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A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

Year:    2024

Author:    Li Ma, Hongfei Fu, Bingyin Zhang, Shusen Xie

Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 2 : pp. 429–462

Abstract

In this article, an $α$-th $(0 < α < 1)$ order time-fractional reaction-diffusion equation with variably diffusion coefficient and initial weak singularity is considered. Combined with the fast $L1$ time-stepping method on graded temporal meshes, we develop and analyze a fourth-order compact block-centered finite difference (BCFD) method. By utilizing the discrete complementary convolution kernels and the $α$-robust fractional Grönwall inequality, we rigorously prove the $α$-robust unconditional stability of the developed fourth-order compact BCFD method whether for positive or negative reaction terms. Optimal sharp error estimates for both the primal variable and its flux are simultaneously derived and carefully analyzed. Finally, numerical examples are given to validate the efficiency and accuracy of the developed method.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.OA-2023-0108

Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 2 : pp. 429–462

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    34

Keywords:    Time-fractional reaction-diffusion equation compact BCFD method fast $L1$ method $α$-robust unconditional stability error estimates.

Author Details

Li Ma

Hongfei Fu

Bingyin Zhang

Shusen Xie