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A Positivity-Preserving and Robust Fast Solver for Time-Fractional Convection–Diffusion Problems

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Abstract

This paper presents a fast solver for time-fractional two-dimensional convection-diffusion problems that maintains non-negativity of numerical solutions. To this end, two new techniques are developed. (i) A three-part decomposition of the L1 discretization for Caputo derivatives is proposed and justified for fast evaluation while maintaining positivity; (ii) A positivity-correction technique is devised for both diffusive and convective fluxes. An upwinding technique for the bilinear finite volume approximation on general quadrilaterals is utilized for enabling the solver robustness in handling convection dominance. The solver attains optimal convergence rates when graded temporal meshes are used. These properties are theoretically justified and numerically illustrated.

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Acknowledgements

Y.Li was partially supported by the National Natural Science Foundation of China (Grant No.12071177). J.Liu was partially supported by US National Science Foundation under Grant DMS-2208590. We sincerely thank the anonymous reviewers, whose comments have helped improve the quality of this paper, and also Prof. Guangwei Yuan, with whom we have meaningful discussion about certain techniques in this paper.

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  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Boyang Yu & Yonghai Li

  2. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    Jiangguo Liu

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Yu, B., Li, Y. & Liu, J. A Positivity-Preserving and Robust Fast Solver for Time-Fractional Convection–Diffusion Problems. J Sci Comput 98, 59 (2024). https://doi.org/10.1007/s10915-024-02454-z

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