Fund from Shandong University (Grant No. 11140082063130).

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  1. School of Mathematics and Statistics, Shandong University, Weihai, 264209, China

    Xiaoqing Chi

  2. School of Mathematics, Shandong University, Jinan, 250100, China

    Hui Zhang & Xiaoyun Jiang

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  1. Xiaoqing Chi
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  2. Hui Zhang
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Correspondence to Hui Zhang.

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Chi, X., Zhang, H. & Jiang, X. The fast method and convergence analysis of the fractional magnetohydrodynamic coupled flow and heat transfer model for the generalized second-grade fluid. Sci. China Math. 67, 919–950 (2024). https://doi.org/10.1007/s11425-021-2063-0

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